NiVRQS91c3IvbGFicmkvYm91c3F1ZXQvYWxnb2xpYi9nZnVuNiJROi91c3IvbG9jYWwvbWFwbGUvbGliL2dmdW5GJFE1L3Vzci9sb2NhbC9tYXBsZS9saWJGJA==

NiZRRC91c3IvbGFicmkvYm91c3F1ZXQvYWxnb2xpYi9hbGdvbGliNiJRQS91c3IvbGFicmkvYm91c3F1ZXQvYWxnb2xpYi9nZnVuRiRROi91c3IvbG9jYWwvbWFwbGUvbGliL2dmdW5GJFE1L3Vzci9sb2NhbC9tYXBsZS9saWJGJA==

NiZRSC91c3IvbGFicmkvYm91c3F1ZXQvYWxnb2xpYi9OZXd0b25HRjEuMzYiUUEvdXNyL2xhYnJpL2JvdXNxdWV0L2FsZ29saWIvZ2Z1bkYkUTovdXNyL2xvY2FsL21hcGxlL2xpYi9nZnVuRiRRNS91c3IvbG9jYWwvbWFwbGUvbGliRiQ=

restart:

liste:=x*y+1/x+1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwoKiZJInhHRigiIiJJInlHRihGMUYxKiRGMCEiIkYxKiRGMkY0RjE3I0Yu

K:=collect((t*liste-1)*x*y,t,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCYqJiwoKiYpSSJ4R0YoIiIjIiIiKUkieUdGKEY0RjVGNUYzRjVGN0Y1RjVJInRHRihGNUY1KiZGM0Y1RjdGNSEiIjcjRi4=

The functional equation eqQ:=K*Q(x,y)+x*y-subs(x=0,K)*Q(0,y) -subs(y=0,K)*Q(x,0)+subs(x=0,y=0,K)*Q(0,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcVFHRigsKiomLCYqJiwoKiYpSSJ4R0YoIiIjIiIiKUkieUdGKEY2RjdGN0Y1RjdGOUY3RjdJInRHRihGN0Y3KiZGNUY3RjlGNyEiIkY3LUkiUUdGKDYkRjVGOUY3RjdGO0Y3KihGOUY3RjpGNy1GPjYkIiIhRjlGN0Y8KihGNUY3RjpGNy1GPjYkRjVGQ0Y3Rjw3I0Yu

Qser:=proc(n) option remember: if n=0 then 1 else normal(series(subs(Q(x,y)=Qser(n-1),Q(x,0)=subs(y=0,Qser(n-1)), Q(0,y)=subs(x=0,Qser(n-1)), expand(eqQ/x/y+Q(x,y))),t,n+1)): fi: end: Qser(4);

Ky1JInRHNiIiIiIiIiEqJkkieEdGJEYlSSJ5R0YkRiVGJSwoKiYpRigiIiNGJSlGKUYtRiVGJUYoRiVGKUYlRi0sKiomKUYoIiIkRiUpRilGMkYlRiUqKEYtRiVGLEYlRilGJUYlKihGLUYlRihGJUYuRiVGJUYtRiVGMiwuKiYpRigiIiVGJSlGKUY5RiVGJSooRjJGJUYxRiVGLkYlRiUqKEYyRiVGLEYlRjNGJUYlKiZGLUYlRixGJUYlKigiIidGJUYoRiVGKUYlRiUqJkYtRiVGLkYlRiVGOQ==

Rser:=proc(n) normal(series(subs(y=0,K)*subs(y=0,Qser(n-1)),t,n+1)):end: Sser:=proc(n) normal(series(subs(x=0,K)*subs(x=0,Qser(n-1)),t,n+1)):end: Rser(4);Sser(4);

KylJInRHNiJJInhHRiQiIiIqJClGJSIiI0YmIiIkLCQqJkYpRiZGJUYmRiYiIiU=

KylJInRHNiJJInlHRiQiIiIqJClGJSIiI0YmIiIkLCQqJkYpRiZGJUYmRiYiIiU=

Checking the functional equation n:=10: normal(series(Qser(n)*K+x*y-Rser(n)-Sser(n)+subs(x=0,Rser(n)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Invariants have been constructed in the Maple session "invariants-syst" Jy:=t/y^2-1/y-t*y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKeUdGKCwoKiZJInRHRigiIiIpSSJ5R0YoIiIjISIiRjEqJEYzRjVGNSomRjBGMUYzRjFGNTcjRi4=

Ix:=subs(y=x,Jy);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeEdGKCwoKiZJInRHRigiIiIpSSJ4R0YoIiIjISIiRjEqJEYzRjVGNSomRjBGMUYzRjFGNTcjRi4=

Sy:=subs(x=0,K*Q(y));Sx:=subs(y=x,Sy);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNTeUdGKCooSSJ0R0YoIiIiSSJ5R0YoRjAtSSJRR0YoNiNGMUYwNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNTeEdGKCooSSJ0R0YoIiIiSSJ4R0YoRjAtSSJRR0YoNiNGMUYwNyNGLg==

Here we take the same decoupling function for x and y Gx:=-1/x+1/(2*t);Gy:=subs(x=y,Gx);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeEdGKCwmKiRJInhHRighIiJGMSomIyIiIiIiI0Y0SSJ0R0YoRjFGNDcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwmKiRJInlHRighIiJGMSomIyIiIiIiI0Y0SSJ0R0YoRjFGNDcjRi4=

factor(Gx+Gy-x*y);

LCQqKiwqKihJInRHNiIiIiIpSSJ4R0YnIiIjRigpSSJ5R0YnRitGKEYoKiZGJkYoRipGKEYoKiZGJkYoRi1GKEYoKiZGKkYoRi1GKCEiIkYoRipGMUYtRjFGJkYxRjE=

factor(%/K);

LCQqKEkieEc2IiEiIkkieUdGJUYmSSJ0R0YlRiZGJg==

The invariant property factor(Ix-Jy);

LCQqKiwmSSJ4RzYiIiIiSSJ5R0YmISIiRicsKiooSSJ0R0YmRicpRiUiIiNGJylGKEYuRidGJyomRixGJ0YlRidGJyomRixGJ0YoRidGJyomRiVGJ0YoRidGKUYnRi1GKUYvRilGKQ==

factor(%/K);

LCQqKCwmSSJ4RzYiIiIiSSJ5R0YmISIiRicpRiUiIiNGKSlGKEYrRilGKQ==

A pole free series Ax:=t^2*(S(x)-Gx)^2-t*Ix; subs(S(0)=0,normal(series(%,x,2)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNBeEdGKCwmKiYpSSJ0R0YoIiIjIiIiKSwoLUkiU0dGKDYjSSJ4R0YoRjMqJEY5ISIiRjMqJiNGM0YyRjNGMUY7RjtGMkYzRjMqJkYxRjMsKComRjFGMylGOUYyRjtGM0Y6RjsqJkYxRjNGOUYzRjtGM0Y7NyNGLg==

KydJInhHNiIsJiooIiIjIiIiLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kiU0dGJDYjIiIhRigpSSJ0R0YkRidGKEYoI0YoIiIlRihGMi1JIk9HRiw2I0YoRig=

We will see that A(x) is independent of x. Let us check this: n:=17: normal(series(subs(S(x)=convert(Rser(n),polynom),Ax),t,n+1));

KzFJInRHNiIjIiIiIiIlIiIhIiIjIiIkRiciIiciI0siIioiJCVRIiM3IiVLYyIjOi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YmIiM9

The expression of C(x,y) exprC:=t^2*(Q(x,y)+1/x/y/t)*(Sx-Gx-Sy+Gy)-t*(y-x)/x^2/y^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZleHByQ0dGKCwmKigpSSJ0R0YoIiIjIiIiLCYtSSJRR0YoNiRJInhHRihJInlHRihGMyooRjghIiJGOUY7RjFGO0YzRjMsKiooRjFGM0Y4RjMtRjY2I0Y4RjNGMyokRjhGO0YzKihGMUYzRjlGMy1GNjYjRjlGM0Y7KiRGOUY7RjtGM0YzKipGMUYzLCZGOEY7RjlGM0YzKUY4RjJGOylGOUYyRjtGOzcjRi4=

We want to prove that this is a series in t with polynomial coeffs in x and y. expand(subs(Q(x,y)=Q(x)+Q(y)-Q00+x*y*W(x,y),Q(x)=Q00+W(x)*x,Q(y)=Q00+y*W(y),exprC));

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

Now we take a simpler decoupling function Gy:=-1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiRJInlHRighIiJGMTcjRi4=

Jy; Ly:=S(y)-Gy;

LCgqJkkidEc2IiIiIilJInlHRiUiIiMhIiJGJiokRihGKkYqKiZGJEYmRihGJkYq

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNMeUdGKCwmLUkiU0dGKDYjSSJ5R0YoIiIiKiRGMiEiIkYzNyNGLg==

We construct an expression with no pole at y=0 (in brief, J must be a polynomial in L) eqcat:=Jy-a*Ly^2-b*Ly-c;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZlcWNhdEdGKCwuKiYpLCYtSSJTR0YoNiNJInlHRigiIiIqJEY1ISIiRjYiIiNGNkkiYUdGKEY2RjgqJkYxRjZJImJHRihGNkY4KiZJInRHRihGNilGNUY5RjhGNkY3RjgqJkY+RjZGNUY2RjhJImNHRihGODcjRi4=

normal(series(eqcat,y,0));

KydJInlHNiIsJkkiYUdGJCEiIkkidEdGJCIiIiEiIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YpRic=

asol:=solve(op(1,%),a);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhc29sR0YoSSJ0R0YoNyNGLg==

normal(series(subs(a=asol,eqcat),y,1));

KydJInlHNiIsKCooIiIjIiIiLUkiU0dGJDYjIiIhRihJInRHRiRGKCEiIkkiYkdGJEYuRihGLkYuLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGLA==

subs(x=0,K);

KiZJInRHNiIiIiJJInlHRiRGJQ==

bsol:=solve(subs(S(0)=0,op(1,%%)),b);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVic29sR0YoISIiNyNGLg==

normal(subs(S(0)=0,series(subs(a=asol,b=bsol,eqcat),y,2)));

KydJInlHNiIsJiooIiIjIiIiLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kiU0dGJDYjIiIhRihJInRHRiRGKCEiIkkiY0dGJEY0RjItSSJPR0YsNiNGKEYo

csol:=solve(op(1,%),c);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVjc29sR0YoLCQqKCIiIyIiIi0tSSJER0YlNiNJIlNHRig2IyIiIUYxSSJ0R0YoRjEhIiI3I0Yu

eqcat1:=numer(factor(subs(a=asol,b=bsol,c=csol,eqcat)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdlcWNhdDFHRigsLCooKS1JIlNHRig2I0kieUdGKCIiIyIiIkkidEdGKEY2RjRGNiEiIioqRjVGNi0tSSJER0YlNiNGMjYjIiIhRjZGN0Y2RjRGNkY2KiZGN0Y2KUY0RjVGNkY4KihGNUY2RjFGNkY3RjZGOComRjFGNkY0RjZGNjcjRi4=

Can this be true ? n:=30: normal(series(subs(S(y)=subs(x=y,Sser(n)), D(S)(0)=subs(y=0,diff(Sser(n),y)), eqcat1),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0o=

Back to Q(0,y) (denoted here Q(y) to lighten notation) Sy:=subs(x=0,K)*Q(y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNTeUdGKCooSSJ0R0YoIiIiSSJ5R0YoRjAtSSJRR0YoNiNGMUYwNyNGLg==

factor(subs(S(y)=Sy,(D(S))(0)=subs(y=0,diff(Sy,y)),eqcat1));

LCQqKEkidEc2IiIiIkkieUdGJUYmLCwqKCktSSJRR0YlNiNGJyIiI0YmKUYkRi5GJilGJ0YuRiZGJiooRi5GJkYkRiZGK0YmRiYqJkYrRiZGJ0YmISIiKihGLkYmRiRGJi1GLDYjIiIhRiZGM0YnRiZGJkYz

eqcatQ:=collect(op(4,%),Q,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdlcWNhdFFHRigsKiooIiIjIiIiSSJ0R0YoRjEtSSJRR0YoNiMiIiFGMSEiIiooKS1GNDYjSSJ5R0YoRjBGMSlGMkYwRjEpRjxGMEYxRjEqJiwmKiZGMEYxRjJGMUYxRjxGN0YxRjpGMUYxRjxGMTcjRi4=

n:=30: normal(series(subs(Q(y)=subs(x=0,Qser(n)), Q(0)=subs(x=0,y=0,Qser(n)),eqcatQ),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0o=

We refer to [BM02] for the solution of this equation.

liste:=1/x/y+x+y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwoKiZJInhHRighIiJJInlHRihGMSIiIkYwRjNGMkYzNyNGLg==

K:=collect((t*liste-1)*x*y,t,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCYqJiwoKiYpSSJ4R0YoIiIjIiIiSSJ5R0YoRjVGNSomRjNGNSlGNkY0RjVGNUY1RjVGNUkidEdGKEY1RjUqJkYzRjVGNkY1ISIiNyNGLg==

The functional equation eqQ:=K*Q(x,y)+x*y-subs(x=0,K)*Q(0,y) -subs(y=0,K)*Q(x,0)+subs(x=0,y=0,K)*Q(0,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcVFHRigsLComLCYqJiwoKiYpSSJ4R0YoIiIjIiIiSSJ5R0YoRjdGNyomRjVGNylGOEY2RjdGN0Y3RjdGN0kidEdGKEY3RjcqJkY1RjdGOEY3ISIiRjctSSJRR0YoNiRGNUY4RjdGN0Y8RjcqJkY7RjctRj82JCIiIUY4RjdGPSomRjtGNy1GPzYkRjVGREY3Rj0qJkY7RjctRj82JEZERkRGN0Y3NyNGLg==

Qser:=proc(n) option remember: if n=0 then 1 else convert(normal(series(subs(Q(x,y)=Qser(n-1),Q(x,0)=subs(y=0,Qser(n-1)), Q(0,0)=subs(y=0,x=0,Qser(n-1)), Q(0,y)=subs(x=0,Qser(n-1)), expand(eqQ/x/y+Q(x,y))),t,n+1)),polynom): fi: end: Qser(4);

LCwiIiJGIyomLCZJInhHNiJGI0kieUdGJ0YjRiNJInRHRidGI0YjKiYpRiUiIiNGIylGKUYsRiNGIyomLCwqJClGJiIiJEYjRiMqKEYyRiMpRiZGLEYjRihGI0YjKihGMkYjRiZGIylGKEYsRiNGIyokKUYoRjJGI0YjRixGI0YjKUYpRjJGI0YjKiYsMCokKUYmIiIlRiNGIyooRj5GI0YxRiNGKEYjRiMqKCIiJ0YjRjRGI0Y2RiNGIyooRj5GI0YmRiNGOEYjRiMqJClGKEY+RiNGIyomIiImRiNGJkYjRiMqJkZGRiNGKEYjRiNGIylGKUY+RiNGIw==

Rser:=proc(n) convert(normal(series(subs(y=0,K)*subs(y=0,Qser(n-1)),t,n+1)),polynom):end: Sser:=proc(n) convert(normal(series(subs(x=0,K)*subs(x=0,Qser(n-1)),t,n+1)),polynom):end: Rser(4);Sser(4);

LCpJInRHNiIiIiIqJkkieEdGJEYlKUYjIiIjRiVGJSomKUYnRilGJSlGIyIiJEYlRiUqJiwmKiQpRidGLUYlRiVGKUYlRiUpRiMiIiVGJUYl

LCpJInRHNiIiIiIqJkkieUdGJEYlKUYjIiIjRiVGJSomKUYnRilGJSlGIyIiJEYlRiUqJiwmKiQpRidGLUYlRiVGKUYlRiUpRiMiIiVGJUYl

Checking the functional equation n:=10: normal(series(Qser(n)*K+x*y-Rser(n)-Sser(n)+subs(x=0,Rser(n)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Invariants have been constructed in the Maple session "invariants-syst" Jy:=t*y^2-y-t/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKeUdGKCwoKiZJInRHRigiIiIpSSJ5R0YoIiIjRjFGMUYzISIiKiZGMEYxRjNGNUY1NyNGLg==

factor(subs(y=x,Jy)-Jy);

KiosJkkieEc2IiIiIkkieUdGJSEiIkYmLCoqKEkidEdGJUYmKUYkIiIjRiZGJ0YmRiYqKEYrRiZGJEYmKUYnRi1GJkYmKiZGJEYmRidGJkYoRitGJkYmRiRGKEYnRig=

factor(%/K);

KigsJkkieEc2IiIiIkkieUdGJSEiIkYmRiRGKEYnRig=

We take the same decoupling function in x and y Gy:=expand(-1/(2*t)*(t*y^2-y+t/y));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwoKiYjIiIiIiIjRjEpSSJ5R0YoRjJGMSEiIiooRjBGMUkidEdGKEY1RjRGMUYxKiZGMEYxRjRGNUY1NyNGLg==

factor(Gy+subs(y=x,Gy)-x*y);

LCQqLiMiIiIiIiNGJSwmSSJ4RzYiRiVJInlHRilGJUYlLCoqKEkidEdGKUYlKUYoRiZGJUYqRiVGJSooRi1GJUYoRiUpRipGJkYlRiUqJkYoRiVGKkYlISIiRi1GJUYlRi1GMkYqRjJGKEYyRjI=

factor(%/K);

LCQqLCMiIiIiIiNGJSwmSSJ4RzYiRiVJInlHRilGJUYlSSJ0R0YpISIiRipGLEYoRixGLA==

Jy; Ly:=S(y)-S(0)/2-Gy;

LCgqJkkidEc2IiIiIilJInlHRiUiIiNGJkYmRighIiIqJkYkRiZGKEYqRio=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNMeUdGKCwsLUkiU0dGKDYjSSJ5R0YoIiIiKiYjRjMiIiNGMy1GMDYjIiIhRjMhIiIqJkY1RjMpRjJGNkYzRjMqKEY1RjNJInRHRihGOkYyRjNGOiomRjVGM0YyRjpGMzcjRi4=

Jx:=subs(y=x,Jy); Lx:=subs(y=x,Ly);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKeEdGKCwoKiZJInRHRigiIiIpSSJ4R0YoIiIjRjFGMUYzISIiKiZGMEYxRjNGNUY1NyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNMeEdGKCwsLUkiU0dGKDYjSSJ4R0YoIiIiKiYjRjMiIiNGMy1GMDYjIiIhRjMhIiIqJkY1RjMpRjJGNkYzRjMqKEY1RjNJInRHRihGOkYyRjNGOiomRjVGM0YyRjpGMzcjRi4=

We construct an expression with no pole at y=0 exprA:=4*t^2*Ly^2-Jy^2+2*t*S(0)*Jy;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZleHByQUdGKCwoKigiIiUiIiIpSSJ0R0YoIiIjRjEpLCwtSSJTR0YoNiNJInlHRihGMSomI0YxRjRGMS1GODYjIiIhRjEhIiIqJkY8RjEpRjpGNEYxRjEqKEY8RjFGM0ZARjpGMUZAKiZGPEYxRjpGQEYxRjRGMUYxKiQpLCgqJkYzRjFGQkYxRjFGOkZAKiZGM0YxRjpGQEZARjRGMUZAKipGNEYxRjNGMUY9RjFGR0YxRjE3I0Yu

normal(series(exprA,y,2));

KydJInlHNiIqJkkidEdGJCIiIiwoKiYpLUkiU0dGJDYjIiIhIiIjRidGJkYnRicqKCIiJUYnLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YsRi1GJ0YmRidGJ0YxISIiRidGLi1JIk9HRjU2I0YnRic=

We will show that A is independent of y n:=10: normal(series(subs(S(y)=Sser(n),S(0)=subs(y=0,Sser(n)),exprA),t,n+1));

Ky1JInRHNiIhIiUiIiIiIiYiIiUiI0MiIigiJFMjIiM1LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiYiIzY=

The expression of C(x,y) exprC:=4*t^2*(Q(x,y)+(x+y)/(2*x*y*t))*(Lx-Ly)-(x-y)/x/y*(Jx+Jy-2*t*S0);

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

subs(x=0,K);

SSJ0RzYi

Let us check that this is a series in t with polynomial coeffs in x and y. expand(subs(Q(x,y)=Q(x)+Q(y)-Q00+x*y*W(x,y), S(x)=t*Q(x),S(y)=t*Q(y),S0=t*Q00,Q(x)=Q00+x*W(x),Q(y)=Q00+y*W(y),exprC));

LFoqKiIiJSIiIilJInRHNiIiIiRGJSlJInhHRigiIiNGJSktSSJXR0YoNiNGK0YsRiVGJSoqRixGJUYnRiVJJFEwMEdGKEYlRitGJSEiIioqRiRGJUYmRiUpSSJ5R0YoRixGJSktRi82I0Y2RixGJUYzKipGLEYlKUYnRixGJSlGK0YpRiVGLkYlRiUqKkYsRiVGO0YlRjJGJUYqRiVGJSoqRixGJUY7RiVGMkYlRjVGJUYzKipGLEYlRjtGJSlGNkYpRiVGOEYlRjMqKkYsRiVGO0YlRitGJS1GLzYkRitGNkYlRjMqKkYsRiVGO0YlRjZGJUZCRiVGJSoqRixGJUYnRiVGNUYlRjhGJUYlKipGLEYlRidGJUYyRiVGNkYlRiUqKkYsRiVGJ0YlRipGJUYuRiVGMyooRiRGJUY7RiVGLkYlRiUqKEYkRiVGO0YlRjhGJUYzKi5GJEYlRiZGJUYrRiVGNUYlRkJGJUY4RiVGMyouRiRGJUYmRiVGKkYlRjZGJUZCRiVGLkYlRiUqKEYsRiVGJ0YlRjZGJUYzKihGLEYlRidGJUYrRiVGJSosRixGJUY7RiVGK0YlRkBGJUZCRiVGMyosRixGJUY7RiVGPEYlRjZGJUZCRiVGJSosRixGJUY7RiVGNkYlRjhGJUYqRiVGJSosRixGJUY7RiVGK0YlRi5GJUY1RiVGMyosRixGJUYnRiVGK0YlRi5GJUY2RiVGJSosRiRGJUYmRiVGMkYlRitGJUYuRiVGJSosRiRGJUYmRiVGMkYlRjZGJUY4RiVGMyosRixGJUYnRiVGNkYlRjhGJUYrRiVGMyosRixGJUYnRiVGKkYlRjZGJUZCRiVGMyosRixGJUYnRiVGK0YlRjVGJUZCRiVGJQ==

Hence A is independent of y normal(series(exprA,y,2));

KydJInlHNiIqJkkidEdGJCIiIiwoKiYpLUkiU0dGJDYjIiIhIiIjRidGJkYnRicqKCIiJUYnLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YsRi1GJ0YmRidGJ0YxISIiRidGLi1JIk9HRjU2I0YnRic=

eqcat:=exprA-convert(%,polynom);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZlcWNhdEdGKCwqKigiIiUiIiIpSSJ0R0YoIiIjRjEpLCwtSSJTR0YoNiNJInlHRihGMSomI0YxRjRGMS1GODYjIiIhRjEhIiIqJkY8RjEpRjpGNEYxRjEqKEY8RjFGM0ZARjpGMUZAKiZGPEYxRjpGQEYxRjRGMUYxKiQpLCgqJkYzRjFGQkYxRjFGOkZAKiZGM0YxRjpGQEZARjRGMUZAKipGNEYxRjNGMUY9RjFGR0YxRjEqJkYzRjEsKComKUY9RjRGMUYzRjFGMSooRjBGMS0tSSJER0YlNiNGOEY+RjFGM0YxRjFGMEZARjFGQDcjRi4=

Back to Q(0,y) (denoted here Q(y) to lighten notation) Sy:=subs(x=0,K)*Q(y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNTeUdGKComSSJ0R0YoIiIiLUkiUUdGKDYjSSJ5R0YoRjA3I0Yu

numer(factor(subs(S(y)=Sy,(D(S))(0)=limit(diff(Sy,y),y=0),S(0)=subs(y=0,Sy),eqcat)));

LCQqKCIiJSIiIilJInRHNiIiIiNGJSwyKigpLUkiUUdGKDYjSSJ5R0YoRilGJUYmRiVGMEYlRiUqKkYtRiUtRi42IyIiIUYlRiZGJUYwRiUhIiIqKEYtRiVGJ0YlKUYwIiIkRiVGJSomRi1GJSlGMEYpRiVGNSooLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRig2I0YuRjNGJUYnRiVGMEYlRjUqJkYnRiVGLUYlRiUqJkYnRiVGMkYlRjUqJEY6RiVGJUYlRiU=

eqcatQ:=collect(op(3,%),Q(y),factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdlcWNhdFFHRigsLCooKS1JIlFHRig2I0kieUdGKCIiIyIiIilJInRHRihGNUY2RjRGNkY2KiYsKiooRjdGNi1GMjYjIiIhRjZGNEY2ISIiKiZGOEY2KUY0IiIkRjZGNiokKUY0RjVGNkY/RjhGNkY2RjFGNkY2KigtLUkiREdGJTYjRjJGPUY2RjhGNkY0RjZGPyomRjhGNkY8RjZGP0ZDRjY3I0Yu

eqcatQ;

LCwqKCktSSJRRzYiNiNJInlHRiciIiMiIiIpSSJ0R0YnRipGK0YpRitGKyomLCoqKEYsRistRiY2IyIiIUYrRilGKyEiIiomRi1GKylGKSIiJEYrRisqJClGKUYqRitGNEYtRitGK0YlRitGKyooLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRic2I0YmRjJGK0YtRitGKUYrRjQqJkYtRitGMUYrRjRGOEYr

eqcat:=collect(subs(Q(y)=Q,Q(0)=A1,D(Q)(0)=A2,eqcatQ),Q,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZlcWNhdEdGKCwsKigpSSJRR0YoIiIjIiIiKUkidEdGKEYyRjNJInlHRihGM0YzKiYsKiooSSNBMUdGKEYzRjRGM0Y2RjMhIiIqJkY1RjMpRjYiIiRGM0YzKiQpRjZGMkYzRjtGNUYzRjNGMUYzRjMqKEkjQTJHRihGM0Y1RjNGNkYzRjsqJkY1RjNGOkYzRjtGP0YzNyNGLg==

two series cancel this derivative edQ:=factor(diff(eqcat,Q));subs(t=0,%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlZFFHRigsLCooSSNBMUdGKCIiIilJInRHRigiIiNGMUkieUdGKEYxISIiKipGNEYxSSJRR0YoRjFGMkYxRjVGMUYxKiZGM0YxKUY1IiIkRjFGMSokKUY1RjRGMUY2RjNGMTcjRi4=

LCQqJClJInlHNiIiIiMiIiIhIiI=

Let us compute their expansions in s=sqrt(t). Of course, one of the series is obtained from the other by s-> -s y0:='y0': Y0ser:=proc(n) add(y0[i]*s^i,i=1..n): end: n:=1: n:=1+n: normal(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)), y=Y0ser(n),t=s^2, edQ),s,n+1));

KydJInNHNiIsJiokKSZJI3kwR0YkNiMiIiIiIiNGKyEiIkYrRitGLC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiRGKiIiJA==

y0[1]:=1: From now on we can iterate the next two lines. Each time we obtain one more coefficient of Y. n:=n+1: normal(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)), A2=subs(x=0,y=0,diff(Qser(n),y)), y=Y0ser(n), t=s^2, edQ),s,n+1));

KydJInNHNiIsJComIiIjIiIiJkkjeTBHRiQ2IyIjNkYoISIiIiM3LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIzg=

y0[n-1]:=solve(coeff(%,s,n),y0[n-1]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTBHRig2IyIjNiIiITcjRjE=

Y0ser(n);

LCxJInNHNiIiIiIqJClGIyIiJUYlRiUqJiMiIiYiIiNGJSlGIyIiKEYlRiUqJiIiKUYlKUYjIiM1RiVGJSomJkkjeTBHRiQ2IyIjN0YlKUYjRjdGJUYl

edy:=factor(diff(eqcat,y));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlZHlHRigsLiooSSNBMUdGKCIiIkkiUUdGKEYxKUkidEdGKCIiI0YxISIiKiYpRjJGNUYxRjNGMUYxKioiIiRGMUYyRjFGNEYxKUkieUdGKEY1RjFGMSomSSNBMkdGKEYxRjRGMUY2KihGNUYxRjJGMUY8RjFGNiomRjVGMUY8RjFGMTcjRi4=

disc:=collect(factor(discrim(eqcat,Q)),y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkaXNjR0YoLDAqJilJInRHRigiIiMiIiIpSSJ5R0YoIiInRjNGMyooRjJGM0YxRjMpRjUiIiZGMyEiIiomLCYqKEYyRjNJI0ExR0YoRjMpRjEiIiRGM0Y6RjNGM0YzKUY1IiIlRjNGMyomLCYqKEYyRjNGPkYzRjBGM0YzKiZGMkYzRjBGM0Y6RjMpRjVGQEYzRjMqJiwoKiYpRj5GMkYzKUYxRkJGM0YzKihGQkYzSSNBMkdGKEYzRj9GM0YzKiZGMkYzRjFGM0Y6RjMpRjVGMkYzRjMqKkYyRjNGPkYzRj9GM0Y1RjNGMyokRjBGM0YzNyNGLg==

The discriminant of this polynomial (with respect to y) vanishes res:=factor(discrim(disc,y));nops(res);

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

IiIl

The factor that is cubed cannot vanish (constant term 1), so here is our first equation between A1 and A2 alg1:=op(4,res);

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

Let us check n:=10: normal(series(subs(A1=subs(x=0,y=0,Qser(n)), A2=subs(y=0,diff(subs(x=0,Qser(n)),y)),alg1),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzc=

A polynomial P in y has d double roots (here, k=2) if the first generalized discriminants defined below vanish for k=0, 1, ... , d-1. with(LinearAlgebra): disciter:=proc(P,k) local M,i,j,d: d:=degree(P,y): M:=Matrix(2*d-1-2*k,2*d-1-2*k): for i from 1 to d-1-k do for j from 0 to min(d, 2*d-1-2*k-i) do M[i,i+j]:=coeff(P,y,d-j):od:od: for i from 0 to d-k- 1 do for j from 1 to min( 2*d-2*k-1-i) do M[i+d-k,i+j]:=coeff(diff(P,y),y,d-j): od: od: M:end: The first one is just the usual discriminant factor(Determinant(disciter(disc,0)));

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

factor(%/alg1);

LCQqKCIlJzQlIiIiKUkidEc2IiIjO0YlKSwmKiZJI0EyR0YoRiUpRiciIiNGJUYlRiUhIiIiIiRGJUYl

factor(Determinant(disciter(disc,1)));nops(%);

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

IiIl

alg2:=op(4,%%): Let us now eliminate A2 algA1cand:=factor(resultant(alg1,alg2,A2));nops(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSphbGdBMWNhbmRHRigsJCouIitbT1taQCIiIilJInRHRigiIzpGMSksJiomIiIkRjFGM0YxRjFGMSEiIiIiI0YxKSwoKiYiIipGMSlGM0Y6RjFGMUY3RjFGMUYxRjpGMSwsKigiIiVGMSlJI0ExR0YoRjhGMSlGMyIiJ0YxRjEqJilGREY6RjEpRjNGOEYxRjEqKCIjPUYxRkRGMUZJRjFGOSomIiNGRjFGSUYxRjlGQkY5RjEpLC4qKCIja0YxRkNGMUZFRjFGMSooIiM7RjFGSEYxRklGMUYxKigiI3NGMUZERjFGSUYxRjkqJiIjYUYxRklGMUYxRkRGMUYxRjlGOkYxRjE3I0Yu

IiIn

Only one factor can vanish (look at the constant term), and we have found the algebraic equation satisfied by A1 n:=14: normal(series(subs(A1=subs(x=0,y=0,Qser(n)),op(1,op(6,algA1cand))),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

algA1:=collect(op(1,op(6,algA1cand)),A1,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdBMUdGKCwsKigiI2siIiIpSSNBMUdGKCIiJEYxKUkidEdGKCIiJ0YxRjEqKCIjO0YxKUYzIiIjRjEpRjZGNEYxRjEqJiwmKiYiI3NGMUY8RjEhIiJGMUYxRjFGM0YxRjEqJiIjYUYxRjxGMUYxRjFGQTcjRi4=

Now for A2 algA2cand:=factor(resultant(alg1,alg2,A1));nops(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSphbGdBMmNhbmRHRigsJCouIi53eGk2Jio0IiIiIilJInRHRigiI1xGMSksJiomIiIkRjFGM0YxRjFGMSEiIiIiJUYxKSwoKiYiIipGMSlGMyIiI0YxRjFGN0YxRjFGMUY6RjEpLDQqKCImT2InRjEpSSNBMkdGKEY4RjEpRjMiIzVGMUYxKigiJjtnKUYxKUZGRkBGMSlGMyIiKUYxRjkqKCIkNyZGMUZLRjEpRjMiIiZGMUYxKigiJiEzQUYxRkZGMSlGMyIiJ0YxRjEqJiImJG8+RjEpRjMiIihGMUY5KigiJCNmRjFGRkYxKUYzRjhGMUY5KiYiJChlRjEpRjNGOkYxRjFGRkYxRjNGOUZARjEpLCYqJkZGRjFGP0YxRjFGMUY5RjhGMUYxNyNGLg==

IiIn

n:=14: normal(series(subs(A2=subs(y=0,diff(subs(x=0,Qser(n)),y)),op(1,op(5,algA2cand))),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzs=

The algebraic equation satisfied by A2 algA2:=collect(op(1,op(5,algA2cand)),A2,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdBMkdGKCwqKigiJk9iJyIiIilJI0EyR0YoIiIkRjEpSSJ0R0YoIiM1RjFGMSoqIiQ3JkYxKUY2IiImRjEsJiomIiRvIkYxKUY2RjRGMUYxRjEhIiJGMSlGMyIiI0YxRkAqKCwmKiYiJF8mRjFGP0YxRjFGMUZARjEsJiomIiNTRjFGP0YxRjFGMUZARjFGM0YxRjEqJkY2RjEsKComIiYkbz5GMSlGNiIiJ0YxRjEqJiIkKGVGMUY/RjFGQEYxRjFGMUZANyNGLg==

Comparison with [MBM-Mishna], Section 6.2 It is convenient to define the series Z, with no constant term, satisfying the equation below: algZ:=t*(2+Z^3)-Z;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdaR0YoLCYqJkkidEdGKCIiIiwmKiQpSSJaR0YoIiIkRjFGMSIiI0YxRjFGMUY1ISIiNyNGLg==

tZ:=isolate(algZ,t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN0WkdGKC9JInRHRigqJkkiWkdGKCIiIiwmKiQpRjEiIiRGMkYyIiIjRjIhIiI3I0Yu

numer(factor(subs(tZ,algA1)));

KiYsKiokKUkiWkc2IiIiJyIiIkYpKiYiIiNGKSlGJiIiJEYpISIiKiYiIilGKUkjQTFHRidGKUYpRjBGLkYpLDYqJkYxRikpRiYiIzdGKUYpKiRGNEYpRi4qKCIjOUYpRjFGKSlGJiIiKkYpRikqJkY4RilGOUYpRi4qKEYwRikpRjFGK0YpRiVGKUYpKigiI0tGKUYxRilGJUYpRikqJiIjVUYpRiVGKUYuKigiIztGKUYxRilGLEYpRikqJkY/RilGLEYpRi5GMEYpRik=

Expression of A1 in terms of Z A1Z:=factor(solve(op(1,%),A1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBMVpHRigsJCooIyIiIiIiKUYxLCYqJClJIlpHRigiIiRGMUYxIiIjRjFGMSwmRjRGMSIiJSEiIkYxRjs3I0Yu

Expression of A2 in terms of Z numer(factor(subs(tZ,algA2)));

KiYsLCokKUkiWkc2IiIjNSIiIkYpKiYiI0VGKSlGJiIiKEYpRikqJiIjO0YpKUYmIiIlRikhIiIqJiIkYyNGKUkjQTJHRidGKUYpKiYiJEciRilGJkYpRjJGKSw+KiZGNUYpKUYmIiM/RilGKSooIiInRilGNUYpKUYmIiM8RilGMiokKUYmIiM9RilGMiooIiRTI0YpRjVGKSlGJiIjOUYpRjIqJiIiKUYpKUYmIiM6RilGKSooRjRGKSlGNSIiI0YpRiVGKUYpKigiJWM1RilGNUYpKUYmIiM2RilGMiomIiQ+I0YpKUYmIiM3RilGKSooIiUlPSJGKUY1RikpRiZGSEYpRjIqJiIkeSdGKSlGJiIiKkYpRikqKCIja0YpRjVGKSlGJiIiJkYpRikqJiIkQCVGKSlGJkY9RilGKSomIiR3I0YpKUYmIiIkRilGMkYxRilGKQ==

A2Z:=factor(solve(op(1,%),A2));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBMlpHRigsJCoqIyIiIiIkYyNGMUkiWkdGKEYxLCYqJClGMyIiJEYxRjEiIiNGMUYxLCgqJClGMyIiJ0YxRjEqJiIjQ0YxRjZGMUYxIiNrISIiRjFGQDcjRi4=

Back to Q(0,y) collect(numer(factor(subs(A1=A1Z,A2=A2Z,tZ,eqcat))),Q,factor);

LCgqKiIkYyMiIiIpSSJRRzYiIiIjRiUpSSJaR0YoRilGJUkieUdGKEYlRiUqKiIjS0YlLC4qJilGKyIiJkYlRixGJUYlKigiIilGJSlGKyIiJEYlKUYsRilGJSEiIiooRjRGJUYrRiUpRixGNkYlRiUqKCIiJUYlRipGJUYsRiVGOComIiM7RiVGN0YlRjgqJkY0RiVGK0YlRiVGJUYnRiUsJiokRjVGJUYlRilGJUYlRiUqJilGQEYpRiUsLiomKUYrRjRGJUYsRiVGJSooIiNDRiVGMUYlRixGJUYlKiZGLkYlKUYrRjxGJUYlKigiI2tGJUYqRiVGLEYlRjgqJkYkRiVGN0YlRiUqJiIkRyJGJUYrRiVGOEYlRiU=

simplify(factor([solve(%,Q)]),symbolic);

NyQsJCosIyIiIiIjO0YmLC4qJiwoKigiIiVGJilJIlpHNiIiIiNGJkkieUdGL0YmISIiKiYiIilGJilGMUYwRiZGJiomRixGJkYuRiZGMkYmKSwoRixGJiomRi1GJkY1RiZGJiomLCYqJClGLkYsRiZGMkY2RjJGJkYxRiZGJiNGJkYwRiZGJiomKUYuIiImRiZGMUYmRiYqKEY0RiYpRi4iIiRGJkY1RiZGMkYrRjIqJiwmKiZGNEYmKUYxRkRGJkYmRjRGJkYmRi5GJkYmKiZGJ0YmRjVGJkYyRiYsJiokRkNGJkYmRjBGJkYmRi1GMkYxRjJGMiwkKixGJUYmLC4qJiwoRitGJkYzRjJGNkYmRiZGN0YmRiZGP0YmRkJGMkYrRjJGRUYmRklGMkYmRkpGJkYtRjJGMUYyRjI=

map(normal,map(series,%,Z,2));

NyQrJUkiWkc2Ii1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyIiIiIiISspRiQsJComIiIlRixJInlHRiVGLEYsISIjLCQqKCIiI0YsLCYqJClGMiIiJEYsRixGLEYsRixGMiEiIkY7RjtGJkYt

Q0yZ:=op(1,%%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVRMHlaR0YoLCQqLCMiIiIiIztGMSwuKiYsKCooIiIlRjEpSSJaR0YoIiIjRjFJInlHRihGMSEiIiomIiIpRjEpRjtGOkYxRjEqJkY3RjFGOUYxRjxGMSksKEY3RjEqJkY4RjFGP0YxRjEqJiwmKiQpRjlGN0YxRjxGQEY8RjFGO0YxRjEjRjFGOkYxRjEqJilGOSIiJkYxRjtGMUYxKihGPkYxKUY5IiIkRjFGP0YxRjxGNkY8KiYsJiomRj5GMSlGO0ZORjFGMUY+RjFGMUY5RjFGMSomRjJGMUY/RjFGPEYxLCYqJEZNRjFGMUY6RjFGMUY4RjxGO0Y8Rjw3I0Yu

We want to compare this with the expression of [mbm-Mishna], Proposition 14 Delta:=1-y*W*(4+W^3)/4+y^2*W^2/4;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZEZWx0YUdGKCwoIiIiRi8qKiNGLyIiJUYvSSJ5R0YoRi9JIldHRihGLywmKiQpRjQiIiRGL0YvRjJGL0YvISIiKihGMUYvKUYzIiIjRi8pRjRGPEYvRi83I0Yu

Q0ypap:=W*(4-W^3)/16/t-(t-y^2+t*y^3)/2/t^2/y-(2*y^2-y*W^2-W)*sqrt(Delta)/2/t/y/W;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdRMHlwYXBHRigsKCoqIyIiIiIjO0YxSSJXR0YoRjEsJiokKUYzIiIkRjEhIiIiIiVGMUYxSSJ0R0YoRjhGMSoqI0YxIiIjRjEsKComRjpGMSlJInlHRihGN0YxRjEqJClGQUY9RjFGOEY6RjFGMSlGOkY9RjhGQUY4RjgqLiNGMUY5RjEsKComKUYzRj1GMUZBRjFGOComRj1GMUZDRjFGMUYzRjhGMSksKiomKUYzRjlGMUZBRjFGOComRklGMUZDRjFGMSooRjlGMUYzRjFGQUYxRjhGOUYxRjxGMUY6RjhGQUY4RjNGOEY4NyNGLg==

simplify(subs(tZ,W=Z,Q0yZ/Q0ypap)) assuming 0<Z, Z<1;;

IiIi

liste:=x*y+1/x+1/y+x+y+1/x/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwuKiZJInhHRigiIiJJInlHRihGMUYxKiRGMCEiIkYxKiRGMkY0RjFGMEYxRjJGMSomRjBGNEYyRjRGMTcjRi4=

K:=collect((t*liste-1)*x*y,t,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCYqJiwuKiYpSSJ4R0YoIiIjIiIiKUkieUdGKEY0RjVGNSomRjJGNUY3RjVGNSomRjNGNUY2RjVGNUYzRjVGN0Y1RjVGNUY1SSJ0R0YoRjVGNSomRjNGNUY3RjUhIiI3I0Yu

The functional equation eqQ:=K*Q(x,y)+x*y-subs(x=0,K)*Q(0,y) -subs(y=0,K)*Q(x,0)+subs(x=0,y=0,K)*Q(0,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcVFHRigsLComLCYqJiwuKiYpSSJ4R0YoIiIjIiIiKUkieUdGKEY2RjdGNyomRjRGN0Y5RjdGNyomRjVGN0Y4RjdGN0Y1RjdGOUY3RjdGN0Y3SSJ0R0YoRjdGNyomRjVGN0Y5RjchIiJGNy1JIlFHRig2JEY1RjlGN0Y3Rj1GNyooLCZGOUY3RjdGN0Y3RjxGNy1GQDYkIiIhRjlGN0Y+KigsJkY1RjdGN0Y3RjdGPEY3LUZANiRGNUZGRjdGPiomRjxGNy1GQDYkRkZGRkY3Rjc3I0Yu

Qser:=proc(n) option remember: if n=0 then 1 else convert(normal(series(subs(Q(x,y)=Qser(n-1),Q(x,0)=subs(y=0,Qser(n-1)), Q(0,y)=subs(x=0,Qser(n-1)), Q(0,0)=subs(x=0,y=0,Qser(n-1)), expand(eqQ/x/y+Q(x,y))),t,n+1)),polynom): fi: end: Qser(4);

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

Rser:=proc(n) convert(normal(series(subs(y=0,K)*subs(y=0,Qser(n-1)),t,n+1)),polynom):end: Sser:=proc(n) convert(normal(series(subs(x=0,K)*subs(x=0,Qser(n-1)),t,n+1)),polynom):end: Rser(4);Sser(4);

LCoqJiwmSSJ4RzYiIiIiRidGJ0YnSSJ0R0YmRidGJyooRiRGJ0YlRicpRigiIiNGJ0YnKihGJEYnLCgqJClGJUYrRidGJ0YlRiciIiRGJ0YnKUYoRjBGJ0YnKihGJEYnLCoqJClGJUYwRidGJyomRjBGJ0YvRidGJyomIiIpRidGJUYnRiciIiVGJ0YnKUYoRjlGJ0Yn

LCoqJiwmSSJ5RzYiIiIiRidGJ0YnSSJ0R0YmRidGJyooRiRGJ0YlRicpRigiIiNGJ0YnKihGJEYnLCgqJClGJUYrRidGJ0YlRiciIiRGJ0YnKUYoRjBGJ0YnKihGJEYnLCoqJClGJUYwRidGJyomRjBGJ0YvRidGJyomIiIpRidGJUYnRiciIiVGJ0YnKUYoRjlGJ0Yn

Checking the functional equation n:=10: normal(series(Qser(n)*K+x*y-Rser(n)-Sser(n)+subs(x=0,Rser(n)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Invariants have been constructed in the Maple session "invariants-syst" Jy:=t/y-t*y-(1+2*t)/(1+y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKeUdGKCwoKiZJInRHRigiIiJJInlHRighIiJGMSomRjBGMUYyRjFGMyomLCZGMUYxKiYiIiNGMUYwRjFGMUYxLCZGMkYxRjFGMUYzRjM3I0Yu

Gy:=-1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiRJInlHRighIiJGMTcjRi4=

Jy; Ly:=S(y)-Gy;

LCgqJkkidEc2IiIiIkkieUdGJSEiIkYmKiZGJEYmRidGJkYoKiYsJkYmRiYqJiIiI0YmRiRGJkYmRiYsJkYnRiZGJkYmRihGKA==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNMeUdGKCwmLUkiU0dGKDYjSSJ5R0YoIiIiKiRGMiEiIkYzNyNGLg==

We construct an expression with no pole at y=0 nor y=-1. We begin with the pole at y=-1 expr:=(Ly-subs(y=-1,Ly))*Jy;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVleHByR0YoKiYsKi1JIlNHRig2I0kieUdGKCIiIiokRjMhIiJGNC1GMTYjRjZGNkY0RjRGNCwoKiZJInRHRihGNEYzRjZGNComRjtGNEYzRjRGNiomLCZGNEY0KiYiIiNGNEY7RjRGNEY0LCZGM0Y0RjRGNEY2RjZGNDcjRi4=

Note that S(-1)=0 (because K(0,y) has a factor (1+y)) subs(x=0,K);

KiYsJkkieUc2IiIiIkYmRiZGJkkidEdGJUYm

expr:=subs(S(-1)=0,expr);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVleHByR0YoKiYsKC1JIlNHRig2I0kieUdGKCIiIiokRjMhIiJGNEY0RjRGNCwoKiZJInRHRihGNEYzRjZGNComRjlGNEYzRjRGNiomLCZGNEY0KiYiIiNGNEY5RjRGNEY0LCZGM0Y0RjRGNEY2RjZGNDcjRi4=

There remains a double pole at y=0 eqcat:=expr-a*Ly^2-b*Ly-c;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZlcWNhdEdGKCwqKiYpLCYtSSJTR0YoNiNJInlHRigiIiIqJEY1ISIiRjYiIiNGNkkiYUdGKEY2RjgqJkYxRjZJImJHRihGNkY4SSJjR0YoRjgqJiwoRjJGNkY3RjZGNkY2RjYsKComSSJ0R0YoRjZGNUY4RjYqJkZCRjZGNUY2RjgqJiwmRjZGNiomRjlGNkZCRjZGNkY2LCZGNUY2RjZGNkY4RjhGNkY2NyNGLg==

Let us identify a, b and c normal(series(eqcat,y,1));

KylJInlHNiIsJkkiYUdGJCEiIkkidEdGJCIiIiEiIywsKigiIiNGKS1JIlNHRiQ2IyIiIUYpRiZGKUYnKiZGKEYpRi5GKUYpSSJiR0YkRidGKEYnRilGJ0YnLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRilGMQ==

asol:=solve(op(1,%),a);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhc29sR0YoSSJ0R0YoNyNGLg==

normal(series(subs(a=asol,eqcat),y,1));

KydJInlHNiIsKiomSSJ0R0YkIiIiLUkiU0dGJDYjIiIhRighIiJJImJHRiRGLUYnRi1GKEYtRi0tSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYs

bsol:=solve(op(1,%),b);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVic29sR0YoLCgqJkkidEdGKCIiIi1JIlNHRig2IyIiIUYxISIiRjBGNkYxRjY3I0Yu

normal(series(subs(a=asol,b=bsol,eqcat),y,2));

KydJInlHNiIsKiomLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kiU0dGJDYjIiIhIiIiSSJ0R0YkRjEhIiIqJkYyRjEtRi5GL0YxRjNJImNHRiRGM0YyRjNGMC1JIk9HRio2I0YxRjE=

csol:=solve(op(1,%),c);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVjc29sR0YoLCgqJi0tSSJER0YlNiNJIlNHRig2IyIiISIiIkkidEdGKEY3ISIiKiZGOEY3LUY0RjVGN0Y5RjhGOTcjRi4=

eqcat1:=numer(factor(subs(a=asol,b=bsol,c=csol,eqcat)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdlcWNhdDFHRigsQCooKS1JIlNHRig2I0kieUdGKCIiIyIiIkkidEdGKEY2KUY0RjVGNiEiIioqRjFGNi1GMjYjIiIhRjZGN0Y2RjhGNkY2KihGMUY2RjdGNilGNCIiJEY2RjkqKEYwRjZGN0Y2RjRGNkY5KipGMUY2RjtGNkY3RjZGNEY2RjYqKC0tSSJER0YlNiNGMkY8RjZGN0Y2RjhGNkY2KihGN0Y2RjtGNkY4RjZGNiomRjdGNkY/RjZGOSoqRjVGNkYxRjZGN0Y2RjRGNkY5KiZGMUY2RjhGNkY2KihGREY2RjdGNkY0RjZGNioqRjVGNkY7RjZGN0Y2RjRGNkY2KiZGN0Y2RjhGNkY5KiZGMUY2RjdGNkY5KiZGN0Y2RjtGNkY2NyNGLg==

Can this be true ? n:=30: normal(series(subs(S(y)=Sser(n), S(0)=subs(y=0,Sser(n)), D(S)(0)=subs(y=0,diff(Sser(n),y)), eqcat1),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0o=

Baxk to Q(0,y) (denoted here Q(y) to lighten notation) Sy:=subs(x=0,K)*Q(y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNTeUdGKCooLCZJInlHRigiIiJGMUYxRjFJInRHRihGMS1JIlFHRig2I0YwRjE3I0Yu

factor(subs(S(y)=Sy,S(0)=subs(y=0,Sy),(D(S))(0)=limit(diff(Sy,y),y=0),eqcat1));

LCQqKEkidEc2IiIiIiwmSSJ5R0YlRiZGJkYmRiYsPCooKS1JIlFHRiU2I0YoIiIjRiYpRiRGL0YmKUYoIiIkRiZGJioqRi9GJkYrRiZGMEYmKUYoRi9GJkYmKipGLEYmLUYtNiMiIiFGJkYwRiZGNEYmISIiKihGK0YmRjBGJkYoRiZGJioqRixGJkY2RiZGMEYmRihGJkY5KihGLEYmRiRGJkYxRiZGJioqRi9GJkYsRiZGJEYmRihGJkYmKiZGLEYmRjRGJkY5KipGL0YmRjZGJkYkRiZGKEYmRjkqKC0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGLUY3RiZGJEYmRihGJkY5KiZGJEYmRixGJkYmKiZGJEYmRjZGJkY5KiRGNEYmRiZGJkY5

eqcatQ:=collect(op(4,%),Q(y),factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdlcWNhdFFHRigsLioqSSJ5R0YoIiIiKUkidEdGKCIiI0YxKSwmRjBGMUYxRjFGNEYxKS1JIlFHRig2I0YwRjRGMUYxKiYsLiooRjJGMS1GOTYjIiIhRjEpRjBGNEYxISIiKihGMkYxRj5GMUYwRjFGQiomRjNGMSlGMCIiJEYxRjEqKEY0RjFGM0YxRjBGMUYxKiRGQUYxRkJGM0YxRjFGOEYxRjEqKkY0RjFGPkYxRjNGMUYwRjFGQiooLS1JIkRHRiU2I0Y5Rj9GMUYzRjFGMEYxRkIqJkYzRjFGPkYxRkJGSEYxNyNGLg==

eqcatQ;

LC4qKkkieUc2IiIiIilJInRHRiUiIiNGJiksJkYkRiZGJkYmRilGJiktSSJRR0YlNiNGJEYpRiZGJiomLC4qKEYnRiYtRi42IyIiIUYmKUYkRilGJiEiIiooRidGJkYzRiZGJEYmRjcqJkYoRiYpRiQiIiRGJkYmKihGKUYmRihGJkYkRiZGJiokRjZGJkY3RihGJkYmRi1GJkYmKipGKUYmRjNGJkYoRiZGJEYmRjcqKC0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGLkY0RiZGKEYmRiRGJkY3KiZGKEYmRjNGJkY3Rj1GJg==

eqcat:=subs(Q(y)=Q,Q(0)=A1,D(Q)(0)=A2,eqcatQ);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZlcWNhdEdGKCwuKipJInlHRigiIiIpSSJ0R0YoIiIjRjEpLCZGMEYxRjFGMUY0RjEpSSJRR0YoRjRGMUYxKiYsLiooSSNBMUdGKEYxRjJGMSlGMEY0RjEhIiIqKEY8RjFGMkYxRjBGMUY+KiZGM0YxKUYwIiIkRjFGMSooRjRGMUYzRjFGMEYxRjEqJEY9RjFGPkYzRjFGMUY4RjFGMSoqRjRGMUY8RjFGM0YxRjBGMUY+KihJI0EyR0YoRjFGM0YxRjBGMUY+KiZGM0YxRjxGMUY+RkRGMTcjRi4=

Two series cancel this derivative edQ:=factor(diff(eqcat,Q));subs(t=0,%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlZFFHRigsNCoqIiIjIiIiSSJRR0YoRjEpSSJ0R0YoRjBGMSlJInlHRigiIiRGMUYxKihJI0ExR0YoRjFGM0YxKUY2RjBGMSEiIioqIiIlRjFGMkYxRjNGMUY6RjFGMSooRjlGMUYzRjFGNkYxRjsqKkYwRjFGMkYxRjNGMUY2RjFGMSomRjRGMUY1RjFGMSooRjBGMUY0RjFGNkYxRjEqJEY6RjFGO0Y0RjE3I0Yu

LCQqJClJInlHNiIiIiMiIiIhIiI=

Let us compute their expansions Y0ser:=proc(n) add(y0[i]*t^i,i=1..n): end: n:=1: normal(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)), y=Y0ser(n), edQ),t,n+1));

KydJInRHNiIiIiJGJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIj

Y0 is in fact a series in s=sqrt(t) (and Y1 will be obtained by replacing sqrt(t) by -sqrt(t) y0:='y0': Y0ser:=proc(n) add(y0[i]*s^i,i=1..n): end: n:=1: normal(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)), y=Y0ser(n), t=s^2, edQ),s,n+1));

KyVJInNHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiM=

n:=n+1: normal(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)), y=Y0ser(n), t=s^2, edQ),s,n+1));

KydJInNHNiIsJiokKSZJI3kwR0YkNiMiIiIiIiNGKyEiIkYrRitGLC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiRGKiIiJA==

y0[1]:=1;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTBHRig2IyIiIkYwNyNGMA==

From now on we execute the next two lines several times to obtain coefficients of Y0 n:=n+1: normal(series(normal(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)), t=s^2, y=Y0ser(n),edQ)),s,n+1));

KydJInNHNiIsJiomIiIjIiIiJkkjeTBHRiQ2IyIiKEYoISIiIyIkVCIiIilGKEYwLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIio=

y0[n-1]:=solve(coeff(%,s,n),y0[n-1]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTBHRig2IyIiKCMiJFQiIiM7NyNGMQ==

Y0ser(n);

LDJJInNHNiIiIiIqJClGIyIiI0YlRiUqJiNGJUYoRiUpRiMiIiRGJUYlKiQpRiMiIiVGJUYlKiYjIiNGIiIpRiUpRiMiIiZGJUYlKiYiIidGJSlGI0Y3RiVGJSomIyIkVCIiIztGJSlGIyIiKEYlRiUqJiZJI3kwR0YkNiNGM0YlKUYjRjNGJUYl

edy:=factor(diff(eqcat,y));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlZHlHRigsOCoqIiIkIiIiKUkiUUdGKCIiI0YxKUkidEdGKEY0RjEpSSJ5R0YoRjRGMUYxKixGNEYxSSNBMUdGKEYxRjNGMUY1RjFGOEYxISIiKioiIiVGMUYyRjFGNUYxRjhGMUYxKihGOkYxRjNGMUY1RjFGOyomRjJGMUY1RjFGMSoqRjBGMUYzRjFGNkYxRjdGMUYxKihGNEYxRjpGMUY2RjFGOyomSSNBMkdGKEYxRjZGMUY7KihGNEYxRjNGMUY2RjFGMSooRjRGMUYzRjFGOEYxRjsqJkY0RjFGOEYxRjE3I0Yu

disc:=collect(factor(discrim(eqcat,Q)),y);

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

The discriminant of this polynomial (with respect to y) vanishes res:=factor(discrim(disc,y));nops(res);

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

IiIm

The factor that is cubed cannot vanish (constant term 1), so here is our first equation between A1 and A2 alg1:=op(4,res);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGcxR0YoLGJ0KiYpSSNBMUdGKCIiKSIiIilJInRHRigiIzZGM0YzKigiI0tGMylGMSIiKEYzKUY1IiM1RjNGMyoqIiM7RjMpRjEiIidGM0kjQTJHRihGM0Y7RjNGMyooIiIlRjNGOUYzKUY1IiIqRjNGMyooIiRbJEYzRj9GM0ZERjNGMyoqIiQlUUYzKUYxIiImRjNGQUYzRkRGM0YzKioiIycqRjMpRjFGQ0YzKUZBIiIjRjNGREYzRjMqKCIja0YzRj9GMylGNUYyRjNGMyoqIiNbRjNGSkYzRkFGM0ZTRjNGMyooRkBGM0Y/RjMpRjVGOkYzRjMqKCIlczlGM0ZKRjNGU0YzRjMqKiIlR0hGM0ZORjNGQUYzRlNGM0YzKioiJU86RjMpRjEiIiRGM0ZPRjNGU0YzRjMqKiIkYyNGMylGMUZQRjMpRkFGaW5GM0ZTRjNGMyooIiMlKUYzRkpGM0ZXRjNGMyoqIiR3JkYzRk5GM0ZBRjNGV0YzRjMqKiIkIz5GM0ZobkYzRk9GM0ZXRjNGMyooRjJGM0ZKRjMpRjVGQEYzRjMqKiIjY0YzRk5GM0ZBRjNGZW9GM0YzKigiJXU8RjNGTkYzRldGM0YzKioiJSM+KUYzRmhuRjNGQUYzRldGM0YzKioiJT9uRjNGXG9GM0ZPRjNGV0YzRjMqKiIlWz9GM0YxRjNGXW9GM0ZXRjNGMyooRltvRjMpRkFGQ0YzRldGM0YzKihGQ0YzRkpGMylGNUZLRjNGMyooIiUzP0YzRk5GM0Zlb0YzISIiKioiJVs3RjNGaG5GM0ZBRjNGZW9GM0YzKipGZ25GM0Zcb0YzRk9GM0Zlb0YzRjMqKkZbb0YzRjFGM0Zdb0YzRmVvRjNGMyooIiRPJ0YzRk5GM0ZjcEYzRmZwKipGY29GM0ZobkYzRkFGM0ZjcEYzRjMqKCIlUzlGM0ZobkYzRmVvRjNGMyoqIiRnIkYzRlxvRjNGT0YzRmNwRjNGMyoqIiYrMyJGM0Zcb0YzRkFGM0Zlb0YzRjMqKiIlOyMqRjNGMUYzRk9GM0Zlb0YzRjMqKCIlL0JGM0Zdb0YzRmVvRjNGMyooRlJGM0ZORjMpRjVGQ0YzRmZwKipGOEYzRmhuRjNGQUYzRmlxRjNGMyooIiUrXEYzRmhuRjNGY3BGM0ZmcCoqRmNvRjNGXG9GM0ZBRjNGY3BGM0YzKioiJVtPRjNGMUYzRk9GM0ZjcEYzRjMqKCIlQzVGM0Zdb0YzRmNwRjNGMyomRk5GMylGNUZpbkYzRjMqKCIlPz5GM0ZobkYzRmlxRjNGZnAqKiIkcydGM0Zcb0YzRkFGM0ZpcUYzRmZwKigiJS9dRjNGXG9GM0ZjcEYzRjMqKiIkUydGM0YxRjNGT0YzRmlxRjNGMyoqIiVnZEYzRjFGM0ZBRjNGY3BGM0YzKigiJEciRjNGXW9GM0ZpcUYzRjMqKCIkKUdGM0ZPRjNGY3BGM0ZmcCooIiQpUUYzRmhuRjNGY3JGM0ZmcCooIiVvPkYzRlxvRjNGaXFGM0YzKipGUkYzRjFGM0ZPRjNGY3JGM0YzKioiJSlbIkYzRjFGM0ZBRjNGaXFGM0ZmcCooRmVyRjNGT0YzRmlxRjNGZnAqKEZnb0YzRmhuRjMpRjVGUEYzRmZwKipGMkYzRlxvRjNGQUYzRlt0RjNGMyooIiVZQ0YzRlxvRjNGY3JGM0YzKioiJSFHIkYzRjFGM0ZBRjNGY3JGM0ZmcCooIiQvKEYzRk9GM0ZjckYzRmZwKigiJUNJRjNGQUYzRmlxRjNGMyooIiRDKUYzRlxvRjNGW3RGM0YzKipGPkYzRjFGM0ZBRjNGW3RGM0ZmcCooIiVPT0YzRjFGM0ZjckYzRmZwKigiJV82RjNGQUYzRmNyRjNGMyoqRjhGM0YxRjNGQUYzRjVGM0YzKigiJTs5RjNGMUYzRlt0RjNGZnAqKEY+RjNGT0YzRjVGM0YzKigiI0NGM0ZBRjNGW3RGM0ZmcComIiUkZiJGM0ZjckYzRjMqJkY+RjNGXG9GM0ZmcCooRjhGM0ZBRjNGNUYzRmZwKiYiJFsnRjNGW3RGM0YzKiZGOEYzRjFGM0YzRj5GZnA3I0Yu

n:=10: normal(series(subs(A1=subs(x=0,y=0,Qser(n)), A2=subs(y=0,diff(subs(x=0,Qser(n)),y)), alg1),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

A polynomial P in y has d double roots (here, d=2) if the generalized discriminants defined below vanish for k=0, 1, ..., d-1 with(LinearAlgebra): disciter:=proc(P,k) local M,i,j,d: d:=degree(P,y): M:=Matrix(2*d-1-2*k,2*d-1-2*k): for i from 1 to d-1-k do for j from 0 to min(d, 2*d-1-2*k-i) do M[i,i+j]:=coeff(P,y,d-j):od:od: for i from 0 to d-k- 1 do for j from 1 to min( 2*d-2*k-1-i) do M[i+d-k,i+j]:=coeff(diff(P,y),y,d-j): od: od: M:end: The first one is just the usual discriminant factor(Determinant(disciter(disc,0)));

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factor(%/alg1);

LCQqKiIlJzQlIiIiKUkidEc2IiIjOUYlKSwmRiVGJSomIiIjRiVGJ0YlRiVGLUYlKSwqKiZJI0ExR0YoRiUpRidGLUYlRiUqJkkjQTJHRihGJUYyRiVGJUYnISIiRiVGNSIiJEYlRiU=

factor(Determinant(disciter(disc,1)));nops(%);

LCQqKiIkNyYiIiIpSSJ0RzYiIiM1RiUsaFtsKigiIiZGJSlJI0ExR0YoIiIpRiUpRiciIzlGJUYlKigiIiVGJUYtRiUpRiciIzhGJUYlKiZGLUYlKUYnIiM3RiVGJSooIiQlPUYlKUYuIiIoRiVGNEYlRiUqKiIjISlGJSlGLiIiJ0YlSSNBMkdGKEYlRjRGJUYlKigiJG8iRiVGO0YlRjdGJUYlKioiI2tGJUY/RiVGQUYlRjdGJUYlKigiI19GJUY7RiUpRiciIzZGJUYlKioiIztGJUY/RiVGQUYlRkhGJUYlKigiJW9DRiVGP0YlRjdGJUYlKioiJTNBRiUpRi5GLEYlRkFGJUY3RiVGJSoqIiQhW0YlKUYuRjNGJSlGQSIiI0YlRjdGJUYlKihGM0YlRjtGJUYmRiVGJSooIiVLQ0YlRj9GJUZIRiVGJSoqIiU7P0YlRlBGJUZBRiVGSEYlRiUqKiIkJVFGJUZTRiVGVEYlRkhGJUYlKigiJFspRiVGP0YlRiZGJUYlKioiJEMnRiVGUEYlRkFGJUYmRiVGJSooIiZ3ViJGJUZQRiVGSEYlRiUqKiIjJypGJUZTRiVGVEYlRiZGJUYlKioiJlMtI0YlRlNGJUZBRiVGSEYlRiUqKiIlSykpRiUpRi4iIiRGJUZURiVGSEYlRiUqKiIlIUciRiUpRi5GVUYlKUZBRmRvRiVGSEYlRiUqKCIkMSJGJUY/RiUpRiciIipGJUYlKioiI1tGJUZQRiVGQUYlRltwRiVGJSooIiZrUyJGJUZQRiVGJkYlRiUqKiImJzQ/RiVGU0YlRkFGJUYmRiVGJSoqIiVrISlGJUZjb0YlRlRGJUYmRiVGJSoqIiVDNUYlRmdvRiVGaG9GJUYmRiVGJSooRkBGJUY/RiUpRidGL0YlRiUqKCIlKXklRiVGUEYlRltwRiVGJSoqIiVPckYlRlNGJUZBRiVGW3BGJUYlKigiJjElR0YlRlNGJUYmRiVGJSoqIiUnXCNGJUZjb0YlRlRGJUZbcEYlRiUqKiImZ1wnRiVGY29GJUZBRiVGJkYlRiUqKiIkYyNGJUZnb0YlRmhvRiVGW3BGJUYlKioiJmNNJUYlRmdvRiVGVEYlRiZGJUYlKioiJnc8IkYlRi5GJUZob0YlRiZGJUYlKihGZm9GJSlGQUYzRiVGJkYlRiUqKCIkMydGJUZQRiVGaHBGJUYlKioiJE8qRiVGU0YlRkFGJUZocEYlRiUqKCImV3YiRiVGU0YlRltwRiVGJSoqIiQjPkYlRmNvRiVGVEYlRmhwRiVGJSoqIiZTWSdGJUZjb0YlRkFGJUZbcEYlRiUqKiImS1MlRiVGZ29GJUZURiVGW3BGJUYlKioiJl8yIkYlRi5GJUZob0YlRltwRiVGJSooRmZwRiVGanFGJUZbcEYlRiUqKCIjY0YlRlBGJSlGJ0Y8RiVGJSoqRltzRiVGU0YlRkFGJUZcc0YlRiUqKCIlPTpGJUZTRiVGaHBGJSEiIioqIiZbTyNGJUZjb0YlRkFGJUZocEYlRiUqKCImczMkRiVGY29GJUZbcEYlRmBzKioiJiVRO0YlRmdvRiVGVEYlRmhwRiVGJSoqIiZbVyJGJUZnb0YlRkFGJUZbcEYlRiUqKiIlR0xGJUYuRiVGaG9GJUZocEYlRiUqKiImQylIRiVGLkYlRlRGJUZbcEYlRiUqKEZkcUYlRmpxRiVGaHBGJUYlKigiJU96RiVGaG9GJUZbcEYlRiUqKEYzRiVGUEYlKUYnRkBGJUYlKigiJUNJRiVGU0YlRlxzRiVGYHMqKiIld1BGJUZjb0YlRkFGJUZcc0YlRiUqKCImS3kqRiVGY29GJUZocEYlRmBzKioiJStDRiVGZ29GJUZURiVGXHNGJUYlKioiJkc0I0YlRmdvRiVGQUYlRmhwRiVGYHMqKkZkcUYlRi5GJUZob0YlRlxzRiVGJSoqIiZPTiRGJUYuRiVGVEYlRmhwRiVGJSooIiZTLSJGJUZob0YlRmhwRiVGJSooIiRaJkYlRlNGJUZhdEYlRmBzKioiJFslRiVGY29GJUZBRiVGYXRGJUYlKigiJik9JSlGJUZjb0YlRlxzRiVGYHMqKiIkZyJGJUZnb0YlRlRGJUZhdEYlRiUqKiImOyFHRiVGZ29GJUZBRiVGXHNGJUZgcyooIiczbDVGJUZnb0YlRmhwRiVGYHMqKiImJSl6IkYlRi5GJUZURiVGXHNGJUYlKioiJ0szNUYlRi5GJUZBRiVGaHBGJUZgcyooIiVLY0YlRmhvRiVGXHNGJUYlKigiJi8+JEYlRlRGJUZocEYlRmBzKigiI0tGJUZTRiUpRidGLEYlRmBzKipGZnZGJUZjb0YlRkFGJUZndkYlRiUqKCImOz0kRiVGY29GJUZhdEYlRmBzKipGYHVGJUZnb0YlRkFGJUZhdEYlRmBzKigiJykzOCNGJUZnb0YlRlxzRiVGYHMqKiIld2BGJUYuRiVGVEYlRmF0RiVGJSoqIidLN0BGJUYuRiVGQUYlRlxzRiVGYHMqKCIlMzlGJUZob0YlRmF0RiVGJSooIiYpR2dGJUZURiVGXHNGJUZgcyomRlNGJSlGJ0YzRiVGJSooIiUpKWVGJUZjb0YlRmd2RiVGYHMqKiIlWzVGJUZnb0YlRkFGJUZndkYlRmBzKigiJz9AOkYlRmdvRiVGYXRGJUZgcyoqIiQnKilGJUYuRiVGVEYlRmd2RiVGJSoqIic3PDtGJUYuRiVGQUYlRmF0RiVGYHMqKCIlOzlGJUYuRiVGXHNGJUZgcyooIiRHIkYlRmhvRiVGZ3ZGJUYlKigiJl8+JUYlRlRGJUZhdEYlRmBzKigiJmNNJEYlRkFGJUZcc0YlRiUqKCIkSydGJUZjb0YlRmd3RiVGYHMqKkZFRiVGZ29GJUZBRiVGZ3dGJUYlKigiJkU8JkYlRmdvRiVGZ3ZGJUZgcyoqRkVGJUYuRiVGVEYlRmd3RiVGJSoqIiY7J2ZGJUYuRiVGQUYlRmd2RiVGYHMqKCImby4lRiVGLkYlRmF0RiVGJSooIiZvTiJGJUZURiVGZ3ZGJUZgcyooIiYpKSlwRiVGQUYlRmF0RiVGJSooIiNXRiVGY29GJSlGJ0Zkb0YlRmBzKipGL0YlRmdvRiVGQUYlRmp5RiVGJSooIiUxJipGJUZnb0YlRmd3RiVGYHMqKiImQzEiRiVGLkYlRkFGJUZnd0YlRmBzKigiJmNGKUYlRi5GJUZndkYlRiUqKCIlUz1GJUZURiVGZ3dGJUZgcyooRmV3RiVGQUYlRmd2RiVGJSomIiZCRiVGJUZhdEYlRmBzKigiJTc4RiVGZ29GJUZqeUYlRmBzKioiJHMnRiVGLkYlRkFGJUZqeUYlRmBzKigiJiEpeiZGJUYuRiVGZ3dGJUYlKigiJjNfI0YlRkFGJUZnd0YlRiUqJiImYyJ5RiVGZ3ZGJUZgcyooIiQnPUYlRmdvRiUpRidGVUYlRmBzKipGS0YlRi5GJUZBRiVGY1tsRiVGJSooIiYhMzxGJUYuRiVGanlGJUYlKihGS0YlRlRGJUZjW2xGJUYlKigiJVtZRiVGQUYlRmp5RiVGJSomIiYuPyZGJUZnd0YlRmBzKihGMUYlRmdvRiVGJ0YlRmBzKigiJTc5RiVGLkYlRmNbbEYlRiUqKEZFRiVGQUYlRmNbbEYlRiUqJiImX1kiRiVGanlGJUZgcyooIiRfI0YlRi5GJUYnRiVGYHMqKCIjKSlGJUZBRiVGJ0YlRmBzKiYiJTU1RiVGY1tsRiVGYHMqJiIjU0YlRi5GJUZgcyomRi9GJUZBRiVGYHMqJiIkdSNGJUYnRiVGJUZpXGxGJUYlKSwqKiZGLkYlRmNbbEYlRiUqJkZBRiVGY1tsRiVGJUYnRmBzRiVGYHNGVUYlRiU=

IiIl

alg2:=op(3,%%): algA1cand:=factor(resultant(alg1,alg2,A2));nops(%);

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

IiIp

Only one factor can vanish (look at the constant term), and we have found the algebraic equation satisfied by A1 (degree 4) res:=factor(resultant(disc,algA1,A1)): nops(%);

IiIj

op(1,res);

KiQpSSJ0RzYiIiM3IiIi

res1:=factor(resultant(op(2,res),algA2,A2)):nops(res1);

IiIn

seq(indets(op(i,res1)),i=1..nops(res1)); seq(nops(op(i,res1)),i=1..nops(res1));

Nig8IjwjSSJ0RzYiPCRGJUkieUdGJkYnRidGJw==

NigiIiIiIiMiJEciIiRmJkYkRiQ=

n:=14: normal(series(subs(y=Y0ser(n),t=s^2,op(3,res1)),s,n+1));

KydJInNHNiIiJGMjIiM5LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiM6

n:=20: normal(series(subs(y=Y0ser(n),t=s^2,op(4,res1)),s,n+1));

Ky9JInNHNiIiIioiIzshIycqIiM8ISUzNyIjPSIlZ2wiIz4iJldxKiIjPy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIjQA==

op(5,res1);

KiQpLDIqJilJInRHNiIiIiMiIiIpSSJ5R0YoIiIlRipGKiooRi1GKkYmRiopRiwiIiRGKiEiIiooIiInRipGJkYqKUYsRilGKkYxKihGKUYqRidGKkYvRipGMSooRi1GKkYmRipGLEYqRjEqJEYmRipGKiooRilGKkYnRipGLEYqRjEqJEY0RipGKkYpRio=

n:=10: normal(series(subs(y=Y0ser(n),t=s^2,op(1,op(5,res1))),s,n+1));

KzVJInNHNiIiIiIiIiNGJSIiJSEiJSIiJiEiKiIiJyEjOSIiKCEjPiIiKSwmIyIkUiJGJiEiIiomRiZGJSZJI3kwR0YkNiNGL0YlRiUiIiosJiMiJXIqKiIja0YzKiZGJkYlJkY2NiNGOEYlRiUiIzUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNg==

op(6,res1);

KiQpLCwqJkkidEc2IiIiIilJInlHRiciIiVGKEYoKigiIiNGKEYmRigpRioiIiRGKEYoKihGLUYoRiZGKEYqRihGKCokKUYqRi1GKCEiIkYmRihGK0Yo

n:=10: normal(series(subs(y=Y0ser(n),t=s^2,op(1,op(6,res1))),s,n+1));

KylJInNHNiIsJiIjUSIiIiomIiIjRicmSSN5MEdGJDYjIiIpRichIiIiIiosJiMiJXZlIiNrRicqJkYpRicmRis2I0YvRidGLiIjNS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YnIiM2

algY:=subs(y=Y,collect(op(1,op(6,res1)),y,factor));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdZR0YoLCwqJilJIllHRigiIiUiIiJJInRHRihGM0YzKigiIiNGMylGMSIiJEYzRjRGM0YzKiQpRjFGNkYzISIiKihGNkYzRjFGM0Y0RjNGM0Y0RjM3I0Yu

tY:=factor(solve(algY,t));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN0WUdGKComKUkiWUdGKCIiIyIiIiwqKiQpRjAiIiVGMkYyKiZGMUYyKUYwIiIkRjJGMiomRjFGMkYwRjJGMkYyRjIhIiI3I0Yu

n:=14: normal(series(subs(A1=subs(x=0,y=0,Qser(n)),op(1,op(7,algA1cand))),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

algA1:=collect(op(1,op(7,algA1cand)),A1,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdBMUdGKCwwKigiIiUiIiIpSSNBMUdGKEYwRjEpSSJ0R0YoIiInRjFGMSoqIiIpRjEpRjVGMEYxLCYqJkYwRjFGNUYxRjFGMUYxRjEpRjMiIiRGMUYxKigpRjUiIiNGMSwoKiYiI2dGMUY/RjFGMSomIiNDRjFGNUYxRjEiIiZGMUYxKUYzRkBGMUYxKihGOkYxLCgqJkYwRjFGP0YxRjEqJkY4RjFGNUYxRjFGMSEiIkYxRjNGMUZMKiYiI0dGMUY/RjFGMUY7RjFGMUZMNyNGLg==

algA2cand:=factor(resultant(alg1,alg2,A1));nops(%);

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

IiIp

n:=14: normal(series(subs(A2=subs(y=0,diff(subs(x=0,Qser(n)),y)),op(1,op(8,algA2cand))),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

The algebraic equation satisfied by A2 (also quartic) algA2:=collect(op(1,op(8,algA2cand)),A2,factor);

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

res:=factor(resultant(disc,algA1,A1)): nops(%);

IiIj

op(1,res);

KiQpSSJ0RzYiIiM3IiIi

res1:=factor(resultant(op(2,res),algA2,A2)):nops(res1);

IiIn

seq(indets(op(i,res1)),i=1..nops(res1)); seq(nops(op(i,res1)),i=1..nops(res1));

Nig8IjwjSSJ0RzYiPCRGJUkieUdGJkYnRidGJw==

NigiIiIiIiMiJEciIiRmJkYkRiQ=

n:=14: normal(series(subs(y=Y0ser(n),t=s^2,op(3,res1)),s,n+1));

KydJInNHNiIiJGMjIiM5LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiM6

n:=20: normal(series(subs(y=Y0ser(n),t=s^2,op(4,res1)),s,n+1));

Ky9JInNHNiIiIioiIzshIycqIiM8ISUzNyIjPSIlZ2wiIz4iJldxKiIjPy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIjQA==

op(5,res1);

KiQpLDIqJilJInRHNiIiIiMiIiIpSSJ5R0YoIiIlRipGKiooRi1GKkYmRiopRiwiIiRGKiEiIiooIiInRipGJkYqKUYsRilGKkYxKihGKUYqRidGKkYvRipGMSooRi1GKkYmRipGLEYqRjEqJEYmRipGKiooRilGKkYnRipGLEYqRjEqJEY0RipGKkYpRio=

n:=10: normal(series(subs(y=Y0ser(n),t=s^2,op(1,op(5,res1))),s,n+1));

KzVJInNHNiIiIiIiIiNGJSIiJSEiJSIiJiEiKiIiJyEjOSIiKCEjPiIiKSwmIyIkUiJGJiEiIiomRiZGJSZJI3kwR0YkNiNGL0YlRiUiIiosJiMiJXIqKiIja0YzKiZGJkYlJkY2NiNGOEYlRiUiIzUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNg==

op(6,res1);

KiQpLCwqJkkidEc2IiIiIilJInlHRiciIiVGKEYoKigiIiNGKEYmRigpRioiIiRGKEYoKihGLUYoRiZGKEYqRihGKCokKUYqRi1GKCEiIkYmRihGK0Yo

n:=10: normal(series(subs(y=Y0ser(n),t=s^2,op(1,op(6,res1))),s,n+1));

KylJInNHNiIsJiIjUSIiIiomIiIjRicmSSN5MEdGJDYjIiIpRichIiIiIiosJiMiJXZlIiNrRicqJkYpRicmRis2I0YvRidGLiIjNS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YnIiM2

algY:=subs(y=Y,collect(op(1,op(6,res1)),y,factor));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdZR0YoLCwqJilJIllHRigiIiUiIiJJInRHRihGM0YzKigiIiNGMylGMSIiJEYzRjRGM0YzKiQpRjFGNkYzISIiKihGNkYzRjFGM0Y0RjNGM0Y0RjM3I0Yu

tY:=factor(solve(algY,t));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN0WUdGKComKUkiWUdGKCIiIyIiIiwqKiQpRjAiIiVGMkYyKiZGMUYyKUYwIiIkRjJGMiomRjFGMkYwRjJGMkYyRjIhIiI3I0Yu

JSFH

Comparison with [MBM-Mishna], Section 6.3. It is convenient to define the series Z, with no constant term, satisfying the equation below: algZ:=t*(1 - 2*Z + 6*Z^2 - 2*Z^3 + Z^4)-Z*(1-Z)^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdaR0YoLCYqJkkidEdGKCIiIiwsKiQpSSJaR0YoIiIlRjFGMSomIiIjRjEpRjUiIiRGMSEiIiomIiInRjEpRjVGOEYxRjEqJkY4RjFGNUYxRjtGMUYxRjFGMSomRjVGMSksJkYxRjFGNUY7RjhGMUY7NyNGLg==

Then it is said in [MBM-Mishna] that A1Zpap:=Z*(1 - 2* Z - Z^2 )/t/(1-Z)^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdBMVpwYXBHRigqKkkiWkdGKCIiIiwoKiQpRi8iIiNGMCEiIiomRjRGMEYvRjBGNUYwRjBGMEkidEdGKEY1KSwmRjBGMEYvRjVGNEY1NyNGLg==

tZ:=factor(solve(algZ,t));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN0WkdGKCooSSJaR0YoIiIiKSwmRjAhIiJGL0YwIiIjRjAsLCokKUYvIiIlRjBGMComRjRGMClGLyIiJEYwRjMqJiIiJ0YwKUYvRjRGMEYwKiZGNEYwRi9GMEYzRjBGMEYzNyNGLg==

let us check numer(factor(subs(t=tZ,algA1)));

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

subs(Z=0,op(2,%));

IiIi

A1Z:=factor(solve(op(1,%%),A1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBMVpHRigsJCooLCgqJClJIlpHRigiIiMiIiJGNSomRjRGNUYzRjVGNUY1ISIiRjUsLCokKUYzIiIlRjVGNSomRjRGNSlGMyIiJEY1RjcqJiIiJ0Y1RjJGNUY1RjZGN0Y1RjVGNSksJkY1RjdGM0Y1RjtGN0Y3NyNGLg==

factor(subs(t=tZ,A1Z/A1Zpap));

IiIi

Now the series A2 numer(factor(subs(t=tZ,algA2)));

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

subs(Z=0,op(2,%));

ISIi

A2Z:=factor(solve(op(1,%%),A2));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBMlpHRigqKkkiWkdGKCIiIiwoKiQpRi8iIiRGMEYwKiZGNEYwRi9GMCEiIkYwRjBGMCwsKiQpRi8iIiVGMEYwKiYiIiNGMEYzRjBGNiomIiInRjApRi9GPEYwRjAqJkY8RjBGL0YwRjZGMEYwRjApLCZGMEY2Ri9GMEY+RjY3I0Yu

Back to Q(0,y) collect(numer(factor(subs(A1=A1Z,A2=A2Z,t=tZ,eqcat))),Q,factor);

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

simplify(factor([solve(%,Q)]),symbolic);

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

map(normal,map(series,%,Z,2));

NyQrKUkiWkc2IiomSSJ5R0YlIiIiKSwmRidGKEYoRigiIiMhIiIhIiMsJCooLCgqJClGJyIiJEYoRigqJkYrRihGJ0YoRihGKEYoRihGJ0YsRilGLEYsRiwtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGKCIiISslRiRGNUY7

Q0yZ:=op(2,%%);

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

We want to compare this with the expression of [mbm-Mishna], Proposition 15 Delta:=1-2*Z*(1+Z^2)/(1-Z)^2*y+y^2*Z^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZEZWx0YUdGKCwoIiIiRi8qLCIiI0YvSSJaR0YoRi8sJiokKUYyRjFGL0YvRi9GL0YvKSwmRi9GL0YyISIiRjFGOEkieUdGKEYvRjgqJkY1Ri8pRjlGMUYvRi83I0Yu

Q0ypap:=(Z*(1 - Z) + 2*y*Z - (1 - Z) *y^2)*sqrt(Delta)/2/t/y/Z/(1-Z)/(1+y)^2 -(Z*(1-Z)^2+Z*(Z^3+4*Z^2-5*Z+2)*y-(1-2*Z+7*Z^2-4*Z^3)*y^2+y^3*Z*(1-Z)^2)/2/t/y/Z/(1-Z)^2/(1+y)^2;

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

simplify(subs(t=tZ,Q0yZ/Q0ypap)) assuming 0<Z, Z<1;;

IiIi

liste:=x*y+1/x+x+1/x/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwqKiZJInhHRigiIiJJInlHRihGMUYxKiRGMCEiIkYxRjBGMSomRjBGNEYyRjRGMTcjRi4=

K:=collect((t*liste-1)*x*y,t,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCYqKCwmSSJ5R0YoIiIiRjJGMkYyLCYqJilJInhHRigiIiNGMkYxRjJGMkYyRjJGMkkidEdGKEYyRjIqJkY2RjJGMUYyISIiNyNGLg==

The functional equation eqQ:=K*Q(x,y)+x*y-subs(x=0,K)*Q(0,y) -subs(y=0,K)*Q(x,0)+subs(x=0,y=0,K)*Q(0,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcVFHRigsLComLCYqKCwmSSJ5R0YoIiIiRjRGNEY0LCYqJilJInhHRigiIiNGNEYzRjRGNEY0RjRGNEkidEdGKEY0RjQqJkY4RjRGM0Y0ISIiRjQtSSJRR0YoNiRGOEYzRjRGNEY7RjQqKEYyRjRGOkY0LUY+NiQiIiFGM0Y0RjwqJkY6RjQtRj42JEY4RkNGNEY8KiZGOkY0LUY+NiRGQ0ZDRjRGNDcjRi4=

Qser:=proc(n) option remember: if n=0 then 1 else convert(normal(series(subs(Q(x,y)=Qser(n-1),Q(x,0)=subs(y=0,Qser(n-1)), Q(0,y)=subs(x=0,Qser(n-1)), Q(0,0)=subs(x=0,y=0,Qser(n-1)), expand(eqQ/x/y+Q(x,y))),t,n+1)),polynom): fi: end: Qser(4);

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

Rser:=proc(n) convert(normal(series(subs(y=0,K)*subs(y=0,Qser(n-1)),t,n+1)),polynom):end: Sser:=proc(n) convert(normal(series(subs(x=0,K)*subs(x=0,Qser(n-1)),t,n+1)),polynom):end: Rser(4);Sser(4);

LCpJInRHNiIiIiIqJkkieEdGJEYlKUYjIiIjRiVGJSomLCYqJClGJ0YpRiVGJUYpRiVGJSlGIyIiJEYlRiUqKEYnRiUsJkYsRiUiIiZGJUYlKUYjIiIlRiVGJQ==

LCYqJiwmSSJ5RzYiIiIiRidGJ0YnSSJ0R0YmRidGJyooRiRGJywmRiVGJyIiI0YnRicpRigiIiRGJ0Yn

Checking the functional equation n:=10: normal(series(Qser(n)*K+x*y-Rser(n)-Sser(n)+subs(x=0,Rser(n)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Invariants have been constructed in the Maple session "invariants-syst" (and first of all, in Section 3) Jy:=y/t/(1+y)^2+t/y*(1+y)^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKeUdGKCwmKihJInlHRigiIiJJInRHRighIiIpLCZGMEYxRjFGMSIiI0YzRjEqKEYyRjFGMEYzRjRGMUYxNyNGLg==

A decoupling function Gy:=-1/t/(1+y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiZJInRHRighIiIsJkkieUdGKCIiIkY0RjRGMUYxNyNGLg==

Jy; Ly:=S(y)-Gy;

LCYqKEkieUc2IiIiIkkidEdGJSEiIiksJkYkRiZGJkYmIiIjRihGJiooRidGJkYkRihGKUYmRiY=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNMeUdGKCwmLUkiU0dGKDYjSSJ5R0YoIiIiKiZJInRHRighIiIsJkYyRjNGM0YzRjZGMzcjRi4=

We construct an expression with no pole at y=0 nor y=-1. We begin with the pole at y=0 expr:=(Ly-subs(y=0,Ly))*Jy;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVleHByR0YoKiYsKi1JIlNHRig2I0kieUdGKCIiIiomSSJ0R0YoISIiLCZGM0Y0RjRGNEY3RjQtRjE2IyIiIUY3KiRGNkY3RjdGNCwmKihGM0Y0RjZGNylGOCIiI0Y3RjQqKEY2RjRGM0Y3Rj9GNEY0RjQ3I0Yu

There remains a triple pole at y=-1 eqcat:=expr-a*Ly^3-b*Ly^2-c*Ly-d;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZlcWNhdEdGKCwsKiYpLCYtSSJTR0YoNiNJInlHRigiIiIqJkkidEdGKCEiIiwmRjVGNkY2RjZGOUY2IiIkRjZJImFHRihGNkY5KiYpRjEiIiNGNkkiYkdGKEY2RjkqJkYxRjZJImNHRihGNkY5SSJkR0YoRjkqJiwqRjJGNkY3RjYtRjM2IyIiIUY5KiRGOEY5RjlGNiwmKihGNUY2RjhGOSlGOkY/RjlGNiooRjhGNkY1RjlGTEY2RjZGNkY2NyNGLg==

normal(series(eqcat,y=-1,1));

KycsJkkieUc2IiIiIkYmRiYsJComLCZJImFHRiVGJkkidEdGJUYmRiYpRisiIiQhIiJGLiEiJC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YmISIj

asol:=solve(op(1,%),a);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhc29sR0YoLCRJInRHRighIiI3I0Yu

normal(series(subs(a=asol,eqcat),y=-1,2));

KyksJkkieUc2IiIiIkYmRiYqJiwqKigiIiNGJi1JIlNHRiU2IyEiIkYmSSJ0R0YlRiZGJiomLUYsNiMiIiFGJkYvRiZGJkkiYkdGJUYuRipGJkYmKUYvRipGLiEiIyomLDAqKCIiJEYmKUYrRipGJkY1RiZGJioqRipGJkYrRiZGNEYmRi9GJkYuKiZGK0YmRi9GJkYmRjBGLiooRipGJi0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGLEYtRiZGL0YmRiYqJkkiY0dGJUYmRi9GJkYuRiZGLkYmRjVGLkYuLUkiT0dGQjYjRiZGMw==

bsol:=solve(op(1,%),b);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVic29sR0YoLCgqKCIiIyIiIi1JIlNHRig2IyEiIkYxSSJ0R0YoRjFGMSomLUYzNiMiIiFGMUY2RjFGMUYwRjE3I0Yu

subs(x=0,K);

KiYsJkkieUc2IiIiIkYmRiZGJkkidEdGJUYm

bsol:=subs(S(-1)=0,bsol);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVic29sR0YoLCYiIiMiIiIqJi1JIlNHRig2IyIiIUYwSSJ0R0YoRjBGMDcjRi4=

normal(subs(S(-1)=0,series(subs(a=asol,b=bsol,eqcat),y=-1,1)));

KycsJkkieUc2IiIiIkYmRiYsJComLCoqJi1JIlNHRiU2IyIiIUYmSSJ0R0YlRiZGJiooIiIjRiYtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiw2IyEiIkYmRi9GJkY6KiZJImNHRiVGJkYvRiZGJkYmRiZGJilGL0YxRjpGOkY6LUkiT0dGNTYjRiZGLg==

csol:=solve(op(1,%),c);collect(%,S,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVjc29sR0YoLCQqJiwoKiYtSSJTR0YoNiMiIiEiIiJJInRHRihGNkY2KigiIiNGNi0tSSJER0YlNiNGMzYjISIiRjZGN0Y2Rj9GNkY2RjZGN0Y/Rj83I0Yu

LCgtSSJTRzYiNiMiIiEhIiIqJiIiIyIiIi0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJDYjRihGK0YrKiRJInRHRiVGKEYo

normal(subs(S(-1)=0,series(subs(a=asol,b=bsol,c=csol,eqcat),y=-1,3)));

KycsJkkieUc2IiIiIkYmRiYqJiwqKioiIiNGJi1JIlNHRiU2IyIiIUYmLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YsNiMhIiJGJkkidEdGJUYmRjcqJkkiZEdGJUYmRjhGJkY3LS0tSSNAQEdGMjYkRjFGKkY1RjZGJiomIiIkRiZGL0YmRjdGJkY4RjdGLi1JIk9HRjI2I0YmRiY=

dsol:=solve(op(1,%),d);collect(%,S,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkc29sR0YoKiYsKCoqIiIjIiIiLUkiU0dGKDYjIiIhRjItLUkiREdGJTYjRjQ2IyEiIkYySSJ0R0YoRjJGPC0tLUkjQEBHRiU2JEY5RjFGOkY7RjIqJiIiJEYyRjdGMkY8RjJGPUY8NyNGLg==

LCgqKCIiIyIiIi0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJIlNHRiw2IyEiIkYlLUYuNiMiIiFGJUYwKigiIiRGJUkidEdGLEYwRiZGJUYwKiZGNkYwLS0tSSNAQEdGKTYkRihGJEYtRi9GJUYl

eqcat1:=numer(factor(subs(a=asol,b=bsol,c=csol,d=dsol,eqcat)));indets(%);

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

PCVJInRHNiJJInlHRiQtSSJTR0YkNiNGJQ==

Can this be true ? n:=30: normal(series(subs(S(y)=Sser(n), S(0)=subs(y=0,Sser(n)), D(S)(-1)=subs(y=-1,diff(Sser(n),y)),(D@@2)(S)(-1)=subs(y=-1,diff(Sser(n),y$2)), eqcat1),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0o=

Back to Q(0,y) (denoted here Q(y) to lighten notation) Sy:=subs(x=0,K)*Q(y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNTeUdGKCooLCZJInlHRigiIiJGMUYxRjFJInRHRihGMS1JIlFHRig2I0YwRjE3I0Yu

factor(subs(S(y)=Sy,S(0)=subs(y=0,Sy),(D(S))(-1)=limit(diff(Sy,y),y=-1),(D@@2)(S)(-1)=limit(diff(Sy,y$2),y=-1),eqcat1));

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

eqcatQ:=collect(op(4,%),Q(y),factor);

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

eqcatQ;

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

eqcat:=subs(Q(y)=Q,Q(0)=A1,Q(-1)=A2,D(Q)(-1)=A3,eqcatQ);

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

Three series cancel this derivative: two have constant term 0, the third one constant term 1 edQ:=factor(diff(eqcat,Q));factor(subs(t=0,%));

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

LCQqJilJInlHNiIiIiMiIiIsJkYlRihGKCEiIkYoRio=

Let us compute their expansions. We start with the two series with no constant term y0:='y0': Y0ser:=proc(n) add(y0[i]*t^i,i=1..n): end: n:=2: normal(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)), y=Y0ser(n), edQ),t,n+1));

KydJInRHNiIsJiokKSZJI3kwR0YkNiMiIiIiIiNGK0YrRishIiJGLC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiRGKiIiJA==

y0[1]:=1;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTBHRig2IyIiIkYwNyNGMA==

From now on we execute the next two lines several times to obtain coefficients of Y0 n:=n+1: normal(series(normal(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)), y=Y0ser(n),edQ)),t,n+1));

KydJInRHNiIsJiomIiIjIiIiJkkjeTBHRiQ2IyIiJ0YoRigiJGsjISIiIiIoLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIik=

y0[n-1]:=solve(coeff(%,t,n),y0[n-1]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTBHRig2IyIiJyIkSyI3I0Yx

We recognize the Catalan series Y0ser(n);

LDAqJilJInRHNiIiIigiIiImSSN5MEdGJjYjRidGKEYoKiYiJEsiRigpRiUiIidGKEYoKiYiI1VGKClGJSIiJkYoRigqJiIjOUYoKUYlIiIlRihGKComRjNGKClGJSIiJEYoRigqJiIiI0YoKUYlRjxGKEYoRiVGKA==

Another series starting at -t y1:='y1': Y1ser:=proc(n) add(y1[i]*t^i,i=1..n): end: n:=2: normal(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)), y=Y1ser(n), edQ),t,n+1));

KydJInRHNiIsJiokKSZJI3kxR0YkNiMiIiIiIiNGK0YrRishIiJGLC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiRGKiIiJA==

y1[1]:=-1;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTFHRig2IyIiIiEiIjcjRjE=

From now on we execute the next two lines several times to obtain coefficients of Y1 n:=n+1: normal(series(normal(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)), y=Y1ser(n),edQ)),t,n+1));

KydJInRHNiIsJiomIiIjIiIiJkkjeTFHRiQ2IyIiJkYoISIiIiMlKUYtIiInLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIig=

y1[n-1]:=solve(coeff(%,t,n),y1[n-1]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTFHRig2IyIiJiEjVTcjRjE=

We recognize the (signed) Catalan series Y1ser(n);

LC4qJilJInRHNiIiIiciIiImSSN5MUdGJjYjRidGKEYoKiYiI1VGKClGJSIiJkYoISIiKiYiIzlGKClGJSIiJUYoRigqJkYvRigpRiUiIiRGKEYwKiYiIiNGKClGJUY5RihGKEYlRjA=

Another series starts with constant term 1 y2:='y2': Y2ser:=proc(n) add(y2[i]*t^i,i=0..n): end: n:=0: factor(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)), y=Y2ser(n), edQ),t,n+1));

KydJInRHNiIsJComKSZJI3kyR0YkNiMiIiEiIiMiIiIsJkYoRi1GLSEiIkYtRi9GKy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YtRiw=

y2[0]:=1;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTJHRig2IyIiISIiIjcjRjE=

From now on we execute the next two lines several times to obtain coefficients of Y2 n:=n+1: normal(series(normal(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)), y=Y2ser(n),edQ)),t,n+1));

KydJInRHNiIsJCZJI3kyR0YkNiMiIiUhIiJGKS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIiJw==

y2[n]:=solve(coeff(%,t,n),y2[n]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTJHRig2IyIiJSIiITcjRjE=

We recognize a very simple series Y2ser(n);

IiIi

Let us check that y=1 is indeed a solution of edQ n:=20: normal(series(normal(subs(Q=subs(x=0,y=1,Qser(n)), A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)), y=1, edQ)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0M=

edy:=factor(diff(eqcat,y)): Why should this be true? Let us eliminate A2 and A3 between the three equations res:=factor(resultant(eqcat,edy,A3));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyZXNHRigsZG8qKiIiJyIiIilJIlFHRigiIiRGMSlJInRHRigiIiVGMSlJInlHRihGMEYxISIiKixGN0YxSSNBMUdGKEYxKUYzIiIjRjFGNUYxKUY5IiImRjFGMSoqIiNDRjFGMkYxRjVGMUY/RjFGOiosIiM3RjFGPEYxRj1GMUY1RjEpRjlGN0YxRjEqKiIjT0YxRjJGMUY1RjFGRUYxRjoqLEZERjFGPEYxRj1GMUY1RjEpRjlGNEYxRjEqKkZCRjFGMkYxRjVGMUZJRjFGOiosRjdGMUY8RjFGPUYxRjVGMSlGOUY+RjFGMSoqRjBGMUYyRjFGNUYxRkxGMUY6KioiIilGMUY9RjEpRjZGPkYxRj9GMUYxKixGPkYxRjxGMUYzRjFGUEYxRkVGMUY6KixGN0YxSSNBMkdGKEYxRjNGMUZQRjFGRUYxRjEqKiIjPUYxRj1GMUZQRjFGRUYxRjEqKkY3RjFGM0YxRlBGMUY/RjFGOiosRjdGMUY8RjFGM0YxRlBGMUZJRjFGOioqRj5GMUY8RjFGUEYxRkVGMUYxKixGT0YxRlNGMUYzRjFGUEYxRklGMUYxKipGREYxRj1GMUZQRjFGSUYxRjEqKiIjOUYxRjNGMUZQRjFGRUYxRjoqLEY+RjFGPEYxRjNGMUZQRjFGTEYxRjoqKkY3RjFGPEYxRlBGMUZJRjFGMSosRjdGMUZTRjFGM0YxRlBGMUZMRjFGMSoqRj5GMUY9RjFGUEYxRkxGMUYxKioiIztGMUYzRjFGUEYxRklGMUY6KipGN0YxRjNGMUZQRjFGTEYxRjoqKEY+RjFGM0YxRkVGMUY6KipGN0YxRjxGMUZQRjFGOUYxRjoqKkY3RjFGM0YxRlBGMUY5RjFGMSooRjdGMUYzRjFGSUYxRjoqJkY+RjFGRUYxRjEqKEY+RjFGPEYxRlBGMUY6KihGN0YxRlNGMUZMRjFGOiooRj5GMUYzRjFGUEYxRjEqKEY+RjFGM0YxRkxGMUYxKiZGN0YxRklGMUYxKiZGPkYxRkxGMUYxNyNGLg==

factor(resultant(res,edQ,A2));

LCQqLiIiJSIiIkkieUc2IkYlLCZGJkYlRiUhIiJGJSwqKiZJInRHRidGJSlGJiIiI0YlRiUqKEYuRiVGLEYlRiZGJUYlRixGJUYmRiVGJSwqRitGJUYvRiVGLEYlRiZGKUYlLC4qKEkiUUdGJ0YlKUYsRi5GJUYtRiVGKSooSSNBMUdGJ0YlRjRGJUYmRiVGJSoqRi5GJUYzRiVGNEYlRiZGJUYpKiZGNkYlRjRGJUYlKiZGM0YlRjRGJUYpRiZGJUYlRik=

collect(-Q*t^2*y^2+A1*t^2*y-2*Q*t^2*y+A1*t^2-Q*t^2+y,[Q,A1],factor);

LCgqKClJInRHNiIiIiMiIiIpLCZJInlHRiZGKEYoRihGJ0YoSSJRR0YmRighIiIqKEYkRihGKkYoSSNBMUdGJkYoRihGK0Yo

This shows that the series Y that starts with CT=1 must be 1, and gives the minimal polynomial of the other two, provided we check that they do not cancel the last term : but since Y0 and Y1 have a first term of the form ct, this cannot happen. Y0ser(4);Y1ser(4);

LCoqJiIjOSIiIilJInRHNiIiIiVGJUYlKiYiIiZGJSlGJyIiJEYlRiUqJiIiI0YlKUYnRi9GJUYlRidGJQ==

LCoqJiIjOSIiIilJInRHNiIiIiVGJUYlKiYiIiZGJSlGJyIiJEYlISIiKiYiIiNGJSlGJ0YwRiVGJUYnRi4=

algY0:=t*y^2+2*t*y+t-y; algY1:=t*y^2+2*t*y+t+y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdZMEdGKCwqKiZJInRHRigiIiIpSSJ5R0YoIiIjRjFGMSooRjRGMUYwRjFGM0YxRjFGMEYxRjMhIiI3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdZMUdGKCwqKiZJInRHRigiIiIpSSJ5R0YoIiIjRjFGMSooRjRGMUYwRjFGM0YxRjFGMEYxRjNGMTcjRi4=

disc:=factor(discrim(eqcat,Q)): nops(disc);

IiIl

seq(nops(op(i,disc)),i=1..nops(disc));

NiYiIiMiJChHRiMiIiI=

op(1,disc); op(3,disc);op(4,disc);

KiQpLCZJInlHNiIiIiJGJ0YnIiIlRic=

KiQpSSJ0RzYiIiIlIiIi

SSJ5RzYi

discs:=op(2,disc): symmetry in y and 1/y factor(subs(y=1/y,discs)/discs);

KiQpSSJ5RzYiIiM3ISIi

degree(discs,y);

IiM3

Write discs as a polynomial in s:=y+1/y (times y^6) res:=discs/y^6: discsum:=0: while res<>0 do d:=degree(res,y): discsum:=s^d*coeff(res,y,d)+discsum: res:=expand(res-coeff(res,y,d)*(y+1/y)^d):od: collect(discsum,s,factor);

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

This vanishes at s=2 (double root at y=1), and has a root at s=1/t-2, and also at s=-1/t-2 factor(subs(s=2,discsum));

LCQqKCIjayIiIilJInRHNiIiIiNGJSxcbyoqIiIpRiUpSSNBMUdGKCIiJUYlSSNBMkdGKEYlKUYnRixGJUYlKigiIztGJUYtRiVGMUYlISIiKioiIz9GJSlGLiIiJEYlRjBGJSlGJyIiJ0YlRiUqKkYsRiVGN0YlSSNBM0dGKEYlRjlGJUY0KioiI0tGJSlGLkYpRiUpRjBGKUYlRjlGJUY0KihGPkYlRjdGJUY5RiVGNCoqIiRHIkYlRj9GJUYwRiVGOUYlRiUqKEZDRiVGP0YlRjlGJUY0KioiIz1GJUY/RiVGMEYlKUYnRi9GJUYlKioiIzdGJUY/RiVGPEYlRkdGJUY0KioiJC8iRiVGLkYlRkBGJUZHRiVGNCosIiRXIkYlRi5GJUYwRiVGPEYlRkdGJUYlKihGPkYlKUYwRjhGJUZHRiVGJSooIiNDRiVGP0YlRkdGJUY0KioiJHMjRiVGLkYlRjBGJUZHRiVGJSoqIiQpR0YlRi5GJUY8RiVGR0YlRjQqKCIkIz5GJUZARiVGR0YlRjQqKEZDRiVGLkYlRkdGJUY0KigiJCVRRiVGMEYlRkdGJUYlKioiIihGJUYuRiVGMEYlRiZGJUYlKipGOkYlRi5GJUY8RiVGJkYlRjQqKCIjckYlRkBGJUYmRiVGNCoqIiQhPUYlRjBGJUY8RiVGJkYlRiUqKCIkMyJGJSlGPEYpRiVGJkYlRjQqJiIkYyNGJUZHRiVGNCooRixGJUYuRiVGJkYlRjQqKEZLRiVGMEYlRiZGJUYlKihGTUYlRjxGJUYmRiVGNComRj5GJUYmRiVGNEYwRiVGPEY0RiVGNEYlRiU=

alg1:=op(3,%);

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

n:=10: normal(series(subs(A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)), alg1),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzc=

alg2:=numer(factor(subs(s=1/t-2,discsum)));

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

n:=10: normal(series(subs(A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)), alg2),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

alg3:=numer(factor(subs(s=-1/t-2,discsum)));

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

n:=10: normal(series(subs(A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)), alg3),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

map(degree,[alg1,alg2,alg3],A1); map(degree,[alg1,alg2,alg3],A2); map(degree,[alg1,alg2,alg3],A3);

NyUiIiVGI0Yj

NyUiIiRGI0Yj

NyUiIiNGI0Yj

We eliminate A3 by taking resultants res1:=op(3,factor(resultant(alg2,alg3,A3)));

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

res2:=op(4,factor(resultant(alg1,alg2,A3)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVyZXMyR0YoLGR0KigiIiUiIiIpSSNBMUdGKCIiKUYxKUkidEdGKCIjOUYxRjEqKCIjO0YxKUYzIiIoRjEpRjYiIzdGMUYxKioiI2tGMSlGMyIiJ0YxSSNBMkdGKEYxRjxGMUYxKihGP0YxRkBGMUY8RjEhIiIqKCIjS0YxRkBGMSlGNiIjNkYxRkQqKEZGRjFGQEYxKUY2IiM1RjFGMSoqIiQjPkYxKUYzIiImRjFGQkYxRkpGMUYxKioiJClHRjEpRjNGMEYxKUZCIiIjRjFGSkYxRjEqKEZNRjFGTkYxRkpGMUZEKioiJHcmRjFGUkYxRkJGMUZKRjFGRCooIiMnKkYxRk5GMSlGNiIiKkYxRkQqKkZRRjFGUkYxRkJGMUZaRjFGRCooIiQnKilGMUZSRjFGSkYxRkQqKCIjU0YxRk5GMSlGNkY0RjFGMSoqIiR3I0YxRlJGMUZCRjFGW29GMUYxKigiJXM5RjFGUkYxRlpGMUYxKipGV0YxKUYzIiIkRjFGU0YxRltvRjFGMSooIiRbJ0YxRlJGMUZbb0YxRkQqKiIlXzZGMUZhb0YxRkJGMUZbb0YxRkQqKEY/RjFGUkYxKUY2RjtGMUZEKipGV0YxRmFvRjFGQkYxRmhvRjFGRCooIiUjeiJGMUZhb0YxRltvRjFGRCoqIiVzSUYxKUYzRlRGMUZCRjFGW29GMUYxKigiI0hGMUZSRjEpRjZGQUYxRjEqKiIkSyNGMUZhb0YxRkJGMUZhcEYxRjEqKCIlV0hGMUZhb0YxRmhvRjFGMSoqIiRLJUYxRl5wRjFGU0YxRmFwRjFGMSoqRl1wRjFGXnBGMUZCRjFGaG9GMUZEKihGXXBGMUZecEYxRltvRjFGRCooIiVHPEYxKUZCRjBGMUZhcEYxRkQqKCIkdypGMUZhb0YxRmFwRjFGRCoqRlFGMUZecEYxRkJGMUZhcEYxRjEqKCIlTzpGMUZecEYxRmhvRjFGMSooIiU3cEYxKUZCRmJvRjFGYXBGMUYxKihGRkYxRmFvRjEpRjZGT0YxRjEqKiIkQydGMUZecEYxRkJGMUZmcUYxRkQqKCIkbyhGMUZecEYxRmFwRjFGRCoqRl1wRjFGM0YxRkJGMUZhcEYxRjEqKCIlY01GMUZkcUYxRmZxRjFGMSooIiY/OiJGMUZTRjFGYXBGMUZEKihGS0YxRmFvRjEpRjZGMEYxRjEqKiIkRSJGMUZecEYxRkJGMUZhckYxRjEqKCIlSztGMUZecEYxRmZxRjFGMSoqIiRXIkYxRjNGMUZTRjFGYXJGMUYxKipGXXBGMUYzRjFGQkYxRmZxRjFGRCooRl1wRjFGM0YxRmFwRjFGRCooRmdwRjFGZHFGMUZhckYxRkQqKCIlOyMqRjFGU0YxRmZxRjFGRCooRlxzRjFGQkYxRmFwRjFGMSooIiQ7JkYxRl5wRjFGYXJGMUZEKioiJGspRjFGM0YxRkJGMUZhckYxRjEqKEZhcUYxRjNGMUZmcUYxRjEqKEZbcUYxRlNGMUZhckYxRkQqKEZcc0YxRkJGMUZmcUYxRjEqJkZdcEYxRmFwRjFGRCooIiNPRjFGXnBGMSlGNkZib0YxRjEqKiIkTyRGMUYzRjFGQkYxRmhzRjFGRCooIiRHIkYxRjNGMUZhckYxRjEqKCIkPyhGMUZTRjFGaHNGMUYxKigiJUNVRjFGQkYxRmFyRjFGMSomRl1wRjFGZnFGMUZEKiZGXnBGMSlGNkZURjFGMSoqIiNVRjFGM0YxRkJGMUZjdEYxRjEqKCIkZyJGMUYzRjFGaHNGMUYxKigiI0ZGMUZTRjFGY3RGMUZEKihGanFGMUZCRjFGaHNGMUZEKiYiJSlvI0YxRmFyRjFGRCooIiNnRjFGM0YxRmN0RjFGRCooIiQzIkYxRkJGMUZjdEYxRkQqJkZNRjFGaHNGMUYxKihGMEYxRjNGMUY2RjFGMSooRj1GMUZCRjFGNkYxRkQqJiIkWyJGMUZjdEYxRjEqJkYwRjFGQkYxRjEqJkY0RjFGNkYxRjFGMEZENyNGLg==

res3:=factor(resultant(res1,res2,A2)):nops(%);

IiIm

map(indets,[seq(op(i,res3),i=1..nops(res3))]);

Nyc8IjwjSSJ0RzYiRiQ8JEkjQTFHRiZGJUYn

nops(op(4,res3));nops(op(5,res3));

IiNB

IiRGIg==

n:=10: normal(series(subs(A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)),op(4,res3)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzc=

n:=10: normal(series(subs(A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)),op(5,res3)),t,n+1));

KztJInRHNiIiIikiIiEhJGMlIiIiIiZfWiIiIiMhJ29uPyIiJCIoX05uIiIiJSEoTyJRJSoiIiYiKWs3UlMiIichKjsrek4iIiIoIipHYmcjUkYlIStfUDEjNCIiIioiKyM+N3MhSCIjNS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiM2

The algebraic equation for A1=Q(0,0) has degree 8 algA1:=collect(op(4,res3),A1,factor);

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

Now A2 res4:=factor(resultant(res1,res2,A1)):nops(%);

IiIm

map(indets,[seq(op(i,res4),i=1..nops(res4))]);

Nyc8IjwjSSJ0RzYiRiQ8JEkjQTJHRiZGJUYn

nops(op(4,res4));nops(op(5,res4));

IiIj

IiIj

nops(op(1,op(4,res4)));nops(op(1,op(5,res4)));

IiM5

IiNz

n:=10: normal(series(subs(A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)),op(1,op(4,res4))),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzc=

normal(series(subs(A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)),op(1,op(5,res4))),t,n+1));

KztJInRHNiIiIzciIiEhJTs2IiIiIiZnJFIiIiMhJyc0MikiIiQiKS9IIjQiIiIlISovbysuIiIiJiIqIUdkYnAiIichK1c8UCpRJCIiKCIsKywoSCw3IiIpISxjKkcnbzwkIiIqIixnaE4mSG4iIzUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIjNg==

Degree 4 only for A2=Q(0,-1) algA2:=collect(op(1,op(4,res4)),A2,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdBMkdGKCwyKigiJEgoIiIiKUkjQTJHRigiIiVGMSlJInRHRigiIidGMUYxKioiJFYjRjEpRjZGNEYxLCYqJiIiKUYxKUY2IiIjRjFGMUYxISIiRjEpRjMiIiRGMUZAKioiI0ZGMUY+RjEsKComIiNTRjFGOkYxRjEqJiIjWUYxRj5GMUZARjFGMUYxKUYzRj9GMUYxKiosJiomIiNjRjFGPkYxRjFGMUZARjEsKComRj9GMUY+RjFGMSomRjdGMUY2RjFGMUYxRkBGMSwoRlBGMUZRRkBGMUZARjFGM0YxRkAqJiIjO0YxRjVGMUYxKiYiJTM3RjFGOkYxRkAqJiIjb0YxRj5GMUYxRjFGQDcjRi4=

Finally A3 res1:=factor(resultant(algA1,alg1,A1)): nops(%);;

IiIj

op(1,res1);

KiQpSSJ0RzYiIiNjIiIi

res2:=factor(resultant(res1,algA2,A2)): nops(res2);

IiIm

map(indets,[seq(op(i,res2),i=1..nops(res2))]);

Nyc8IjwjSSJ0RzYiPCRJI0EzR0YmRiVGJ0Yn

nops(op(3,res2));nops(op(4,res2));nops(op(5,res2));

IiNj

IiNo

IiVyPA==

n:=10: normal(series(subs(A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)),op(3,res2)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzc=

normal(series(subs(A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)),op(4,res2)),t,n+1));

KzFJInRHNiIhK3RKK3RTIiIhIi1TbWV0TUsiIiMiLm9ONz1obiciIiUhMTtjZGBbQ0c+IiInIjInXD1FSDtDdyEpIiIpITRnOGdWMjlSUkkiIiM1LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiM3

n:=4:normal(series(subs(A1=subs(x=0,y=0,Qser(n)), A2=subs(y=-1,x=0,Qser(n)), A3=subs(y=-1,diff(subs(x=0,Qser(n)),y)),op(5,res2)),t,n+1));

KytJInRHNiIiTCUpPl8rVDclXCkqZSs5ZDVGY1FyR0ZBVyIiISFOJil5ZjAlUixxdFUpPl8kb2sqejd5VCdlIz0jIiIjIlJhJ1skXE9PZVM7LiZ6Lmt5cG1XW1koZmBPXyIiIiUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIic=

degree 8 again for A3=Q_y(0,-1) algA3:=collect(op(3,res2),A3,factor);

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

The usual parametrization algT:=(T-1)*(T+3)^3-256*t^2*T^3;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdUR0YoLCYqJiwmSSJUR0YoIiIiRjIhIiJGMiksJkYxRjIiIiRGMkY2RjJGMiooIiRjI0YyKUkidEdGKCIiI0YyKUYxRjZGMkYzNyNGLg==

tT:=op(1,[solve(algT,t)]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN0VEdGKCwkKiojIiIiIiM7RjEpSSJUR0YoIiIjISIiKSomRjRGMSwoKiRGM0YxRjEqJkY1RjFGNEYxRjEiIiRGNkYxI0YxRjVGMSwmRjRGMUY8RjFGMUYxNyNGLg==

Parametrization for A2 numer(factor(subs(t=tT,algA2)));

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

subs(T=1,op(2,%));

IidXQEU=

A2T:=factor(solve(op(1,%%),A2));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBMlRHRigsJCooIiIjIiIiLCoqJClJIlRHRigiIiRGMUYxKiQpRjVGMEYxRjEqJiIjRkYxRjVGMUYxRjZGMUYxKSwmRjVGMUY2RjFGNiEiIkYxNyNGLg==

numer(factor(subs(t=tT,algA1)));

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

subs(T=1,op(2,%));

Ii4vNl5ZISlSJQ==

Now A1 A1T:=simplify(factor([solve(op(1,%%),A1)]),symbolic);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBMVRHRig3JCwkKiwiI0siIiIsKiokKUkiVEdGKCMiIiYiIiNGMkYyKiYiIidGMilGNiMiIiRGOUYyRjIqJiIiJUYyKUY2RjlGMiEiIiomRj5GMilGNiNGMkY5RjJGQkYyRjZGMiwmRjZGMkYyRkJGQiksJkY2RjJGPkYyRj5GQkYyLCQqLEYxRjIsKkY0RjJGOkYyRj9GMkZDRkJGMkY2RjJGRkZCRkdGQkZCNyNGLg==

factor(simplify(subs(T=Z^2,A1T),symbolic));

NyQsJCosIiNLIiIiLCoqJClJIlpHNiIiIiRGJkYmKiZGLEYmKUYqIiIjRiYhIiIqJkYsRiZGKkYmRiZGLEYmRiZGKUYmKSwmKiRGLkYmRiZGLEYmRixGMCwmRipGJkYmRiZGMEYmLCQqLEYlRiYsKkYoRiZGLUYmRjFGJkYsRjBGJkYpRiZGMkYwLCZGKkYmRiZGMEYwRjA=

Recall that Z has constant term 1 to choose the correct value A1Z:=op(1,%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBMVpHRigsJCosIiNLIiIiLCoqJClJIlpHRigiIiRGMUYxKiZGNkYxKUY1IiIjRjEhIiIqJkY2RjFGNUYxRjFGNkYxRjFGNEYxKSwmKiRGOEYxRjFGNkYxRjZGOiwmRjVGMUYxRjFGOkYxNyNGLg==

Finally, A3 numer(factor(subs(t=tT,algA3))): nops(%);

IiIj

subs(T=1,op(2,%%));

IjI7UyxcYiwpZjo=

A3T:=simplify(factor([solve(op(1,%%%),A3)]),symbolic);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBM1RHRig3JCooLDQqJiIiIyIiIilJIlRHRigjIiIqRjJGM0YzKiQpRjUiIiZGM0YzKiYiIilGMylGNSMiIihGMkYzRjMqJClGNSIiJUYzRjMqJiIjN0YzKUY1I0Y6RjJGM0YzKiYiI0ZGMylGNSIiJEYzRjMqJkY8RjMpRjUjRkpGMkYzRjMqJkZKRjMpRjVGMkYzRjMqJkYyRjMpRjUjRjNGMkYzRjNGM0ZPISIiKSwmRjVGM0ZKRjNGSkZTKigsNEYxRlNGOEYzRjtGU0ZARjNGQ0ZTRkdGM0ZLRlNGTkYzRlBGU0YzRk9GU0ZURlM3I0Yu

A3sol:=factor(simplify(subs(T=Z^2,A3T),symbolic));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZBM3NvbEdGKDckKiosJkkiWkdGKCIiIkYyRjJGMiwyKiQpRjEiIilGMkYyKiQpRjEiIihGMkYyKiZGNkYyKUYxIiImRjJGMiomIiM+RjIpRjEiIiVGMkYyKiZGOUYyKUYxIiIkRjIhIiIqJiIjNUYyKUYxIiIjRjJGMiomRkhGMkYxRjJGREZIRjJGMkZCRkQpLCYqJEZHRjJGMkZDRjJGQ0ZEKiosJkYxRjJGMkZERjIsMkY0RjJGN0ZERjpGREY9RjJGQUYyRkVGMkZJRjJGSEYyRjJGQkZERkpGRDcjRi4=

subs(Z=1,%);

NyQiIiIiIiE=

We want no constant term in A3 n:=4:subs(y=-1,diff(subs(x=0,Qser(n)),y));

LCYqJiIiJSIiIilJInRHNiJGJEYlRiUqJClGJyIiI0YlRiU=

A3Z:=op(2,A3sol);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBM1pHRigqKiwmSSJaR0YoIiIiRjEhIiJGMSwyKiQpRjAiIilGMUYxKiQpRjAiIihGMUYyKiZGNkYxKUYwIiImRjFGMiomIiM+RjEpRjAiIiVGMUYxKiZGOUYxKUYwIiIkRjFGMSomIiM1RjEpRjAiIiNGMUYxKiZGR0YxRjBGMUYxRkdGMUYxRkJGMiksJiokRkZGMUYxRkNGMUZDRjI3I0Yu

tT;

LCQqKiMiIiIiIztGJSlJIlRHNiIiIiMhIiIpKiZGKEYlLCgqJEYnRiVGJSomRipGJUYoRiVGJSIiJEYrRiUjRiVGKkYlLCZGKEYlRjFGJUYlRiU=

eqcatQZ:=collect(numer(factor(subs(A1=A1Z,A2=A2T,A3=A3Z,t=tT,T=Z^2,eqcat))),Q,factor);nops(factor(%));

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

IiQ/Iw==

Parametrization of y by a series V with constant term 0 algV:=1 - T + 3*V + V* T - y*V^2 *(3 + V + T - V *T );

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdWR0YoLCwiIiJGL0kiVEdGKCEiIiomIiIkRi9JIlZHRihGL0YvKiZGNEYvRjBGL0YvKihJInlHRihGLylGNCIiI0YvLCoqJkYwRi9GNEYvRjFGMEYvRjRGL0YzRi9GL0YxNyNGLg==

yV:=solve(%,y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN5VkdGKCwkKigsKiomSSJUR0YoIiIiSSJWR0YoRjNGM0YyISIiKiYiIiRGM0Y0RjNGM0YzRjNGMylGNCIiI0Y1LCpGMUYzRjJGNUY0RjVGN0Y1RjVGNTcjRi4=

solve(numer(yV),V);

KiYsJkkiVEc2IiIiIkYmISIiRiYsJkYkRiYiIiRGJkYn

res:=numer(factor(subs(y=yV,T=Z^2,eqcatQZ))): nops(res);

IiIj

We determine the correct factor by setting y=0 factor(subs(V=(T-1)/(T+3),T=Z^2,op(1,res)));factor(subs(V=(T-1)/(T+3),T=Z^2,op(2,res)));

LCQqLiIjOyIiIiwmSSJaRzYiRiVGJSEiIkYlKSwmKiQpRiciIiNGJUYlRiVGJUYuRiUpLCZGJ0YlRiVGJSIiJUYlLDoqJkkiUUdGKEYlKUYnIiIoRiVGJSomRjRGJSlGJyIiJ0YlRiUqKCIiKkYlRjRGJSlGJyIiJkYlRiUqJiIjS0YlRjhGJUYpKihGO0YlRjRGJSlGJ0YxRiVGJSomIiMnKkYlRjxGJUYlKigiI0ZGJUY0RiUpRiciIiRGJUYlKiZGQ0YlRkFGJUYpKihGRUYlRjRGJUYtRiVGJSomRkNGJUZGRiVGKSooRkVGJUY0RiVGJ0YlRiUqJkZFRiVGNEYlRiVGJSksJkYsRiVGR0YlRjFGKUYl

LCQqLiInKUdDJiIiIilJIlpHNiIiIidGJSksJiokKUYnIiIjRiVGJUYlRiUiIiRGJSksJkYnRiVGJUYlRilGJSksJkYnRiVGJSEiIiIjNUYlKSwmRixGJUYvRiUiIilGNEYl

collect(op(1,res),Q,factor);

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

factor(solve(%,Q));

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

Let us check that this coincides with the known solution (e.g. MBM, Adv. Math. Thm 1) collect(op(5,numer(%)),V,factor);

LCoqKCksJkkiWkc2IiIiIkYoISIiIiIjRigsJiokKUYmRipGKEYoIiIkRihGKClJIlZHRidGLkYoRigqKkYlRigsJkYmRihGKEYoRigsKEYsRigqJkYqRihGJkYoRigiIihGKUYoKUYwRipGKEYoKipGJUYoRjJGKCwoRixGKEY0RilGNUYpRihGMEYoRigqJkYrRigpRjJGKkYoRig=

Back to Q(x,0), using R(x)+S(Y)-R(0)=xy if (x,y) cancels the kernel eqxy:=subs(Q(x,y)=0,Q(0,0)=Q00,Q(x,0)=Qx,Q(0,y)=Qy,eqQ);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcXh5R0YoLCoqJkkieEdGKCIiIkkieUdGKEYxRjEqKCwmRjJGMUYxRjFGMUkidEdGKEYxSSNReUdGKEYxISIiKiZGNUYxSSNReEdGKEYxRjcqJkY1RjFJJFEwMEdGKEYxRjE3I0Yu

The value of Q(0,Y0) solve(eqxy,Qy);

KigsKComSSRRMDBHNiIiIiJJInRHRiZGJ0YnKiZJI1F4R0YmRidGKEYnISIiKiZJInhHRiZGJ0kieUdGJkYnRidGJywmRi5GJ0YnRidGK0YoRis=

We plug this in the algebraic equation satisfied by Q(0,y) res1:=numer(factor(subs(Q=%,Q00=A1Z, eqcatQZ))): nops(%);

IiRwJw==

indets(res1);

PCdJI1F4RzYiSSJaR0YkSSJ0R0YkSSJ4R0YkSSJ5R0Yk

Eliminate y (=Y0) res2:=factor(resultant(res1,K,y)): nops(res2);

IiIk

map(indets,[seq(op(i,res2),i=1..nops(res2))]);

NyU8I0kidEc2IjwjSSJ4R0YlPCZJI1F4R0YlSSJaR0YlRiRGJw==

nops(op(3,res2));

IiVHYw==

res3:=op(3,res2): degree(res3,Qx);

IiIn

We now consider Q(tx,0), which is an even series in t res4:=numer(factor(subs(x=t*x,t=tT,T=Z^2,Qx=Qxt,res3))): nops(res4);

IiIm

map(indets,[seq(op(i,res4),i=1..nops(res4))]);

Nyc8I0kiWkc2IkYjRiM8I0kieEdGJTwlSSRReHRHRiVGJEYn

nops(op(5,res4));

IiIj

algQxt:=collect(op(1,op(5,res4)),Qxt,factor);

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

Parametrization for x in terms of a series U with constant term 1 algU:=16*T ^ 2 *(U^ 2 - T ) - x*(U + U* T - 2*T )*(U^ 2 - 9*T + 8*T* U + T^ 2 - T* U^ 2 );

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdVR0YoLCYqKCIjOyIiIilJIlRHRigiIiNGMSwmKiQpSSJVR0YoRjRGMUYxRjMhIiJGMUYxKihJInhHRihGMSwoKiZGM0YxRjhGMUYxKiZGNEYxRjNGMUY5RjhGMUYxLCwqJkYzRjFGN0YxRjkqJEYyRjFGMSooIiIpRjFGM0YxRjhGMUYxRjZGMSomIiIqRjFGM0YxRjlGMUY5NyNGLg==

xU:=solve(%,x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN4VUdGKCwkKiwiIzsiIiIpSSJUR0YoIiIjRjEsJiokKUkiVUdGKEY0RjEhIiJGM0YxRjEsKComRjNGMUY4RjFGMSomRjRGMUYzRjFGOUY4RjFGOSwsKiZGM0YxRjdGMUY5KiRGMkYxRjEqKCIiKUYxRjNGMUY4RjFGMUY2RjEqJiIiKkYxRjNGMUY5RjlGOTcjRi4=

res:=numer(factor(subs(x=xU,T=Z^2,algQxt))): nops(res);indets(res);

IiIl

PCVJJFF4dEc2IkkiVUdGJEkiWkdGJA==

op(1,res);op(2,res);

IiRjIw==

KiQpSSJaRzYiIiIpIiIi

nops(op(3,res));nops(op(4,res));

IiNo

IiRYJQ==

We choose the correct factor by looking at the case x=0 factor(subs(U=Z,op(3,res)));factor(subs(U=Z,op(4,res)));

LCQqKiIjOyIiIilJIlpHNiIiIiRGJSksJkYnRiVGJSEiIiIiI0YlLDoqJkkkUXh0R0YoRiUpRiciIihGJUYlKiZGMEYlKUYnIiInRiVGJSooIiIqRiVGMEYlKUYnIiImRiVGJSomIiNLRiVGNEYlRiwqKEY3RiVGMEYlKUYnIiIlRiVGJSomIiMnKkYlRjhGJUYlKigiI0ZGJUYwRiVGJkYlRiUqJkZARiVGPUYlRiwqKEZCRiVGMEYlKUYnRi1GJUYlKiZGQEYlRiZGJUYsKihGQkYlRjBGJUYnRiVGJSomRkJGJUYwRiVGJUYlRiw=

LCQqKCInczU4IiIiKUkiWkc2IiIjOEYlKSwmRidGJUYlISIiIiM1RiVGLA==

collect(op(3,res),Qxt,factor);

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

factor(solve(%,Qxt));

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

collect(op(4,numer(%)),U,factor);

LCoqKCksJkkiWkc2IiIiIkYoISIiIiIjRigpLCZGJkYoRihGKEYqRigpSSJVR0YnIiIkRihGKCosRiZGKEYlRihGLEYoLCgqJClGJkYqRihGKComIiM7RihGJkYoRilGKEYpRigpRi5GKkYoRigqKEYzRigsLCokKUYmIiIlRihGKComRjVGKClGJkYvRihGKComIiMjKUYoRjNGKEYpRjRGKSIjPEYoRihGLkYoRikqJkY9RigsKkY5RigqJiIjPUYoRjNGKEYpKiYiJEciRihGJkYoRigiIyIpRihGKEYp

This is the result given in [MBM, Adv. Math]

liste:=x*y+1/x+lambda/y+2*x+x/y+1/x/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwuKiZJInhHRigiIiJJInlHRihGMUYxKiRGMCEiIkYxKiZJJ2xhbWJkYUdGKEYxRjJGNEYxKiYiIiNGMUYwRjFGMSomRjBGMUYyRjRGMSomRjBGNEYyRjRGMTcjRi4=

K:=collect((t*liste-1)*x*y,t,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCYqJiwuKiYpSSJ4R0YoIiIjIiIiKUkieUdGKEY0RjVGNSooRjRGNUYyRjVGN0Y1RjUqJkknbGFtYmRhR0YoRjVGM0Y1RjUqJEYyRjVGNUY3RjVGNUY1RjVJInRHRihGNUY1KiZGM0Y1RjdGNSEiIjcjRi4=

The functional equation eqQ:=K*Q(x,y)+x*y-subs(x=0,K)*Q(0,y) -subs(y=0,K)*Q(x,0)+subs(x=0,y=0,K)*Q(0,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcVFHRigsLComLCYqJiwuKiYpSSJ4R0YoIiIjIiIiKUkieUdGKEY2RjdGNyooRjZGN0Y0RjdGOUY3RjcqJkknbGFtYmRhR0YoRjdGNUY3RjcqJEY0RjdGN0Y5RjdGN0Y3RjdJInRHRihGN0Y3KiZGNUY3RjlGNyEiIkY3LUkiUUdGKDYkRjVGOUY3RjdGP0Y3KigsJkY5RjdGN0Y3RjdGPkY3LUZCNiQiIiFGOUY3RkAqKCwoRjtGN0Y9RjdGN0Y3RjdGPkY3LUZCNiRGNUZIRjdGQComRj5GNy1GQjYkRkhGSEY3Rjc3I0Yu

Qser:=proc(n) option remember: if n=0 then 1 else convert(normal(series(subs(Q(x,y)=Qser(n-1),Q(x,0)=subs(y=0,Qser(n-1)), Q(0,y)=subs(x=0,Qser(n-1)), Q(0,0)=subs(x=0,y=0,Qser(n-1)), expand(eqQ/x/y+Q(x,y))),t,n+1)),polynom): fi: end: Qser(4);

LCwiIiJGIyomLCYqJkkieEc2IkYjSSJ5R0YoRiNGIyomIiIjRiNGJ0YjRiNGI0kidEdGKEYjRiMqJiwuKiYpRidGK0YjKUYpRitGI0YjKigiIiVGI0YwRiNGKUYjRiMqJkknbGFtYmRhR0YoRiNGJ0YjRiMqJiIiJkYjRjBGI0YjRilGIyIiJEYjRiMpRixGK0YjRiMqJiw2KiYpRidGOEYjKUYpRjhGI0YjKigiIidGI0Y9RiNGMUYjRiMqKkYrRiNGNUYjRjBGI0YpRiNGIyooIiM5RiNGPUYjRilGI0YjKihGQEYjRjVGI0YwRiNGIyomRkNGI0Y9RiNGIyooRitGI0YnRiNGMUYjRiMqKCIjNUYjRidGI0YpRiNGIyomRitGI0Y1RiNGIyomIiM7RiNGJ0YjRiNGIylGLEY4RiNGIyomLEYqJilGJ0YzRiMpRilGM0YjRiMqKCIiKUYjRlBGI0Y+RiNGIyoqRjhGI0Y1RiNGPUYjRjFGI0YjKigiI0ZGI0ZQRiNGMUYjRiMqKkZLRiNGNUYjRj1GI0YpRiNGIyooIiNbRiNGUEYjRilGI0YjKihGOEYjRjBGI0Y+RiNGIyooRitGIylGNUYrRiNGMEYjRiMqKCIjR0YjRjVGI0Y9RiNGIyomIiNVRiNGUEYjRiMqKCIjQEYjRjBGI0YxRiNGIyoqRkBGI0Y1RiNGJ0YjRilGI0YjKigiI2VGI0YwRiNGKUYjRiMqKCIjQUYjRjVGI0YnRiNGIyomIiNxRiNGMEYjRiMqJkYrRiNGMUYjRiMqJiIjN0YjRilGI0YjIiNFRiNGIylGLEYzRiNGIw==

Rser:=proc(n) convert(normal(series(subs(y=0,K)*subs(y=0,Qser(n-1)),t,n+1)),polynom):end: Sser:=proc(n) convert(normal(series(subs(x=0,K)*subs(x=0,Qser(n-1)),t,n+1)),polynom):end: Rser(4);Sser(4);

LCoqJiwoKiZJJ2xhbWJkYUc2IiIiIkkieEdGJ0YoRigqJClGKSIiI0YoRihGKEYoRihJInRHRidGKEYoKipGLEYoRiRGKEYpRigpRi1GLEYoRigqKEYkRigsKEYlRigqJiIiJkYoRitGKEYoIiIkRihGKClGLUY0RihGKCoqRixGKEYkRigsKiooRjRGKEYmRihGK0YoRigqJiIiKEYoKUYpRjRGKEYoRiZGKComIiIpRihGKUYoRihGKClGLSIiJUYoRig=

LCgqJiwmSSJ5RzYiIiIiRidGJ0YnSSJ0R0YmRidGJyooRiRGJywmRiVGJyIiJEYnRicpRihGK0YnRicqKiIiI0YnRiRGJ0knbGFtYmRhR0YmRicpRigiIiVGJ0Yn

Checking the functional equation n:=10: normal(series(Qser(n)*K+x*y-Rser(n)-Sser(n)+subs(x=0,Rser(n)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Invariants have been constructed in the Maple session "invariants-syst" Jy:=t^2*y+(1+lambda*t)/(1+y)-(1+lambda*t)^2/(1+y)^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKeUdGKCwoKiYpSSJ0R0YoIiIjIiIiSSJ5R0YoRjNGMyomLCYqJkknbGFtYmRhR0YoRjNGMUYzRjNGM0YzRjMsJkY0RjNGM0YzISIiRjMqJilGNkYyRjMpRjlGMkY6Rjo3I0Yu

Gy:=-(1+lambda*t)/t/(1+y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKigsJiomSSdsYW1iZGFHRigiIiJJInRHRihGM0YzRjNGM0YzRjQhIiIsJkkieUdGKEYzRjNGM0Y1RjU3I0Yu

Jy; Ly:=S(y)-Gy;

LCgqJilJInRHNiIiIiMiIiJJInlHRiZGKEYoKiYsJiomSSdsYW1iZGFHRiZGKEYlRihGKEYoRihGKCwmRilGKEYoRighIiJGKComKUYrRidGKClGLkYnRi9GLw==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNMeUdGKCwmLUkiU0dGKDYjSSJ5R0YoIiIiKigsJiomSSdsYW1iZGFHRihGM0kidEdGKEYzRjNGM0YzRjNGOCEiIiwmRjJGM0YzRjNGOUYzNyNGLg==

We construct an expression with no pole at y=-1. eqcat:=Jy-a*Ly^2-b*Ly-c;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZlcWNhdEdGKCwuKiYpLCYtSSJTR0YoNiNJInlHRigiIiIqKCwmKiZJJ2xhbWJkYUdGKEY2SSJ0R0YoRjZGNkY2RjZGNkY7ISIiLCZGNUY2RjZGNkY8RjYiIiNGNkkiYUdGKEY2RjwqJkYxRjZJImJHRihGNkY8KiYpRjtGPkY2RjVGNkY2KiZGOEY2Rj1GPEY2KiYpRjhGPkY2KUY9Rj5GPEY8SSJjR0YoRjw3I0Yu

normal(series(eqcat,y=-1,0));

KycsJkkieUc2IiIiIkYmRiYsJCooKSwmKiZJJ2xhbWJkYUdGJUYmSSJ0R0YlRiZGJkYmRiYiIiNGJiwmKiQpRi1GLkYmRiZJImFHRiVGJkYmRjEhIiJGMyEiIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YmRjM=

asol:=solve(op(1,%),a);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhc29sR0YoLCQqJClJInRHRigiIiMiIiIhIiI3I0Yu

normal(series(subs(a=asol,eqcat),y=-1,1));

KycsJkkieUc2IiIiIkYmRiYqJiwuKioiIiNGJi1JIlNHRiU2IyEiIkYmSSdsYW1iZGFHRiVGJilJInRHRiUiIiRGJkYmKihGKkYmKUYxRipGJkYrRiZGJiooSSJiR0YlRiZGL0YmRjFGJkYuKiZGL0YmRjRGJkYmRjZGLkYxRiZGJkYxRi5GLi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YmIiIh

bsol:=solve(op(1,%),b);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVic29sR0YoKiYsJiooIiIjIiIiLUkiU0dGKDYjISIiRjJJInRHRihGMkYyRjJGMkYyRjdGMjcjRi4=

Note that S(y) is a multiple of (1+y), and hence vanishes at y=-1 subs(x=0,K);

KiYsJkkieUc2IiIiIkYmRiZGJkkidEdGJUYm

bsol:=subs(S(-1)=0,bsol);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVic29sR0YoSSJ0R0YoNyNGLg==

normal(subs(S(-1)=0,series(subs(a=asol,b=bsol,eqcat),y=-1,2)));

KycsJkkieUc2IiIiIkYmRiYsKioqIiIjRiYtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSSJTR0YlNiMhIiJGJkknbGFtYmRhR0YlRiYpSSJ0R0YlRilGJkYmKihGKUYmRipGJkY2RiZGJiokRjVGJkYzSSJjR0YlRjMiIiEtSSJPR0YtNiNGJkYm

csol:=solve(coeff(%,y+1,0),c);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVjc29sR0YoLCgqKiIiIyIiIi0tSSJER0YlNiNJIlNHRig2IyEiIkYxSSdsYW1iZGFHRihGMSlJInRHRihGMEYxRjEqKEYwRjFGMkYxRjtGMUYxKiRGOkYxRjg3I0Yu

collect(%,S,factor);

LCYqKiIiIyIiIiwmKiZJJ2xhbWJkYUc2IkYlSSJ0R0YpRiVGJUYlRiVGJUYqRiUtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKTYjSSJTR0YpNiMhIiJGJUYlKiQpRipGJEYlRjQ=

eqcat1:=numer(factor(subs(a=asol,b=bsol,c=csol,eqcat)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdlcWNhdDFHRigqJiw6KigpLUkiU0dGKDYjSSJ5R0YoIiIjIiIiSSJ0R0YoRjdGNUY3RjcqLEY2RjctLUkiREdGJTYjRjM2IyEiIkY3SSdsYW1iZGFHRihGN0Y4RjdGNUY3Rj8qJkYxRjdGOEY3RjcqKkY2RjdGMkY3RkBGN0Y4RjdGNyoqRjZGN0Y6RjdGQEY3RjhGN0Y/KiZGOEY3KUY1RjZGN0Y3KiZGMkY3RjVGN0Y/KihGNkY3RjpGN0Y1RjdGPyooRjZGN0Y4RjdGNUY3RjdGMkY3KiZGNkY3RjpGN0Y/RjhGN0Y3RjhGNzcjRi4=

eqcat1:=eqcat1/t;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdlcWNhdDFHRigsOiooKS1JIlNHRig2I0kieUdGKCIiIyIiIkkidEdGKEY2RjRGNkY2KixGNUY2LS1JIkRHRiU2I0YyNiMhIiJGNkknbGFtYmRhR0YoRjZGN0Y2RjRGNkY+KiZGMEY2RjdGNkY2KipGNUY2RjFGNkY/RjZGN0Y2RjYqKkY1RjZGOUY2Rj9GNkY3RjZGPiomRjdGNilGNEY1RjZGNiomRjFGNkY0RjZGPiooRjVGNkY5RjZGNEY2Rj4qKEY1RjZGN0Y2RjRGNkY2RjFGNiomRjVGNkY5RjZGPkY3RjY3I0Yu

Can this be true ? n:=30: normal(series(subs(S(y)=Sser(n), D(S)(-1)=subs(y=-1,diff(Sser(n),y)),eqcat1),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0o=

Baxk to Q(0,y) (denoted here Q(y) to lighten notation) Sy:=subs(x=0,K)*Q(y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNTeUdGKCooLCZJInlHRigiIiJGMUYxRjFJInRHRihGMS1JIlFHRig2I0YwRjE3I0Yu

factor(subs(S(y)=Sy,(D(S))(-1)=limit(diff(Sy,y),y=-1),eqcat1));

KihJInRHNiIiIiIsJkkieUdGJEYlRiVGJUYlLDYqKCktSSJRR0YkNiNGJyIiI0YlKUYjRi5GJSlGJ0YuRiVGJSoqRi5GJUYqRiVGL0YlRidGJUYlKiZGKkYlRi9GJUYlKipGLkYlRitGJUknbGFtYmRhR0YkRiVGI0YlRiUqKkYuRiUtRiw2IyEiIkYlRjRGJUYjRiVGOComRitGJUYnRiVGOEYrRiUqJkYuRiVGNkYlRjhGJ0YlRiVGJUYl

eqcatQ:=collect(-op(3,%),Q,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdlcWNhdFFHRigsLComLCYqKCIiIyIiIkknbGFtYmRhR0YoRjNJInRHRihGM0YzRjJGM0YzLUkiUUdGKDYjISIiRjNGMyooKSwmSSJ5R0YoRjNGM0YzRjJGMylGNUYyRjMpLUY3NiNGPUYyRjNGOSomLChGMUY5Rj1GM0YzRjlGM0ZARjNGM0Y9RjlGM0Y5NyNGLg==

eqcatQ;

LCwqJiwmKigiIiMiIiJJJ2xhbWJkYUc2IkYnSSJ0R0YpRidGJ0YmRidGJy1JIlFHRik2IyEiIkYnRicqKCksJkkieUdGKUYnRidGJ0YmRicpRipGJkYnKS1GLDYjRjJGJkYnRi4qJiwoRiVGLkYyRidGJ0YuRidGNUYnRidGMkYuRidGLg==

eqcat:=subs(Q(y)=Q,Q(-1)=A1,eqcatQ);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZlcWNhdEdGKCwsKiYsJiooIiIjIiIiSSdsYW1iZGFHRihGM0kidEdGKEYzRjNGMkYzRjNJI0ExR0YoRjNGMyooKSwmSSJ5R0YoRjNGM0YzRjJGMylGNUYyRjMpSSJRR0YoRjJGMyEiIiomLChGMUY+RjpGM0YzRj5GM0Y9RjNGM0Y6Rj5GM0Y+NyNGLg==

One series cancels this derivative edQ:=factor(diff(eqcat,Q));subs(t=0,%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlZFFHRigsLioqIiIjIiIiSSJRR0YoRjEpSSJ0R0YoRjBGMSlJInlHRihGMEYxISIiKioiIiVGMUYyRjFGM0YxRjZGMUY3KihGMEYxRjJGMUYzRjFGNyooRjBGMUknbGFtYmRhR0YoRjFGNEYxRjdGNkYxRjFGNzcjRi4=

LCZJInlHNiIiIiJGJSEiIg==

Let us compute its expansion y0:='y0': Y0ser:=proc(n) add(y0[i]*t^i,i=0..n): end: n:=-1: From now on we execute the next two lines several times to obtain coefficients of Y0 n:=n+1: normal(series(normal(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=-1,Qser(n)), y=Y0ser(n),edQ)),t,n+1));

KydJInRHNiIsKComIiIpIiIiKUknbGFtYmRhR0YkIiIjRighIiImSSN5MEdGJDYjIiIlRigiIycqRixGMC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIm

y0[n]:=solve(coeff(%,t,n),y0[n]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTBHRig2IyIiJSwmKiYiIikiIiIpSSdsYW1iZGFHRigiIiNGNEY0IiMnKkY0NyNGMQ==

Y0ser(n);

LCwiIiJGIyooIiIjRiNJJ2xhbWJkYUc2IkYjSSJ0R0YnRiNGIyomIiIpRiMpRihGJUYjRiMqKCIjO0YjRiZGIylGKCIiJEYjRiMqJiwmKiZGKkYjKUYmRiVGI0YjIiMnKkYjRiMpRigiIiVGI0Yj

disc:=collect(factor(discrim(eqcat,Q)),y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkaXNjR0YoLDQqKCIiJSIiIilJInRHRigiIiNGMSlJInlHRigiIiRGMSEiIiomLCoqKiIiKUYxSSNBMUdGKEYxSSdsYW1iZGFHRihGMSlGM0Y3RjFGMSooRjxGMUY9RjFGMkYxRjEqJiIjN0YxRjJGMUY4RjFGMUYxKUY2RjRGMUYxKiYsLCoqIiM7RjFGPUYxRj5GMUY/RjFGMSooRkdGMUY9RjFGMkYxRjEqKEYwRjFGPkYxRjNGMUY4RkFGOEY0RjhGMUY2RjFGMUY7RjEqKEYwRjEpRj5GNEYxRjJGMUYxRkBGMUZJRjEqJkYwRjFGMkYxRjhGMUYxNyNGLg==

The discriminant of disc (with respect to y) vanishes res:=factor(discrim(disc,y));nops(res);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyZXNHRigsJCoqIiRHIiIiIilJInRHRigiIiNGMSksJiomSSdsYW1iZGFHRihGMUYzRjFGMUYxRjEiIiRGMSw4KioiI2tGMSlJI0ExR0YoRjlGMSlGOEY0RjEpRjMiIidGMUYxKipGMEYxRj1GMUY4RjEpRjMiIiZGMUYxKihGPEYxRj1GMSlGMyIiJUYxRjEqKiIjO0YxKUY+RjRGMUY4RjEpRjNGOUYxRjEqKiIjc0YxRj5GMUY4RjFGS0YxISIiKihGSUYxRkpGMUYyRjFGMSooIiNhRjFGOEYxRktGMUYxKihGTUYxRj5GMUYyRjFGTiomRlFGMUYyRjFGMUY+RjFGMUZORjFGTjcjRi4=

IiIl

here is the (cubic) equation satisfied by A1 algA1:=collect(op(4,res),A1,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdBMUdGKCwuKioiI2siIiIpSSJ0R0YoIiIlRjEpLCYqJkknbGFtYmRhR0YoRjFGM0YxRjFGMUYxIiIjRjEpSSNBMUdGKCIiJEYxRjEqKiIjO0YxRjZGMSlGM0Y5RjEpRjtGOUYxRjEqJiwoKigiI3NGMUY4RjEpRjNGPEYxISIiKiZGREYxRj9GMUZGRjFGMUYxRjtGMUYxKigiI2FGMUY4RjFGRUYxRjEqJkZJRjFGP0YxRjFGMUZGNyNGLg==

n:=10: normal(series(subs(A1=subs(x=0,y=-1,Qser(n)),algA1),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

let us compare this to [Kauers-Yatchak], Section 5.3, where the solution is expressed in terms of the series Z, with Z(0)=0, satisfying: algZ:=Z*(1+4*Z)-t*(1+6*Z+12*Z^2+4*(2+lambda)*Z^3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdaR0YoLCYqJkkiWkdGKCIiIiwmRjFGMSomIiIlRjFGMEYxRjFGMUYxKiZJInRHRihGMSwqRjFGMSomIiInRjFGMEYxRjEqJiIjN0YxKUYwIiIjRjFGMSooRjRGMSwmRj1GMUknbGFtYmRhR0YoRjFGMSlGMCIiJEYxRjFGMSEiIjcjRi4=

tZ:=factor(solve(algZ,t));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN0WkdGKCooSSJaR0YoIiIiLCZGMEYwKiYiIiVGMEYvRjBGMEYwLCwqKEYzRjApRi8iIiRGMEknbGFtYmRhR0YoRjBGMComIiIpRjBGNkYwRjAqJiIjN0YwKUYvIiIjRjBGMComIiInRjBGL0YwRjBGMEYwISIiNyNGLg==

res:=numer(factor(subs(t=tZ,algA1)));

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

the second factor does not vanish subs(Z=0,op(1,res)); subs(Z=0,op(2,res));

LCZJI0ExRzYiIiIiRiUhIiI=

IiIi

A1Z:=factor(solve(op(1,res),A1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBMVpHRigsJCoqLCwqKCIiJSIiIilJIlpHRigiIiRGM0knbGFtYmRhR0YoRjNGMyomIiIpRjNGNEYzRjMqJiIjN0YzKUY1IiIjRjNGMyomIiInRjNGNUYzRjNGM0YzRjMsLCooRj1GM0Y0RjNGN0YzRjMqJkYyRjNGNEYzRjMqJkY/RjNGPEYzISIiRj5GREYzRkRGMyksJiomRj1GM0Y1RjNGM0YzRjNGPUZEKSwmRjNGMyomRjJGM0Y1RjNGM0Y9RkRGRDcjRi4=

with(gfun): Zser:=op(2,algeqtoseries(numer(t-tZ),t,Z,10,true));

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

Back to Q(0,0) subs(Q=Q00,convert(series(eqcat,y,1),polynom));

LCoqJiwmKigiIiMiIiJJJ2xhbWJkYUc2IkYnSSJ0R0YpRidGJ0YmRidGJ0kjQTFHRilGJ0YnKiYpRipGJkYnKUkkUTAwR0YpRiZGJyEiIiomLCZGJUYwRidGMEYnRi9GJ0YnRidGMA==

We get rid of t algQ00Z:=collect(numer(factor(subs(t=tZ,A1=A1Z,%))),Q00,factor);

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

factor(simplify(factor([solve(algQ00Z,Q00)]),symbolic));

NyQsJCosIyIiIiIiI0YmLDoqKiIiJUYmKSwmRiZGJiomRipGJkkiWkc2IkYmRiZGJUYmKUYuRidGJkknbGFtYmRhR0YvRiZGJiooRipGJilGLiIiJEYmRjFGJiEiIiooIiIpRiZGK0YmRjBGJkYmKipGJ0YmRitGJkYuRiZGMUYmRiYqJkY3RiZGM0YmRjUqKEY3RiZGMUYmRjBGJkY1KihGKkYmRitGJkYuRiZGJiomIiM3RiZGMEYmRjUqKEYnRiZGMUYmRi5GJkY1KiRGK0YmRiYqJiIiJ0YmRi5GJkY1RiZGNUYmLCxGMkYmRjlGJkY8RiZGQEYmRiZGJkYmRjBGNSlGLEYnRjVGJiwkKixGJUYmLDpGKUYmRjJGJkY2RiZGOEYmRjlGJkY6RiZGO0YmRjxGJkY+RiZGP0YmRkBGJkYmRiZGJkZCRiZGMEY1RkNGNUY1

The degree over Q(t,lambda) factor(resultant(algQ00Z,algZ,Z));

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

algQ00:=collect(op(4,%),Q00,factor);

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

degree(%,Q00);

IiIn

n:=8: normal(series(subs(Q00=subs(x=0,y=0,Qser(n)),algQ00),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIio=

with(algcurves):genus(algQ00,t,Q00);

IiIh

Back to Q(0,y) eqcat;

LCwqJiwmKigiIiMiIiJJJ2xhbWJkYUc2IkYnSSJ0R0YpRidGJ0YmRidGJ0kjQTFHRilGJ0YnKigpLCZJInlHRilGJ0YnRidGJkYnKUYqRiZGJylJIlFHRilGJkYnISIiKiYsKEYlRjNGL0YnRidGM0YnRjJGJ0YnRi9GM0YnRjM=

algQy:=collect(op(3,factor(resultant(eqcat,algA1,A1))),Q,factor);degree(%,Q);

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

IiIn

n:=8: normal(series(subs(Q=subs(x=0,Qser(n)),algQy),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIio=

algQyZ:=collect(numer(factor(subs(t=tZ,A1=A1Z,eqcat))),Q,factor);

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

let us check that this coincides with the Kauers-Yatchak solution (watch out: their x-variable is our y-variable....) eqKY:=t^4*(y+1)^2*U^2+(2*t*lambda+1-y)*t^2*U+t*(4*t+1)+t^2*y-Z*(3*t^2*(lambda-4)+3*t+1)+t*(1+6*t)*(lambda+2)*Z^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcUtZR0YoLC4qKClJInRHRigiIiUiIiIpLCZJInlHRihGM0YzRjMiIiNGMylJIlVHRihGN0YzRjMqKCwoKihGN0YzSSdsYW1iZGFHRihGM0YxRjNGM0Y2ISIiRjNGM0YzKUYxRjdGM0Y5RjNGMyomRjFGMywmKiZGMkYzRjFGM0YzRjNGM0YzRjMqJkY/RjNGNkYzRjMqJkkiWkdGKEYzLCgqKCIiJEYzRj9GMywmRj1GM0YyRj5GM0YzKiZGSEYzRjFGM0YzRjNGM0YzRj4qKkYxRjMsJkYzRjMqJiIiJ0YzRjFGM0YzRjMsJkY3RjNGPUYzRjMpRkVGN0YzRjM3I0Yu

factor(resultant(eqKY,algZ,Z));

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

OK factor(subs(U=Q,%)/algQy);

KiYpLCYiIiMiIiJJJ2xhbWJkYUc2IkYmRiVGJilJInRHRigiIiVGJg==

Parametrisation in y factor(discrim(algQyZ,Q));

LCQqKiksJiIiIkYmKiYiIiVGJkkiWkc2IkYmRiZGKEYmLCgqKEYoRiYpRikiIiNGJkkieUdGKkYmRiZGJyEiIkYmRjBGJiksLCooRihGJilGKSIiJEYmSSdsYW1iZGFHRipGJkYmKiYiIilGJkY0RiZGJiomIiM3RiZGLUYmRiYqJiIiJ0YmRilGJkYmRiZGJkYuRiYpLDAqKEYoRiZGLUYmRjZGJkYmKiZGOEYmRi1GJkYmKihGLkYmRilGJkY2RiZGJiooRihGJkYpRiZGL0YmRjBGJ0YmRi9GMEYmRiZGLkYmRjA=

We introduce a new series V with CT zero, and polynomial coefficients in lambda and y, satisfying: algV:=V-Z*(y*Z-1)*(1+V)^2;solve(algV,V);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdWR0YoLCZJIlZHRigiIiIqKEkiWkdGKEYwLCYqJkYyRjBJInlHRihGMEYwRjAhIiJGMCksJkYwRjBGL0YwIiIjRjBGNjcjRi4=

NiQsJCoqIyIiIiIiI0YmLCoqKEYnRiYpSSJaRzYiRidGJkkieUdGLEYmISIiKiZGJ0YmRitGJkYmRiZGJiokKSwoKigiIiVGJkYqRiZGLUYmRi4qJkY0RiZGK0YmRiZGJkYmRiVGJkYmRiZGK0YuLCYqJkYrRiZGLUYmRiZGJkYuRi5GJiwkKipGJUYmLCpGKUYmRjBGJkYvRi5GJkYuRiZGK0YuRjZGLkYu

yV:=factor(solve(algV,y));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN5VkdGKCooLCoqJilJIlZHRigiIiMiIiJJIlpHRihGNEY0KihGM0Y0RjJGNEY1RjRGNEYyRjRGNUY0RjQpRjVGMyEiIiksJkY0RjRGMkY0RjNGODcjRi4=

Vser:=op(algeqtoseries(algV,Z,V,10,true));

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

res:=numer(factor(subs(t=tZ,y=yV,algQy))): nops(%);;

IiIk

subs(Z=0,V=0,op(1,res));subs(Z=0,V=0,op(2,res));subs(Z=0,V=0,op(3,res));

IiIh

LCZJIlFHNiIiIiJGJSEiIg==

IiIi

op(1,res);op(2,res);

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

normal(series(subs(Q=subs(x=0,Qser(n)),V=Vser,Z=Zser,op(1,res)),t,5)); normal(series(subs(Q=subs(x=0,Qser(n)),V=Vser,Z=Zser,op(2,res)),t,5));

KytJInRHNiIsJiIiIkYmSSJ5R0YkISIiIiIjLCgqJiIjNUYmRidGJkYoRixGJiomRilGJkknbGFtYmRhR0YkRiZGJiIiJCwqKiZGKUYmKUYnRilGJkYoKiYiIz9GJkYuRiZGJiomIiNqRiZGJ0YmRigiI3RGJiIiJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YmIiIm

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiY=

QyZV:=factor(solve(op(2,res),Q));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVReVpWR0YoLCQqLCwsKigiIiUiIiIpSSJaR0YoIiIkRjNJJ2xhbWJkYUdGKEYzRjMqJiIiKUYzRjRGM0YzKiYiIzdGMylGNSIiI0YzRjMqJiIiJ0YzRjVGM0YzRjNGM0YzLCZGM0YzSSJWR0YoRjNGMyw0KipGPUYzRkFGM0Y0RjNGN0YzRjMqKEYyRjNGQUYzRjRGM0YzKihGPUYzRjRGM0Y3RjNGMyooRj1GM0ZBRjNGPEYzISIiKiZGMkYzRjRGM0YzKiZGQUYzRjVGM0ZHKiZGPUYzRjxGM0ZHKiYiIiZGM0Y1RjNGR0YzRkdGMyksKEZJRjNGNUYzRjNGM0Y9RkcpLCZGM0YzKiZGMkYzRjVGM0YzRj1GR0ZHNyNGLg==

normal(series(subs(V=convert(Vser,polynom),Z=Zser,QyZV),t,6));

Ky9JInRHNiIiIiIiIiEsJkkieUdGJEYlIiIkRiUiIiMsJComRipGJUknbGFtYmRhR0YkRiVGJUYpLCgqJkYqRiUpRihGKkYlRiUqJiIjN0YlRihGJUYlIiNFRiUiIiUsJiooIiIpRiVGLUYlRihGJUYlKiYiI1NGJUYtRiVGJSIiJi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIn

subs(x=0,Qser(5));

LCwiIiJGIyomLCZJInlHNiJGIyIiJEYjRiMpSSJ0R0YnIiIjRiNGIyooRitGI0knbGFtYmRhR0YnRiMpRipGKEYjRiMqJiwoKiZGK0YjKUYmRitGI0YjKiYiIzdGI0YmRiNGIyIjRUYjRiMpRioiIiVGI0YjKiYsJiooIiIpRiNGLUYjRiZGI0YjKiYiI1NGI0YtRiNGI0YjKUYqIiImRiNGIw==

Back to Q(x,0), using R(x)+S(Y)-R(0)=xy if (x,y) cancels the kernel eqxy:=subs(Q(x,y)=0,Q(0,0)=Q00,Q(x,0)=Qx,Q(0,y)=Qy,eqQ);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcXh5R0YoLCoqJkkieEdGKCIiIkkieUdGKEYxRjEqKCwmRjJGMUYxRjFGMUkidEdGKEYxSSNReUdGKEYxISIiKigsKComSSdsYW1iZGFHRihGMUYwRjFGMSokKUYwIiIjRjFGMUYxRjFGMUY1RjFJI1F4R0YoRjFGNyomRjVGMUkkUTAwR0YoRjFGMTcjRi4=

Expression of Q(0,Y0) (with y = Y0) solve(eqxy,Qy);

KigsLCoqSSNReEc2IiIiIkknbGFtYmRhR0YmRidJInRHRiZGJ0kieEdGJkYnISIiKihGJUYnRilGJylGKiIiI0YnRisqJkkkUTAwR0YmRidGKUYnRicqJkYlRidGKUYnRisqJkYqRidJInlHRiZGJ0YnRicsJkYzRidGJ0YnRitGKUYr

res1:=numer(factor(subs(Q=%,algQy))): nops(%);

IiU3Sg==

indets(res1);

PChJJFEwMEc2IkkjUXhHRiRJJ2xhbWJkYUdGJEkidEdGJEkieEdGJEkieUdGJA==

Let us eliminate Y (which cancels the kernel) res2:=factor(resultant(res1,K,y)): nops(res2);

IiIl

map(indets,[seq(op(i,res2),i=1..nops(res2))]);

NyY8J0kkUTAwRzYiSSNReEdGJUknbGFtYmRhR0YlSSJ0R0YlSSJ4R0YlPCNGKDwjRik8JEYnRig=

nops(op(1,res2));

IiIj

An equation satisfied by Q(x,0) res3:=op(1,op(1,res2)):indets(%);

PCdJJFEwMEc2IkkjUXhHRiRJJ2xhbWJkYUdGJEkidEdGJEkieEdGJA==

in terms of Z res4:=factor(resultant(subs(t=tZ,res3),algQ00Z,Q00)): nops(res4);

IiIn

map(indets,[seq(op(i,res4),i=1..nops(res4))]);

Nyg8I0kiWkc2IkYjPCRJJ2xhbWJkYUdGJUkieEdGJTwmSSNReEdGJUYkRidGKEYpPCRGJEYn

nops(op(4,res4));nops(op(5,res4));

IiQmcA==

IiZ2IUc=

normal(series(subs(Qx=subs(y=0,Qser(n)),Z=Zser, op(4,res4)),t));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIic=

n:=2: normal(series(subs(Qx=subs(y=0,Qser(n)),Z=Zser,op(5,res4)),t,n+1));

KytJInRHNiIqJClJInhHRiQiIiUiIiIiIiEsKiooIiM7RilJJ2xhbWJkYUdGJEYpRiZGKUYpKiZGLUYpKUYnIiImRilGKSomIiMpKUYpRiZGKUYpKiZGLUYpKUYnIiIkRilGKUYpLDQqKCIjJypGKSlGLiIiI0YpRiZGKUYpKigiJHciRilGLkYpRjBGKUYpKiYiIyEpRikpRiciIidGKUYpKigiJTM5RilGLkYpRiZGKUYpKiZGQ0YpRjBGKUYpKigiJGciRilGLkYpRjVGKUYpKiYiJUtTRilGJkYpRikqJkZDRilGNUYpRikqJiIja0YpKUYnRjtGKUYpRjstSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKUY2

algQxZ:=collect(op(4,res4),Qx,factor); degree(%,Qx);

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

IiIl

n:=10:normal(series(subs(Z=Zser,Qx=subs(y=0,Qser(n)),algQxZ),t));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIic=

Equation over t, lambda and x res:=factor(resultant(algQxZ,numer(t-tZ),Z)): nops(res);

IiIl

seq(indets(op(i,res)),i=1..nops(res));

NiY8IjwjSSdsYW1iZGFHNiJGJDwmSSNReEdGJkYlSSJ0R0YmSSJ4R0Ym

nops(op(4,res));

IiRzKA==

algQx:=op(4,res): degree(%,Qx);

IiM3

Trying to parametrise x factor(discrim(algQxZ,Qx));

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

simplify(op(1,factor([solve(algQxZ,Qx)])),symbolic);

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

Introduce U such that algU:=U-Z*(-1+x+x*Z*2+lambda*x*Z)*(1+U)^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdVR0YoLCZJIlVHRigiIiIqKEkiWkdGKEYwLCoqKEYyRjBJJ2xhbWJkYUdGKEYwSSJ4R0YoRjBGMCooIiIjRjBGMkYwRjZGMEYwRjZGMEYwISIiRjApLCZGMEYwRi9GMEY4RjBGOTcjRi4=

User:=op(1,algeqtoseries(algU,Z,U,5,true));

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

xU:=solve(algU,x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN4VUdGKCoqLCoqJilJIlVHRigiIiMiIiJJIlpHRihGNEY0KihGM0Y0RjJGNEY1RjRGNEYyRjRGNUY0RjRGNSEiIiwoKiZGNUY0SSdsYW1iZGFHRihGNEY0KiZGM0Y0RjVGNEY0RjRGNEY3KSwmRjRGNEYyRjRGM0Y3NyNGLg==

res:=numer(factor(subs(x=xU,algQxZ))):nops(%);

IiIj

Choosing the correct factor normal(series(subs(U=User,Z=Zser,Qx=subs(y=0,Qser(n)),op(1,res)),t,5)); normal(series(subs(U=User,Z=Zser,Qx=subs(y=0,Qser(n)),op(2,res)),t,5));

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

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiY=

algQxZU:=collect(op(2,res),Qx,factor);

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

factor(discrim(%,Qx));

LCQqMCIiJSIiIiksJkYlRiUqJkYkRiVJIlpHNiJGJUYlIiIkRiUpLCwqKEYkRiUpRilGK0YlSSdsYW1iZGFHRipGJUYlKiYiIilGJUYvRiVGJSomIiM3RiUpRikiIiNGJUYlKiYiIidGJUYpRiVGJUYlRiVGNkYlKSwoKiZGKUYlRjBGJUYlKiZGNkYlRilGJUYlRiVGJUYkRiUpLCZGJUYlSSJVR0YqRiVGOEYlKSwoKiZGP0YlRilGJUYlKiZGK0YlRilGJUYlRiVGJUY2RiUpLDgqKClGP0Y2RiVGNUYlRjBGJUYlKihGNkYlRkdGJUY1RiVGJSoqRjZGJUY/RiVGNUYlRjBGJUYlKihGJEYlRj9GJUY1RiVGJSooRj9GJUYpRiVGMEYlRiUqJkY1RiVGMEYlRiUqKEY2RiVGP0YlRilGJUYlKiZGNkYlRjVGJUYlRjtGJUY8RiVGJUYlRjZGJUYl

Qxsol:=simplify(factor([solve(algQxZU,Qx)]),symbolic);

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

normal(map(series,subs(U=convert(User,polynom),Z=Zser,Qxsol),t));

NyQrL0kidEc2IiIiIiIiISwkKiYiIiNGJkkieEdGJUYmRiZGJiwoKiZJJ2xhbWJkYUdGJUYmRitGJkYmKiYiIiZGJilGK0YqRiZGJiIiJEYmRiosKiooIiInRiZGLkYmRjFGJkYmKiYiIzlGJilGK0YyRiZGJiomRipGJkYuRiZGJiomIiM7RiZGK0YmRiZGMiwuKihGKkYmKUYuRipGJkYxRiZGJiooIiNHRiZGLkYmRjhGJkYmKiYiI1VGJilGKyIiJUYmRiYqKCIjQUYmRi5GJkYrRiZGJiomIiNxRiZGMUYmRiYiI0VGJkZELUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiZGMCsvRiQsJComRipGJkYrISIiRlNGU0ZTRicsJEYpRlNGJiwoRi1GU0YvRlNGMkZTRiosKkY0RlNGNkZTRjlGU0Y6RlNGMkZKRkQ=

subs(y=0,Qser(4));

LCwiIiJGIyooIiIjRiNJInRHNiJGI0kieEdGJ0YjRiMqJiwoKiZJJ2xhbWJkYUdGJ0YjRihGI0YjKiYiIiZGIylGKEYlRiNGIyIiJEYjRiMpRiZGJUYjRiMqJiwqKigiIidGI0YsRiNGL0YjRiMqJiIjOUYjKUYoRjBGI0YjKiZGJUYjRixGI0YjKiYiIztGI0YoRiNGI0YjKUYmRjBGI0YjKiYsLiooRiVGIylGLEYlRiNGL0YjRiMqKCIjR0YjRixGI0Y4RiNGIyomIiNVRiMpRigiIiVGI0YjKigiI0FGI0YsRiNGKEYjRiMqJiIjcUYjRi9GI0YjIiNFRiNGIylGJkZGRiNGIw==

QxZU:=op(1,Qxsol);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVReFpVR0YoLCQqMCwmKiYsJiIiIyIiIkknbGFtYmRhR0YoRjRGNEkiWkdGKEY0RjRGNEY0RjQsKiNGNCIiJUY0KiZGMkY0KUY2IiIkRjRGNComRjxGNClGNkYzRjRGNComI0Y8RjNGNEY2RjRGNEY0LCYqLEY4RjRGMEY0LCZGNEY0SSJVR0YoRjRGNCwmKiYsJkZERjRGPEY0RjRGNkY0RjRGNEY0RjQpLCZGNEY0KiZGOUY0RjZGNEY0I0Y0RjNGNCEiIiomLCZGNkY0RjhGNEY0LChGNEY0KigpRkNGM0Y0RjJGNEY+RjRGNCooRjJGNEZDRjRGNkY0RjRGNEY0RjRGUUY0KUZORjNGTCwmRkRGNComRjZGNEZRRjRGNEZMRk9GTEZMNyNGLg==

factor(%);

KjAsaG4iIiIhIiIqJiksJkYkRiQqJiIiJUYkSSJaRzYiRiRGJCNGJCIiI0YkSSJVR0YsRiRGJCoqRi5GJEYnRiQpRi9GLkYkKUYrRi5GJEYkKihGJ0YkRjFGJEYrRiRGJCoqIiIpRiRGJ0YkRi9GJEYyRiRGJCoqIiInRiRGJ0YkRi9GJEYrRiRGJCooRi5GJEYxRiRGMkYkRiUqKCIjO0YkRi9GJClGKyIiJEYkRiUqKCIjN0YkRi9GJEYyRiRGJSoqRidGJEYxRiRGMkYkSSdsYW1iZGFHRixGJEYkKixGKkYkRidGJEYvRiRGMkYkRkBGJEYkKipGJ0YkRi9GJEYrRiRGQEYkRiQqKEYvRiRGK0YkRkBGJEYlKihGNUYkRjFGJEY7RiRGJSoqRipGJEYxRiRGO0YkRkBGJEYlKihGLkYkRitGJEYvRiRGJSokRidGJEYkKihGN0YkRidGJEYyRiRGJCooIiImRiRGJ0YkRitGJEYkKiYiIzVGJEYyRiRGJSomRjVGJEY7RiRGJSooRipGJEY7RiRGQEYkRiUqKEZKRiRGQEYkRjJGJEYlKiZGN0YkRitGJEYlKipGPEYkRidGJEYyRiRGQEYkRiQqKEYnRiRGK0YkRkBGJEYkKiZGQEYkRitGJEYlKihGMUYkRjJGJEZARiRGJSoqRjVGJEYvRiRGO0YkRkBGJEYlKipGN0YkRi9GJEYyRiRGQEYkRiVGJCksJkYkRiRGL0YkRi5GJCwoKiZGK0YkRkBGJEYkKiZGLkYkRitGJEYkRiRGJEYkLCxGTkYkRk1GJComRj5GJEYyRiRGJEZQRiRGJEYkRiQpRihGLkYlLCoqJkYxRiRGK0YkRiQqKEYuRiRGL0YkRitGJEYkRi9GJEYrRiRGJSw4RlRGJEY4RiQqKkYuRiRGL0YkRjJGJEZARiRGJCooRipGJEYvRiRGMkYkRiRGQ0YkKiZGMkYkRkBGJEYkRltvRiQqJkYuRiRGMkYkRiRGWkYkRmVuRiRGJEYkRiU=

liste:=1/y+x/y+2*x+x*y+2*y+y/x+1/x;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwwKiRJInlHRighIiIiIiIqJkkieEdGKEYyRjBGMUYyKiYiIiNGMkY0RjJGMiomRjRGMkYwRjJGMiomRjZGMkYwRjJGMiomRjBGMkY0RjFGMiokRjRGMUYyNyNGLg==

K:=collect((t*liste-1)*x*y,t,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCYqJiwwKiYpSSJ4R0YoIiIjIiIiKUkieUdGKEY0RjVGNSooRjRGNUYyRjVGN0Y1RjUqKEY0RjVGM0Y1RjZGNUY1KiRGMkY1RjUqJEY2RjVGNUYzRjVGN0Y1RjVJInRHRihGNUY1KiZGM0Y1RjdGNSEiIjcjRi4=

The functional equation eqQ:=K*Q(x,y)+x*y-subs(x=0,K)*Q(0,y) -subs(y=0,K)*Q(x,0)+subs(x=0,y=0,K)*Q(0,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcVFHRigsKiomLCYqJiwwKiYpSSJ4R0YoIiIjIiIiKUkieUdGKEY2RjdGNyooRjZGN0Y0RjdGOUY3RjcqKEY2RjdGNUY3RjhGN0Y3KiRGNEY3RjcqJEY4RjdGN0Y1RjdGOUY3RjdJInRHRihGN0Y3KiZGNUY3RjlGNyEiIkY3LUkiUUdGKDYkRjVGOUY3RjdGP0Y3KigsJkY9RjdGOUY3RjdGPkY3LUZCNiQiIiFGOUY3RkAqKCwmRjxGN0Y1RjdGN0Y+RjctRkI2JEY1RkhGN0ZANyNGLg==

Qser:=proc(n) option remember: if n=0 then 1 else convert(normal(series(subs(Q(x,y)=Qser(n-1),Q(x,0)=subs(y=0,Qser(n-1)), Q(0,y)=subs(x=0,Qser(n-1)), Q(0,0)=subs(x=0,y=0,Qser(n-1)), expand(eqQ/x/y+Q(x,y))),t,n+1)),polynom): fi: end: Qser(4);

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

Rser:=proc(n) convert(normal(series(subs(y=0,K)*subs(y=0,Qser(n-1)),t,n+1)),polynom):end: Sser:=proc(n) convert(normal(series(subs(x=0,K)*subs(x=0,Qser(n-1)),t,n+1)),polynom):end: Rser(4);Sser(4);

LCoqJiwmKiQpSSJ4RzYiIiIjIiIiRipGJ0YqRipJInRHRihGKkYqKipGKUYqRiZGKiwmRidGKkYqRipGKilGK0YpRipGKioqRidGKkYtRiosKComIiImRipGJkYqRioqJiIiJEYqRidGKkYqIiIlRipGKilGK0Y0RipGKiosRilGKkYnRipGLUYqLCoqJiIiKEYqKUYnRjRGKkYqKiYiIipGKkYmRipGKiomIiM3RipGJ0YqRipGNEYqRiopRitGNUYqRio=

LCoqJiwmKiQpSSJ5RzYiIiIjIiIiRipGJ0YqRipJInRHRihGKkYqKipGKUYqRiZGKiwmRidGKkYqRipGKilGK0YpRipGKioqRidGKkYtRiosKComIiImRipGJkYqRioqJiIiJEYqRidGKkYqIiIlRipGKilGK0Y0RipGKiosRilGKkYnRipGLUYqLCoqJiIiKEYqKUYnRjRGKkYqKiYiIipGKkYmRipGKiomIiM3RipGJ0YqRipGNEYqRiopRitGNUYqRio=

Checking the functional equation n:=10: normal(series(Qser(n)*K+x*y-Rser(n)-Sser(n)+subs(x=0,Rser(n)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Invariants have been constructed in the Maple session "invariants-syst" Jy:=t^2/y^2-(1+2*t)*t/y-t*y*(1+3*t)-(1+3*t)*(1+4*t)/(1+y)+(1+3*t)^2/(1+y)^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKeUdGKCwsKiYpSSJ0R0YoIiIjIiIiKUkieUdGKEYyISIiRjMqKCwmKiZGMkYzRjFGM0YzRjNGM0YzRjFGM0Y1RjZGNiooRjFGM0Y1RjMsJkYzRjMqJiIiJEYzRjFGM0YzRjNGNiooRjtGMywmKiYiIiVGM0YxRjNGM0YzRjNGMywmRjVGM0YzRjNGNkY2KiYpRjtGMkYzKUZCRjJGNkYzNyNGLg==

A decoupling function Gy:=-y+1/y-(1+3*t)/t/(1+y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwoSSJ5R0YoISIiKiRGL0YwIiIiKigsJkYyRjIqJiIiJEYySSJ0R0YoRjJGMkYyRjdGMCwmRi9GMkYyRjJGMEYwNyNGLg==

Jy; Ly:=S(y)-Gy;

LCwqJilJInRHNiIiIiMiIiIpSSJ5R0YmRichIiJGKCooLCYqJkYnRihGJUYoRihGKEYoRihGJUYoRipGK0YrKihGJUYoRipGKCwmRihGKComIiIkRihGJUYoRihGKEYrKihGMEYoLCYqJiIiJUYoRiVGKEYoRihGKEYoLCZGKkYoRihGKEYrRisqJilGMEYnRigpRjdGJ0YrRig=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNMeUdGKCwqLUkiU0dGKDYjSSJ5R0YoIiIiRjJGMyokRjIhIiJGNSooLCZGM0YzKiYiIiRGM0kidEdGKEYzRjNGM0Y6RjUsJkYyRjNGM0YzRjVGMzcjRi4=

map(factor,convert((Jy-t^2*(-1/y+(1+3*t)/(t*(y+1)))^2),parfrac,y));

LCgqKEkidEc2IiIiIkkieUdGJUYmLCZGJkYmKiYiIiRGJkYkRiZGJkYmISIiKihGKEYmLCZGJkYmKiYiIidGJkYkRiZGJkYmLCZGJ0YmRiZGJkYrRisqKEYkRiYsJiomIiIlRiZGJEYmRiZGJkYmRiZGJ0YrRiY=

We construct an expression with no pole at y=0. We will see that this also eliminates the pole at -1 eqcat:=Jy-a*Ly^2-b*Ly-c;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZlcWNhdEdGKCwyKiYpLCotSSJTR0YoNiNJInlHRigiIiJGNUY2KiRGNSEiIkY4KigsJkY2RjYqJiIiJEY2SSJ0R0YoRjZGNkY2Rj1GOCwmRjVGNkY2RjZGOEY2IiIjRjZJImFHRihGNkY4KiZGMUY2SSJiR0YoRjZGOComKUY9Rj9GNilGNUY/RjhGNiooLCYqJkY/RjZGPUY2RjZGNkY2RjZGPUY2RjVGOEY4KihGPUY2RjVGNkY6RjZGOCooRjpGNiwmKiYiIiVGNkY9RjZGNkY2RjZGNkY+RjhGOComKUY6Rj9GNilGPkY/RjhGNkkiY0dGKEY4NyNGLg==

normal(series(eqcat,y,1));

KylJInlHNiIsJiokKUkidEdGJCIiIyIiIkYqSSJhR0YkISIiISIjKiYsLioqRilGKi1JIlNHRiQ2IyIiIUYqRitGKkYoRipGKiomRilGKilGKCIiJEYqRiwqKCIiJ0YqRitGKkYoRipGKiomSSJiR0YkRipGKEYqRipGJkYsKiZGKUYqRitGKkYqRipGKEYsRiwtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKkY0

asol:=solve(op(1,%),a);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhc29sR0YoKiQpSSJ0R0YoIiIjIiIiNyNGLg==

normal(series(subs(a=asol,eqcat),y,1));

KydJInlHNiIsKiooIiIjIiIiLUkiU0dGJDYjIiIhRigpSSJ0R0YkRidGKEYoKiYiIiVGKEYtRihGKEkiYkdGJEYoRi5GKCEiIi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRiw=

Note that S(y) is a multiple of y, and hence vanishes at y=0 factor(subs(x=0,K));

KihJInlHNiIiIiIsJkYjRiVGJUYlRiVJInRHRiRGJQ==

bsol:=subs(S(0)=0,solve(op(1,%%),b));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVic29sR0YoLCYqJiIiJSIiIilJInRHRigiIiNGMSEiIkYzRjU3I0Yu

normal(subs(S(0)=0,series(subs(a=asol,b=bsol,eqcat),y,2)));

KydJInlHNiIsKiooIiIjIiIiLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kiU0dGJDYjIiIhRigpSSJ0R0YkRidGKEYoKiYiIiVGKEYzRighIiJJImNHRiRGNyomRidGKEY0RihGN0YyLUkiT0dGLDYjRihGKA==

csol:=solve(coeff(%,y,0),c);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVjc29sR0YoLCgqKCIiIyIiIi0tSSJER0YlNiNJIlNHRig2IyIiIUYxKUkidEdGKEYwRjFGMSomIiIlRjFGOUYxISIiKiZGMEYxRjpGMUY9NyNGLg==

eqcat1:=numer(factor(subs(a=asol,b=bsol,c=csol,eqcat)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdlcWNhdDFHRigsJComLDgqKCktSSJTR0YoNiNJInlHRigiIiMiIiJJInRHRihGOClGNkY3RjhGOCoqRjdGOEYzRjhGOUY4KUY2IiIkRjhGOComRjlGOClGNiIiJUY4RjgqKkY3RjgtLUkiREdGJTYjRjQ2IyIiIUY4RjlGOEY6RjhGOCooRjJGOEY5RjhGNkY4RjgqKkY3RjhGM0Y4RjpGOEY5RjghIiIqKkY3RjhGQkY4RjlGOEY2RjhGOComRjNGOEY6RjhGSiomRjlGOEY6RjhGSiooRjdGOEYzRjhGOUY4RkoqJkYzRjhGNkY4RjhGOEY5RjhGSjcjRi4=

eqcat1:=eqcat1/t;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdlcWNhdDFHRigsOCooKS1JIlNHRig2I0kieUdGKCIiIyIiIkkidEdGKEY2KUY0RjVGNiEiIioqRjVGNkYxRjZGN0Y2KUY0IiIkRjZGOSomRjdGNilGNCIiJUY2RjkqKkY1RjYtLUkiREdGJTYjRjI2IyIiIUY2RjdGNkY4RjZGOSooRjBGNkY3RjZGNEY2RjkqKkY1RjZGMUY2RjhGNkY3RjZGNioqRjVGNkZBRjZGN0Y2RjRGNkY5KiZGMUY2RjhGNkY2KiZGN0Y2RjhGNkY2KihGNUY2RjFGNkY3RjZGNiomRjFGNkY0RjZGOTcjRi4=

Can this be true ? n:=30: normal(series(subs(S(y)=Sser(n), D(S)(0)=subs(y=0,diff(Sser(n),y)),eqcat1),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0o=

Back to Q(0,y) (denoted here Q(y) to lighten notation) Sy:=subs(x=0,K)*Q(y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNTeUdGKCooLCYqJClJInlHRigiIiMiIiJGNEYyRjRGNEkidEdGKEY0LUkiUUdGKDYjRjJGNDcjRi4=

factor(subs(S(y)=Sy,(D(S))(0)=limit(diff(Sy,y),y=0),eqcat1));

LCQqKkkidEc2IiIiIkkieUdGJUYmLCZGJ0YmRiZGJkYmLDgqKCktSSJRR0YlNiNGJyIiI0YmKUYkRi9GJilGJyIiJUYmRiYqKkYvRiZGK0YmRjBGJilGJyIiJEYmRiYqKEYrRiZGMEYmKUYnRi9GJkYmKipGL0YmRixGJkYkRiZGNEYmRiYqKkYvRiZGJEYmRixGJkY3RiYhIiIqJkYsRiZGN0YmRjoqKEYvRiZGJEYmLUYtNiMiIiFGJkYmKihGL0YmRiRGJkYsRiZGOiomRidGJkYsRiZGJiokRjdGJkYmRidGOkYmRjo=

eqcatQ:=collect(op(5,%),Q(y),factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdlcWNhdFFHRigsLCoqKUkieUdGKCIiIyIiIiksJkYxRjNGM0YzRjJGMylJInRHRihGMkYzKS1JIlFHRig2I0YxRjJGM0YzKiYsLCooRjJGM0Y3RjMpRjEiIiRGM0YzKihGMkYzRjdGM0YwRjMhIiIqJEYwRjNGQiomRjJGM0Y3RjNGQkYxRjNGM0Y5RjNGMyooRjJGM0Y3RjMtRjo2IyIiIUYzRjNGQ0YzRjFGQjcjRi4=

At the same time we have got rid of singularities at y=-1 factor(series(eqcatQ,y=-1,2));

KyksJkkieUc2IiIiIkYmRiYsKiooIiIjRiZJInRHRiVGJi1JIlFHRiU2IyIiIUYmRiYqKCIiJ0YmRipGJi1GLDYjISIiRiZGMyomRilGJkYxRiZGM0YpRiZGLiwsKihGMEYmLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YsRjJGJkYqRiZGMyooIiM1RiZGKkYmRjFGJkYmKiZGKUYmRjdGJkYzKiYiIiRGJkYxRiZGJkZCRjNGJi1JIk9HRjo2I0YmRik=

eqcatQ;

LCwqKilJInlHNiIiIiMiIiIpLCZGJUYoRihGKEYnRigpSSJ0R0YmRidGKCktSSJRR0YmNiNGJUYnRihGKComLCwqKEYnRihGLEYoKUYlIiIkRihGKCooRidGKEYsRihGJEYoISIiKiRGJEYoRjcqJkYnRihGLEYoRjdGJUYoRihGLkYoRigqKEYnRihGLEYoLUYvNiMiIiFGKEYoRjhGKEYlRjc=

eqcat:=subs(Q(y)=Q,Q(0)=A1,eqcatQ);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZlcWNhdEdGKCwsKiopSSJ5R0YoIiIjIiIiKSwmRjFGM0YzRjNGMkYzKUkidEdGKEYyRjMpSSJRR0YoRjJGM0YzKiYsLCooRjJGM0Y3RjMpRjEiIiRGM0YzKihGMkYzRjdGM0YwRjMhIiIqJEYwRjNGQComRjJGM0Y3RjNGQEYxRjNGM0Y5RjNGMyooRjJGM0Y3RjNJI0ExR0YoRjNGM0ZBRjNGMUZANyNGLg==

Two series cancel the following derivative edQ:=factor(diff(eqcat,Q));factor(subs(t=0,%));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlZFFHRigsMioqIiIjIiIiSSJRR0YoRjEpSSJ0R0YoRjBGMSlJInlHRigiIiVGMUYxKipGN0YxRjJGMUYzRjEpRjYiIiRGMUYxKipGMEYxRjJGMUYzRjEpRjZGMEYxRjEqKEYwRjFGNEYxRjlGMUYxKihGMEYxRjRGMUY8RjEhIiIqJEY8RjFGPyomRjBGMUY0RjFGP0Y2RjE3I0Yu

LCQqJkkieUc2IiIiIiwmRiRGJkYmISIiRiZGKA==

Let us compute their expansions. First, the series with no constant term y0:='y0': Y0ser:=proc(n) add(y0[i]*t^i,i=1..n): end: n:=1: normal(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=-1,Qser(n)), y=Y0ser(n), edQ),t,n+1));

KydJInRHNiIsJiIiIyEiIiZJI3kwR0YkNiMiIiJGK0YrLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJEYqRiY=

y0[1]:=2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTBHRig2IyIiIiIiIzcjRjE=

From now on we execute the next two lines several times to obtain coefficients of Y0 n:=n+1: normal(series(normal(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=-1,Qser(n)), y=Y0ser(n),edQ)),t,n+1));

KydJInRHNiIsJiZJI3kwR0YkNiMiIiciIiIiJVdYISIiRiktSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKiIiKA==

y0[n]:=solve(coeff(%,t,n),y0[n]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTBHRig2IyIiJyIlV1g3I0Yx

Y0ser(n);

LC4qJiIlV1giIiIpSSJ0RzYiIiInRiVGJSomIiRPKEYlKUYnIiImRiVGJSomIiQ/IkYlKUYnIiIlRiVGJSomIiNDRiUpRiciIiRGJUYlKiZGMUYlKUYnIiIjRiVGJSomRjhGJUYnRiVGJQ==

There is also a series with CT 1 y1:='y1': Y1ser:=proc(n) 1+add(y1[i]*t^i,i=1..n): end: n:=1: normal(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=-1,Qser(n)), y=Y1ser(n), edQ),t,n+1));

KydJInRHNiIsJiIiIyEiIiZJI3kxR0YkNiMiIiJGJ0YrLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJEYqRiY=

y1[1]:=-2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTFHRig2IyIiIiEiIzcjRjE=

From now on we execute the next two lines several times to obtain coefficients of Y0 n:=n+1: normal(series(normal(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=-1,Qser(n)), y=Y1ser(n),edQ)),t,n+1));

KydJInRHNiIsJiZJI3kxR0YkNiMiIichIiIiJUM9RipGKS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIiKA==

y1[n]:=solve(coeff(%,t,n),y1[n]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTFHRig2IyIiJyElQz03I0Yx

Y1ser(n);

LC4qJiIlQz0iIiIpSSJ0RzYiIiInRiUhIiIqJiIkXyRGJSlGJyIiJkYlRioqJiIjU0YlKUYnIiIlRiVGKiomIiM7RiUpRiciIiRGJUYqKiYiIiNGJUYnRiVGKkYlRiU=

disc:=collect(factor(discrim(eqcat,Q)),y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkaXNjR0YoLC4qJiwmKiYiIzciIiIpSSJ0R0YoIiIjRjMhIiIqJiIiJUYzRjVGM0Y3RjMpSSJ5R0YoIiImRjNGMyomLCoqKCIiKUYzSSNBMUdGKEYzKUY1IiIkRjNGNyomRkBGM0Y0RjNGMyomRkBGM0Y1RjNGM0YzRjNGMylGO0Y5RjNGMyomLCoqKCIjO0YzRkFGM0ZCRjNGNyomRjlGM0Y0RjNGN0Y4RjdGNkY3RjMpRjtGQ0YzRjMqJiwqRj9GN0ZERjNGOEYzRjNGM0YzKUY7RjZGM0YzKihGOUYzRjVGM0Y7RjNGN0ZLRjM3I0Yu

The discriminant of disc (with respect to y) vanishes res:=factor(discrim(disc,y));nops(res);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyZXNHRigsJCoqIiYlUTsiIiIpLCZGMUYxKiYiIiRGMUkidEdGKEYxRjEiIiNGMSksPCooIiNrRjEpSSNBMUdGKEY1RjEpRjYiIidGMUYxKigiJEspRjEpRj1GN0YxKUY2IiImRjEhIiIqKCIkJVFGMUZCRjEpRjYiIiVGMUZFKigiIztGMUZCRjEpRjZGNUYxRkUqKCIkcyNGMUY9RjFGSEYxRjEqKCIkRyRGMUY9RjFGTEYxRjEqKCIjU0YxRj1GMSlGNkY3RjFGMSomIiRhI0YxRkxGMUZFKihGS0YxRj1GMUY2RjFGRSomIiNXRjFGU0YxRkVGPUYxKiZGS0YxRjZGMUYxRjFGRUY3RjEpRjYiIilGMUYxNyNGLg==

IiIl

Here is the (cubic) equation satisfied by A1 algA1:=collect(op(1,op(3,res)),A1,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdBMUdGKCwwKigiI2siIiIpSSNBMUdGKCIiJEYxKUkidEdGKCIiJ0YxRjEqKiIjO0YxKUY2RjRGMSwoKiYiI19GMSlGNiIiI0YxRjEqJiIjQ0YxRjZGMUYxRjFGMUYxKUYzRj9GMSEiIiomLCwqJiIkcyNGMSlGNiIiJUYxRjEqJiIkRyRGMUY6RjFGMSomIiNTRjFGPkYxRjEqJkY5RjFGNkYxRkNGMUYxRjFGM0YxRjEqJiIkYSNGMUY6RjFGQyomIiNXRjFGPkYxRkNGTkYxRjFGQzcjRi4=

n:=10: normal(series(subs(A1=subs(x=0,y=0,Qser(n)),algA1),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

We may want to look at the equation satisfied by the Y_i res:=factor(resultant(disc,algA1,A1)); nops(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyZXNHRigsJCosIiVDNSIiIilJInRHRigiIidGMSwwKigiIzdGMSlGMyIiI0YxKUkieUdGKCIiJEYxRjEqKCIiKUYxRjhGMSlGO0Y5RjFGMSooIiIlRjFGM0YxRjpGMUYxKihGQUYxRjhGMUY7RjFGMSokRjhGMSEiIiooRjlGMUYzRjFGO0YxRjEqJEY/RjFGREYxKSwuKihGPEYxRjNGMUY6RjFGMSooRjxGMUYzRjFGP0YxRkQqJEY6RjFGMSomRjNGMUY7RjFGMUZGRkRGM0YxRjlGMSksLComRjNGMUY6RjFGMSooRkFGMUYzRjFGP0YxRjFGRUYxKiZGOUYxRjNGMUYxRjtGREY5RjFGMTcjRi4=

IiIm

Decinding which equation holds n:=4: normal(series(subs(y=Y0ser(n),op(3,res)),t,n+1));

KylJInRHNiIhIiIiIiMiIzsiIiUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiY=

n:=4: normal(series(subs(y=Y0ser(n),op(1,op(4,res))),t,n+1));

Ky1JInRHNiIiIiJGJSEiIyIiIyEjOyIiJCEjayIiJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIm

n:=5: normal(series(subs(y=Y0ser(n),op(1,op(5,res))),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIic=

n:=4: normal(series(subs(y=Y1ser(n),op(3,res)),t,n+1));

Ky9JInRHNiIhIiIiIiEiIzUiIiIhIioiIiMhI0siIiQhI2siIiUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIiJg==

n:=4: normal(series(subs(y=Y1ser(n),op(1,op(4,res))),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiY=

n:=5: normal(series(subs(y=Y1ser(n),op(1,op(5,res))),t,n+1));

KzFJInRHNiIhIiIiIiEiIzYiIiIhI0UiIiMiI1ciIiQhJHciIiIlIiQhRyIiJi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIn

algY0:=subs(y=Y0,collect(op(1,op(5,res)),y,factor)); algY1:=subs(y=Y1,collect(op(1,op(4,res)),y,factor));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdZMEdGKCwqKiZJInRHRigiIiIpSSNZMEdGKCIiJEYxRjEqKCIiJUYxRjBGMSlGMyIiI0YxRjEqJiwmKiZGOEYxRjBGMUYxRjEhIiJGMUYzRjFGMUY7RjE3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdZMUdGKCwqKiYsJiIiIkYxKiYiIiRGMUkidEdGKEYxRjFGMSlJI1kxR0YoRjNGMUYxKiYsJkYyISIiRjFGOUYxKUY2IiIjRjFGMSomRjRGMUY2RjFGMUY0RjE3I0Yu

n:=6: normal(series(subs(Y0=Y0ser(n),algY0),t,n+1)); normal(series(subs(Y1=Y1ser(n),algY1),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIig=

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIig=

tY0:=factor(solve(algY0,t)); tY1:=factor(solve(algY1,t));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR0WTBHRigqJkkjWTBHRigiIiIsKiokKUYvIiIkRjBGMComIiIlRjApRi8iIiNGMEYwKiZGOEYwRi9GMEYwRjhGMCEiIjcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR0WTFHRigsJCooKUkjWTFHRigiIiMiIiIsJkYxRjNGMyEiIkYzLCoqJiIiJEYzKUYxRjhGM0YzKiZGOEYzRjBGM0Y1RjFGM0YzRjNGNUY1NyNGLg==

Do Y0 and Y1 define the same extension of Q(t) ? YES numer(factor(subs(t=tY0,algY1)));

KiYsKComSSNZMEc2IiIiIkkjWTFHRiZGJ0YnRihGJ0YnISIiRicsLiomKUYlIiIjRicpRihGLUYnRicqJkYsRidGKEYnRikqKCIiJEYnRiVGJ0YuRidGJyooRi1GJ0YlRidGKEYnRikqJkYtRidGLkYnRidGJUYpRic=

Y0ser(5);Y1ser(5);

LCwqJiIkTygiIiIpSSJ0RzYiIiImRiVGJSomIiQ/IkYlKUYnIiIlRiVGJSomIiNDRiUpRiciIiRGJUYlKiZGLUYlKUYnIiIjRiVGJSomRjRGJUYnRiVGJQ==

LCwqJiIkXyQiIiIpSSJ0RzYiIiImRiUhIiIqJiIjU0YlKUYnIiIlRiVGKiomIiM7RiUpRiciIiRGJUYqKiYiIiNGJUYnRiVGKkYlRiU=

We recover Y0 series(-1+1/Y1ser(5),t,6);

Ky9JInRHNiIiIiMiIiIiIiVGJSIjQyIiJCIkPyJGJyIkTygiIiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJiIiJw==

res:=numer(factor(subs(t=tY0,algA1)));

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

n:=5: normal(series(subs(Y0=Y0ser(n),A1=subs(x=0,y=0,Qser(n)),op(1,res)),t,n+1)); normal(series(subs(Y0=Y0ser(n),A1=subs(x=0,y=0,Qser(n)),op(2,res)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIic=

KzFJInRHNiIiIikiIiEhI0siIiIhJGciIiIjISRvKCIiJCElR0wiIiUhJilHNyIiJi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIn

A1Y0:=factor(solve(op(1,res),A1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVBMVkwR0YoLCQqKCMiIiIiIilGMSwqKiQpSSNZMEdGKCIiJEYxRjEqJiIiJUYxKUY2IiIjRjFGMSomRjtGMUY2RjFGMUY7RjFGMSwoRjRGMSomRjlGMUY2RjEhIiJGOUYxRjFGMTcjRi4=

A simpler expression involving t AIY0t:=factor(tY0*A1Y0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZBSVkwdEdGKCwkKigjIiIiIiIpRjFJI1kwR0YoRjEsKCokKUYzIiIkRjFGMSomIiIlRjFGM0YxISIiRjlGMUYxRjE3I0Yu

factor(algY0+Y0);

KiYsKiokKUkjWTBHNiIiIiQiIiJGKSomIiIlRikpRiYiIiNGKUYpKiZGLUYpRiZGKUYpRi1GKUYpSSJ0R0YnRik=

Back to Q(0,y) eqcat;

LCwqKilJInlHNiIiIiMiIiIpLCZGJUYoRihGKEYnRigpSSJ0R0YmRidGKClJIlFHRiZGJ0YoRigqJiwsKihGJ0YoRixGKClGJSIiJEYoRigqKEYnRihGLEYoRiRGKCEiIiokRiRGKEY1KiZGJ0YoRixGKEY1RiVGKEYoRi5GKEYoKihGJ0YoRixGKEkjQTFHRiZGKEYoRjZGKEYlRjU=

eqQY:=collect(numer(factor(subs(A1=A1Y0,t=tY0,Y0=Y,eqcat))),Q,factor);

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

QYsolens:=simplify(factor([solve(eqQY,Q)]),symbolic);

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

map(factor,map(series,%,Y,2));

NyQrJUkiWUc2Ii1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyIiIiIiISspRiQsJCoqIiIlRixJInlHRiUhIiIsJkYyRixGLEYzRiwpLCZGMkYsRixGLCIiI0YzRiwhIiMsJCoqRjFGLCwqKiQpRjIiIiRGLEYsKiZGPkYsKUYyRjdGLEYzKiZGN0YsRjJGLEYsRixGM0YsRkBGM0Y1RjNGM0YzRiZGLQ==

QY:=subs(sqrt(delta)=sqrt(-Y^2*y-2*Y*y+1),collect(subs(sqrt(-Y^2*y-2*Y*y+1)=sqrt(delta),op(1,QYsolens)),delta,factor));

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

n:=8: normal(series(subs(Y=Y0ser(n),QY),t,n+1));

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

subs(x=0,Qser(6));

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

liste:=subs(x=1/x,y=1/y,1/y+x/y+2*x+x*y+2*y+y/x+1/x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwwSSJ5R0YoIiIiKiZGL0YwSSJ4R0YoISIiRjAqJiIiI0YwRjJGM0YwKiZGMkYzRi9GM0YwKiZGNUYwRi9GM0YwKiZGMkYwRi9GM0YwRjJGMDcjRi4=

K:=collect((t*liste-1)*x*y,t,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCYqJiwwKiYpSSJ4R0YoIiIjIiIiSSJ5R0YoRjVGNSomRjNGNSlGNkY0RjVGNSokRjJGNUY1KiRGOEY1RjUqJkY0RjVGM0Y1RjUqJkY0RjVGNkY1RjVGNUY1RjVJInRHRihGNUY1KiZGM0Y1RjZGNSEiIjcjRi4=

The functional equation eqQ:=K*Q(x,y)+x*y-subs(x=0,K)*Q(0,y) -subs(y=0,K)*Q(x,0)+subs(x=0,y=0,K)*Q(0,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcVFHRigsLComLCYqJiwwKiYpSSJ4R0YoIiIjIiIiSSJ5R0YoRjdGNyomRjVGNylGOEY2RjdGNyokRjRGN0Y3KiRGOkY3RjcqJkY2RjdGNUY3RjcqJkY2RjdGOEY3RjdGN0Y3RjdJInRHRihGN0Y3KiZGNUY3RjhGNyEiIkY3LUkiUUdGKDYkRjVGOEY3RjdGQEY3KigsKEY8RjdGPkY3RjdGN0Y3Rj9GNy1GQzYkIiIhRjhGN0ZBKigsKEY7RjdGPUY3RjdGN0Y3Rj9GNy1GQzYkRjVGSUY3RkEqJkY/RjctRkM2JEZJRklGN0Y3NyNGLg==

Qser:=proc(n) option remember: if n=0 then 1 else convert(normal(series(subs(Q(x,y)=Qser(n-1),Q(x,0)=subs(y=0,Qser(n-1)), Q(0,y)=subs(x=0,Qser(n-1)), Q(0,0)=subs(x=0,y=0,Qser(n-1)), expand(eqQ/x/y+Q(x,y))),t,n+1)),polynom): fi: end: Qser(4);

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

Rser:=proc(n) convert(normal(series(subs(y=0,K)*subs(y=0,Qser(n-1)),t,n+1)),polynom):end: Sser:=proc(n) convert(normal(series(subs(x=0,K)*subs(x=0,Qser(n-1)),t,n+1)),polynom):end: Rser(4);Sser(4);

LCoqJiwoKiQpSSJ4RzYiIiIjIiIiRioqJkYpRipGJ0YqRipGKkYqRipJInRHRihGKkYqKihGJEYqRidGKilGLEYpRipGKiooRiRGKiwoRiVGKkYnRioiIiVGKkYqKUYsIiIkRipGKiooRiRGKiwqKiQpRidGM0YqRioqJkYzRipGJkYqRioqJiIjNkYqRidGKkYqIiInRipGKilGLEYxRipGKg==

LCoqJiwoKiQpSSJ5RzYiIiIjIiIiRioqJkYpRipGJ0YqRipGKkYqRipJInRHRihGKkYqKihGJ0YqRiRGKilGLEYpRipGKiooRiRGKiwoRiVGKkYnRioiIiVGKkYqKUYsIiIkRipGKiooRiRGKiwqKiQpRidGM0YqRioqJkYzRipGJkYqRioqJiIjNkYqRidGKkYqIiInRipGKilGLEYxRipGKg==

Checking the functional equation n:=10: normal(series(Qser(n)*K+x*y-Rser(n)-Sser(n)+subs(x=0,Rser(n)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Invariants can be constructed from those of the previous model Jy:=map(normal,subs(y=1/y,t^2/y^2-(1+2*t)*t/y-t*y*(1+3*t)-(1+3*t)*(1+4*t)/(1+y)+(1+3*t)^2/(1+y)^2));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKeUdGKCwsKiYpSSJ0R0YoIiIjIiIiKUkieUdGKEYyRjNGMyooLCYqJkYyRjNGMUYzRjNGM0YzRjNGMUYzRjVGMyEiIiooRjFGM0Y1RjksJkYzRjMqJiIiJEYzRjFGM0YzRjNGOSoqRjtGMywmKiYiIiVGM0YxRjNGM0YzRjNGM0Y1RjMsJkY1RjNGM0YzRjlGOSooKUY7RjJGM0Y0RjMpRkJGMkY5RjM3I0Yu

A decoupling function Gy:=-y^2+y*(1+t)/t+(1+3*t)/t/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwoKiQpSSJ5R0YoIiIjIiIiISIiKihGMUYzLCZJInRHRihGM0YzRjNGM0Y3RjRGMyooLCZGM0YzKiYiIiRGM0Y3RjNGM0YzRjdGNEYxRjRGMzcjRi4=

Jy; Ly:=S(y)-Gy;

LCwqJilJInRHNiIiIiMiIiIpSSJ5R0YmRidGKEYoKigsJiomRidGKEYlRihGKEYoRihGKEYlRihGKkYoISIiKihGJUYoRipGLiwmRihGKComIiIkRihGJUYoRihGKEYuKipGMEYoLCYqJiIiJUYoRiVGKEYoRihGKEYoRipGKCwmRipGKEYoRihGLkYuKigpRjBGJ0YoRilGKClGN0YnRi5GKA==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNMeUdGKCwqLUkiU0dGKDYjSSJ5R0YoIiIiKiQpRjIiIiNGM0YzKihGMkYzLCZJInRHRihGM0YzRjNGM0Y5ISIiRjoqKCwmRjNGMyomIiIkRjNGOUYzRjNGM0Y5RjpGMkY6Rjo3I0Yu

We construct an expression with no pole at y=0 nor y=-1. We begin by getting rid of the pole at y=-1 factor(subs(x=0,K));

KiYpLCZJInlHNiIiIiJGJ0YnIiIjRidJInRHRiZGJw==

We note that L(y)-L(-1) has a double zero at y=-1 collect(subs(S(y)=subs(x=0,K)*Q(0,y),Ly)-subs(y=-1,S(-1)=0,Ly),Q,factor);

LCYqKCksJkkieUc2IiIiIkYoRigiIiNGKEkidEdGJ0YoLUkiUUdGJzYkIiIhRiZGKEYoKipGJEYoLCgqJkYqRihGJkYoRigqJiIiJEYoRipGKCEiIkYoRjRGKEYqRjRGJkY0Rig=

expr:=(Ly-subs(y=-1,S(-1)=0,Ly))*Jy;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVleHByR0YoKiYsMC1JIlNHRig2I0kieUdGKCIiIiokKUYzIiIjRjRGNCooRjNGNCwmSSJ0R0YoRjRGNEY0RjRGOiEiIkY7KigsJkY0RjQqJiIiJEY0RjpGNEY0RjRGOkY7RjNGO0Y7RjRGOyomRjlGNEY6RjtGOyomRj1GNEY6RjtGO0Y0LCwqJilGOkY3RjRGNkY0RjQqKCwmKiZGN0Y0RjpGNEY0RjRGNEY0RjpGNEYzRjRGOyooRjpGNEYzRjtGPUY0RjsqKkY9RjQsJiomIiIlRjRGOkY0RjRGNEY0RjRGM0Y0LCZGM0Y0RjRGNEY7RjsqKClGPUY3RjRGNkY0KUZNRjdGO0Y0RjQ3I0Yu

We now have a pole at y=0, of order 2 eqcat:=expr-a*Ly^2-b*Ly-c;

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

normal(series(eqcat,y,1));

KylJInlHNiIsJCooKSwmIiIiRikqJiIiJEYpSSJ0R0YkRilGKSIiI0YpLCYqJClGLEYtRikhIiJJImFHRiRGKUYpRjBGMUYxISIjKihGKEYpLCwqJi1JIlNHRiQ2IyIiIUYpRjBGKUYxKihGLUYpRjdGKUYyRilGKSomIiImRilGMEYpRilJImJHRiRGKSomRi1GKUYsRilGKUYpRixGMUYxLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRilGOg==

asol:=solve(op(1,%),a);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhc29sR0YoKiQpSSJ0R0YoIiIjIiIiNyNGLg==

normal(series(subs(a=asol,eqcat),y,1));

KydJInlHNiIqKCwmIiIiRicqJiIiJEYnSSJ0R0YkRidGJ0YnLCoqJi1JIlNHRiQ2IyIiIUYnKUYqIiIjRidGJyomIiImRidGMUYnRidJImJHRiRGJyomRjJGJ0YqRidGJ0YnRiohIiJGNy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YnRjA=

bsol:=solve(op(1,%),b);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVic29sR0YoLCQqJiwoKiYtSSJTR0YoNiMiIiEiIiJJInRHRihGNkY2KiYiIiZGNkY3RjZGNiIiI0Y2RjZGN0Y2ISIiNyNGLg==

subs(y=0,Sser(4));

LCgqJiIiJyIiIilJInRHNiIiIiVGJUYlKiZGKUYlKUYnIiIkRiVGJUYnRiU=

normal(series(subs(a=asol,b=bsol,eqcat),y,2));

KydJInlHNiIqJkkidEdGJCEiIiw0KigiIiQiIiItLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSJTR0YkNiMiIiFGKylGJkYqRitGKyooIiImRistRjNGNEYrRjZGK0YrKiZGLEYrKUYmIiIjRitGKyooRjxGK0Y5RitGO0YrRisqJiIjUkYrRjZGK0YrKiZJImNHRiRGK0YmRitGJyomIiNNRitGO0YrRisqJiIjNUYrRiZGK0YrRitGK0YrRjUtSSJPR0YvNiNGK0Yr

csol:=solve(coeff(%,y,0),c);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVjc29sR0YoKiZJInRHRighIiIsMiooIiIkIiIiLS1JIkRHRiU2I0kiU0dGKDYjIiIhRjQpRi9GM0Y0RjQqKCIiJkY0LUY5RjpGNEY8RjRGNComRjVGNClGLyIiI0Y0RjQqKEZCRjRGP0Y0RkFGNEY0KiYiI1JGNEY8RjRGNComIiNNRjRGQUY0RjQqJiIjNUY0Ri9GNEY0RjRGNEY0NyNGLg==

collect(%,S,factor);

LCgqKEkidEc2IiIiIiwmKiYiIiZGJkYkRiZGJiIiI0YmRiYtSSJTR0YlNiMiIiFGJkYmKihGJEYmLCZGJkYmKiYiIiRGJkYkRiZGJkYmLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YsRi1GJkYmKihGMEYmLCgqJiIjOEYmKUYkRipGJkYmKiYiIihGJkYkRiZGJkYmRiZGJkYkISIiRiY=

eqcat1:=numer(factor(subs(a=asol,b=bsol,c=csol,eqcat)));

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

Can this be true ? n:=20: normal(series(subs(S(y)=Sser(n), S(0)=subs(y=0,Sser(n)), D(S)(0)=subs(y=0,diff(Sser(n),y)),((D@@2)(S))(0)=subs(y=0,diff(Sser(n),y$2)), eqcat1),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0A=

Back to Q(0,y) (denoted here Q(y) to lighten notation) Sy:=subs(x=0,K)*Q(y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNTeUdGKCooLCgqJClJInlHRigiIiMiIiJGNComRjNGNEYyRjRGNEY0RjRGNEkidEdGKEY0LUkiUUdGKDYjRjJGNDcjRi4=

factor(subs(S(y)=Sy,S(0)=subs(y=0,Sy),(D(S))(0)=limit(diff(Sy,y),y=0),eqcat1));

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

eqcatQ:=collect(op(4,%),Q(y),factor);

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

factor(series(eqcatQ,y=0,3));

KydJInlHNiIsNiooKUkidEdGJCIiJCIiIi1JIlFHRiQ2IyIiIUYqLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YsRi1GKkYqKigiIiNGKkYnRiopRitGN0YqRioqKCNGKUY3RiotLS1JI0BAR0YyNiRGMUY3RjVGLUYqKUYoRjdGKiEiIiooI0YqRjdGKkY7RipGKEYqRkEqKCIjNkYqRi9GKkZARipGQSooIiIlRipGKEYqRi9GKkZBKihGN0YqRihGKkYrRipGKkYrRioqJkY3RipGKEYqRkFGKkZBRjctSSJPR0YyNiNGKkYp

eqcat:=collect(subs(Q(y)=Q,Q(0)=A1,(D(Q))(0)=A2,eqcatQ),[Q,A1,A2],factor);

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

Two series cancel the following derivative edQ:=factor(diff(eqcat,Q));subs(t=0,%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlZFFHRigsSioqIiIjIiIiSSJRR0YoRjEpSSJ0R0YoIiIkRjEpSSJ5R0YoIiImRjFGMSoqIiIpRjFGMkYxRjNGMSlGNyIiJUYxRjEqKEkjQTFHRihGMUYzRjEpRjdGNUYxISIiKioiIzdGMUYyRjFGM0YxRj9GMUYxKiYpRjRGMEYxRjZGMUYxKipGMEYxRj5GMUYzRjEpRjdGMEYxRkAqKkY6RjFGMkYxRjNGMUZGRjFGMSooRjBGMUZERjFGO0YxRjEqKEY+RjFGM0YxRjdGMUZAKipGMEYxRjJGMUYzRjFGN0YxRjEqJkZERjFGP0YxRkAqJkY0RjFGO0YxRkAqJkZERjFGRkYxRkAqKEY1RjFGNEYxRj9GMUZAKigiIzZGMUZERjFGN0YxRkAqJkY0RjFGRkYxRjEqJkY1RjFGREYxRkAqKEY8RjFGNEYxRjdGMUZAKiRGRkYxRjFGNEZANyNGLg==

KiQpSSJ5RzYiIiIjIiIi

Let us compute their expansions y0:='y0': Y0ser:=proc(n) add(y0[i]*t^i,i=1..n): end: n:=1: normal(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)),A2=subs(x=0,y=0,diff(Qser(n),y)), y=Y0ser(n), edQ),t,n+1));

KydJInRHNiIhIiIiIiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJiIiIw==

We need series in s=sqrt(t) instead y0:='y0': Y0ser:=proc(n) add(y0[i]*s^i,i=1..n): end: n:=2: normal(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)),A2=subs(x=0,y=0,diff(Qser(n),y)), y=Y0ser(n), t=s^2,edQ),s,n+1));

KydJInNHNiIsJiokKSZJI3kwR0YkNiMiIiIiIiNGK0YrRishIiJGLC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiRGKiIiJA==

y0[1]:=1: From now on we execute the next two lines several times to obtain coefficients of Y0 n:=n+1: normal(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)),A2=subs(x=0,y=0,diff(Qser(n),y)),y=Y0ser(n), t=s^2,edQ),s,n+1));

KydJInNHNiIsJiIkRyIhIiIqJiIiIyIiIiZJI3kwR0YkNiMiIihGKkYqIiIpLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRioiIio=

y0[n-1]:=solve(coeff(%,s,n),y0[n-1]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTBHRig2IyIiKCIjazcjRjE=

Y0ser(n);

LDJJInNHNiIiIiIqJiIiI0YlKUYjRidGJUYlKiYiIiRGJSlGI0YqRiVGJSomIiImRiUpRiMiIiVGJUYlKiYjIiNCRidGJSlGI0YtRiVGJSomIiNHRiUpRiMiIidGJUYlKiYiI2tGJSlGIyIiKEYlRiUqJiZJI3kwR0YkNiMiIilGJSlGI0ZARiVGJQ==

Of course, the other root is obtained by s-> -s

disc:=collect(factor(discrim(eqcat,Q)),y,factor);

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

The discriminant of disc (with respect to y) vanishes res:=factor(discrim(disc,y));nops(res);

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

IiIm

The factor that is cubed cannot vanish (constant term 1), so here is our first equation between A1 and A2 alg1:=op(1,op(4,res)):nops(alg1);

IiMhKQ==

n:=10: normal(series(subs(A1=subs(x=0,y=0,Qser(n)), A2=subs(y=0,diff(subs(x=0,Qser(n)),y)),alg1),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

A polynomial P in y has d double roots (here, d=2) if the first generalized discriminants defined below vanish, for k=0, 1,... , d-1 with(LinearAlgebra): disciter:=proc(P,k) local M,i,j,d: d:=degree(P,y): M:=Matrix(2*d-1-2*k,2*d-1-2*k): for i from 1 to d-1-k do for j from 0 to min(d, 2*d-1-2*k-i) do M[i,i+j]:=coeff(P,y,d-j):od:od: for i from 0 to d-k- 1 do for j from 1 to min( 2*d-2*k-1-i) do M[i+d-k,i+j]:=coeff(diff(P,y),y,d-j): od: od: M:end: The first one is just the usual discriminant factor(Determinant(disciter(disc,0)));

LCQqLCIodyZbNSIiIilJInRHNiIiI1BGJSksJkYlRiUqJiIiJEYlRidGJUYlIiM+RiUpLFx1KigiIztGJSlJI0ExR0YoIiInRiUpRiciIzZGJUYlKigiJW86RiUpRjQiIiZGJSlGJyIjNUYlISIiKioiJHcmRiUpRjQiIiVGJUkjQTJHRihGJUY8RiVGPiooRkBGJUY6RiUpRiciIipGJUY+KioiJCM+RiVGQUYlRkNGJUZFRiVGPiooIiIpRiVGOkYlKUYnRkpGJUY+KigiJm96JEYlRkFGJUZFRiVGJSoqIiZLdyRGJSlGNEYtRiVGQ0YlRkVGJUYlKioiJTdwRiUpRjQiIiNGJSlGQ0ZURiVGRUYlRiUqKCImT2YjRiVGQUYlRktGJUYlKioiJm9qI0YlRlBGJUZDRiVGS0YlRiUqKiIlM1lGJUZTRiVGVUYlRktGJUYlKigiJTNTRiVGQUYlKUYnIiIoRiVGJSoqIiUrW0YlRlBGJUZDRiVGaG5GJUYlKigiJi9oI0YlRlBGJUZLRiVGJSoqIiRvKEYlRlNGJUZVRiVGaG5GJUYlKioiJ3dsWEYlRlNGJUZDRiVGS0YlRj4qKiInI3pEI0YlRjRGJUZVRiVGS0YlRj4qKCImW3cjRiUpRkNGLUYlRktGJUY+KigiJFciRiVGQUYlKUYnRjVGJUY+KioiI2tGJUZQRiVGQ0YlRmlvRiVGJSooIicnKkc3RiVGUEYlRmhuRiVGJSoqIidzcVlGJUZTRiVGQ0YlRmhuRiVGPioqIidzTUJGJUY0RiVGVUYlRmhuRiVGPiooRmVvRiVGZm9GJUZobkYlRj4qJkZBRiUpRidGO0YlRiUqKCInQzU3RiVGUEYlRmlvRiVGJSoqIidPTjpGJUZTRiVGQ0YlRmlvRiVGPiooIic/PDVGJUZTRiVGaG5GJUY+KioiJk86KUYlRjRGJUZVRiVGaW9GJUY+KioiJj9GJUYlRjRGJUZDRiVGaG5GJUYlKigiJTsjKkYlRmZvRiVGaW9GJUY+KigiJj86IkYlRlVGJUZobkYlRiUqKCImQ1wlRiVGUEYlRmRwRiVGJSoqIiZzVSJGJUZTRiVGQ0YlRmRwRiVGPiooIid3KlsiRiVGU0YlRmlvRiVGPioqIiUlKSoqRiVGNEYlRlVGJUZkcEYlRj4qKiInO3c9RiVGNEYlRkNGJUZpb0YlRiUqKCIlQzVGJUZmb0YlRmRwRiVGPiooIiZjOSZGJUZVRiVGaW9GJUYlKigiJSc0J0YlRlBGJSlGJ0ZCRiVGJSoqIiQ7J0YlRlNGJUZDRiVGY3JGJUYlKigiJy0lNCJGJUZTRiVGZHBGJUY+KioiJEciRiVGNEYlRlVGJUZjckYlRj4qKiInQ1E4RiVGNEYlRkNGJUZkcEYlRiUqKCImKzckRiVGNEYlRmlvRiVGJSooIiZDaSRGJUZVRiVGZHBGJUYlKigiJys7OUYlRkNGJUZpb0YlRj4qKCIkNyJGJUZQRiUpRidGLUYlRiUqKkZKRiVGU0YlRkNGJUZkc0YlRj4qKCImIypSJUYlRlNGJUZjckYlRj4qKiImY1EjRiVGNEYlRkNGJUZjckYlRiUqKCImP3EqRiVGNEYlRmRwRiVGJSooIiUra0YlRlVGJUZjckYlRiUqKCInZ2A2RiVGQ0YlRmRwRiVGPiooIiVLa0YlRlNGJUZkc0YlRj4qKiIlIypIRiVGNEYlRkNGJUZkc0YlRj4qKCImb2onRiVGNEYlRmNyRiVGJSooIiRfKEYlRlVGJUZkc0YlRj4qKCImQzMjRiVGQ0YlRmNyRiVGPiomIiZ2TCRGJUZkcEYlRj4qKCIkTyhGJUZTRiUpRidGVEYlRiUqKiIkUydGJUY0RiVGQ0YlRl51RiVGPiooIiY7PyJGJUY0RiVGZHNGJUYlKigiJGciRiVGVUYlRl51RiVGPiooIiVLSUYlRkNGJUZkc0YlRiUqJiImKydHRiVGY3JGJUY+KihGZHVGJUZTRiVGJ0YlRiUqKkZbcEYlRjRGJUZDRiVGJ0YlRiUqKCIlMzlGJUY0RiVGXnVGJUY+KihGMkYlRlVGJUYnRiVGJSooIiQzJ0YlRkNGJUZedUYlRiUqJiIlIW8mRiVGZHNGJUY+KiZGMkYlRlNGJUY+KigiJD8kRiVGNEYlRidGJUY+KihGW3BGJUZDRiVGJ0YlRj4qJiIkcydGJUZedUYlRiUqJiIjS0YlRjRGJUYlKiZGZHVGJUYnRiVGJUYyRj5GVEYlKSwsKihGVEYlRjRGJUZkc0YlRiUqJkZDRiVGZHNGJUYlKiYiIzhGJUZedUYlRiUqJkZpbkYlRidGJUYlRiVGJUY7RiVGPg==

factor(%/alg1^2);

LCQqKiIodyZbNSIiIilJInRHNiIiI1BGJSksJkYlRiUqJiIiJEYlRidGJUYlIiM+RiUpLCwqKCIiI0YlSSNBMUdGKEYlKUYnRi1GJUYlKiZJI0EyR0YoRiVGNEYlRiUqJiIjOEYlKUYnRjJGJUYlKiYiIihGJUYnRiVGJUYlRiUiIiZGJSEiIg==

res:=factor(Determinant(disciter(disc,1))):nops(%);

IiIn

seq(indets(op(i,res)),i=1..nops(res)); seq(nops(op(i,res)),i=1..nops(res));

Nig8IjwjSSJ0RzYiRiQ8JUkjQTFHRiZJI0EyR0YmRiVGJ0Yn

NigiIiIiIiNGJCIkaCIiIyEpRiQ=

op(6,res);

KiQpLCwqKCIiIyIiIkkjQTFHNiJGJylJInRHRikiIiRGJ0YnKiZJI0EyR0YpRidGKkYnRicqJiIjOEYnKUYrRiZGJ0YnKiYiIihGJ0YrRidGJ0YnRiciIiVGJw==

However, one factor of res is precisely alg1. Hence saying that res vanishes does not bring any new information rem(res,alg1,A1);

IiIh

However, we can obtain other equations between A1 and A2 by saying that the Sylvester matrix of disc and its derivative (which has dimension 19) has rank at most 16. Here are the two matrices in degree 3 disciter(a3*y^3+a2*y^2+a1*y+a0,0);disciter(a3*y^3+a2*y^2+a1*y+a0,1);

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyNYLEkpYW55dGhpbmdHRiZGKEYoW2dsISIlISEhIyoiJCIkSSNhM0dGKCwkRi4iIiQiIiFJI2EyR0YoLCRGMiIiI0YvSSNhMUdGKEY1RjM3Iy1JJ01hdHJpeEdGJTYjL0kkJWlkR0YoIjU9ZEQ9LSVRKW9XPQ==

?LinearAlgebra min1:=factor(Minor(disciter(disc,0),1,1)): nops(%);

IiIn

seq(indets(op(i,min1)),i=1..6); seq(nops(op(i,min1)),i=1..6);

Nig8IjwjSSJ0RzYiRiQ8JUkjQTFHRiZJI0EyR0YmRiVGJ0Yn

NigiIiIiIiNGJCIjISkiJEYkRiQ=

rem(min1,alg1,A1);

IiIh

min1:=factor(Minor(disciter(disc,0),2,2)): nops(%);

IiIn

seq(indets(op(i,min1)),i=1..6); seq(nops(op(i,min1)),i=1..6);

Nig8IjwjSSJ0RzYiRiQ8JUkjQTFHRiZJI0EyR0YmRiVGJ0Yn

NigiIiIiIiNGJCIjISkiJEYkRiQ=

rem(min1,alg1,A1);

IiIh

min1:=factor(Determinant(SubMatrix(disciter(disc,0),[1..17],[1..17]))):nops(%);

IiIm

seq(indets(op(i,min1)),i=1..5); seq(nops(op(i,min1)),i=1..5);

Nic8IjwjSSJ0RzYiRiQ8JUkjQTFHRiZJI0EyR0YmRiVGJw==

NiciIiIiIiNGJCIlRTVGJA==

nops(rem(min1,alg1,A1));

IiIn

op(5,min1);

KiQpLCwqKCIiIyIiIkkjQTFHNiJGJylJInRHRikiIiRGJ0YnKiZJI0EyR0YpRidGKkYnRicqJiIjOEYnKUYrRiZGJ0YnKiYiIihGJ0YrRidGJ0YnRidGLEYn

alg2:=op(4,min1): n:=10: normal(series(subs(A1=subs(x=0,y=0,Qser(n)),A2=subs(x=0,y=0,diff(Qser(n),y)),alg2),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Computing an equation for A1 algA1cand:=factor(resultant(alg1,alg2,A2));nops(%);

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

IiIn

Only one factor can vanish (look at the constant term), and we have found the algebraic equation satisfied by A1 n:=14: normal(series(subs(A1=subs(x=0,y=0,Qser(n)),op(1,op(6,algA1cand))),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

Here is the (cubic) equation satisfied by A1 algA1:=collect(op(1,op(6,algA1cand)),A1,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdBMUdGKCwwKigiI2siIiIpSSNBMUdGKCIiJEYxKUkidEdGKCIiJ0YxRjEqKiIjO0YxKUY2RjRGMSwoKiYiI19GMSlGNiIiI0YxRjEqJiIjQ0YxRjZGMUYxRjFGMUYxKUYzRj9GMSEiIiomLCwqJiIkcyNGMSlGNiIiJUYxRjEqJiIkRyRGMUY6RjFGMSomIiNTRjFGPkYxRjEqJkY5RjFGNkYxRkNGMUYxRjFGM0YxRjEqJiIkYSNGMUY6RjFGQyomIiNXRjFGPkYxRkNGTkYxRjFGQzcjRi4=

Now for A2 algA2cand:=factor(resultant(alg1,alg2,A1));nops(%);

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

IiIn

Only one factor can vanish (look at the constant term), and we have found the algebraic equation satisfied by A2 n:=14: normal(series(subs(A2=subs(x=0,y=0,diff(Qser(n),y)),op(1,op(6,algA2cand))),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

Here is the (cubic) equation satisfied by A2 algA2:=collect(op(1,op(6,algA2cand)),A2,factor);

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

res:=factor(resultant(disc,algA1,A1)): nops(%);

IiIk

op(1,res);op(2,res);nops(op(3,res));

ISIi

KiQpSSJ0RzYiIiM3IiIi

IiR0Jw==

res1:=factor(resultant(op(3,res),algA2,A2)):nops(res1);

IiIo

seq(indets(op(i,res1)),i=1..nops(res1)); seq(nops(op(i,res1)),i=1..nops(res1));

Nik8IjwjSSJ0RzYiRiQ8JEYlSSJ5R0YmRidGJ0Yn

NikiIiIiIiNGJCIlUzZGJEYkRiQ=

n:=14: normal(series(subs(y=Y0ser(n),t=s^2,op(4,res1)),s,n+1));

KydJInNHNiIiIiUiIzktSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

op(5,res1);

KiQpLDAqJilJInRHNiIiIiMiIiIpSSJ5R0YoIiIkRipGKiooIiIlRipGJkYqKUYsRilGKiEiIiooIiIpRipGJkYqRixGKkYxKihGKUYqRidGKkYwRipGMSomIiM3RipGJkYqRjEqJkYvRipGJ0YqRjFGLEYqRilGKg==

n:=10: normal(series(subs(y=Y0ser(n),t=s^2,op(1,op(5,res1))),s,n+1));

KzlJInNHNiIiIiJGJSEiIyIiIyIiJEYoISIqIiIlI0YpRiciIiYhIzciIichIz4iIigsJiIkeSIhIiImSSN5MEdGJDYjIiIpRiVGNywmIiRCJUYzJkY1NiMiIipGJUY8LCYiJVQ1RjMmRjU2IyIjNUYlRkEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNg==

op(6,res1);

KiQpLC4qJkkidEc2IiIiIilJInlHRiciIiRGKEYoKiZGJkYoKUYqIiIjRihGKCooRitGKEYmRihGKkYoISIiKiZGK0YoRiZGKEYoRipGMEYoRigiIiVGKA==

n:=10: normal(series(subs(y=Y0ser(n),t=s^2,op(1,op(6,res1))),s,n+1));

KztJInNHNiIiIiIiIiEhIiJGJUYlIiIjISInIiIkISM1IiIlIyEjSkYoIiImISNGIiInIyEkNiJGKCIiKCwmIiNGRiUmSSN5MEdGJDYjIiIpRidGOiwmIyIkJj5GKEYlJkY4NiMiIipGJ0ZALCgiJGcoRiUqJkYqRiVGN0YlRicmRjg2IyIjNUYnRkYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNg==

op(7,res1);

KiQpLCwqKCIiIyIiIkkidEc2IkYnKUkieUdGKSIiJEYnRicqKEYmRidGKEYnKUYrRiZGJ0YnKigiIiVGJ0YoRidGK0YnRicqJEYuRichIiJGKEYnRjBGJw==

n:=10: normal(series(subs(y=Y0ser(n),t=s^2,op(1,op(7,res1))),s,n+1));

KylJInNHNiIsJiIkJ0giIiIqJiIiI0YnJkkjeTBHRiQ2IyIiKUYnISIiIiIqLCYjIiUmKkciIiVGJyomRilGJyZGKzYjRi9GJ0YuIiM1LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiciIzY=

algY:=subs(y=Y,collect(op(1,op(7,res1)),y,factor));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdZR0YoLCoqKCIiIyIiIkkidEdGKEYxKUkiWUdGKCIiJEYxRjEqJiwmKiZGMEYxRjJGMUYxRjEhIiJGMSlGNEYwRjFGMSooIiIlRjFGMkYxRjRGMUYxRjJGMTcjRi4=

tY:=factor(solve(algY,t));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN0WUdGKComKUkiWUdGKCIiIyIiIiwqKiZGMUYyKUYwIiIkRjJGMiomRjFGMkYvRjJGMiomIiIlRjJGMEYyRjJGMkYyISIiNyNGLg==

JSFH

We recycle the parametrisation of the previous model tZ := Z/(Z^3+4*Z^2+2*Z+2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN0WkdGKComSSJaR0YoIiIiLCoqJClGLyIiJEYwRjAqJiIiJUYwKUYvIiIjRjBGMComRjhGMEYvRjBGMEY4RjAhIiI3I0Yu

with(gfun): Zser:=op(algeqtoseries(numer(t-tZ),t,Z,7,true));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVac2VyR0YoKzNJInRHRigiIiMiIiIiIiVGMCIjQyIiJCIkPyJGMiIkTygiIiYiJVdYIiInIiZrJ0giIigtSSJPR0YlNiNGMSIiKTcjRi4=

We recover the expression of the excursion GF numer(factor(subs(t=tZ,algA1)));

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

A1Z:=factor(solve(op(1,%),A1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBMVpHRigsJCooIyIiIiIiKUYxLCoqJClJIlpHRigiIiRGMUYxKiYiIiVGMSlGNiIiI0YxRjEqJkY7RjFGNkYxRjFGO0YxRjEsKEY0RjEqJkY5RjFGNkYxISIiRjlGMUYxRjE3I0Yu

A1Zt:=factor(A1Z*tZ/t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVBMVp0R0YoLCQqKiMiIiIiIilGMSwoKiQpSSJaR0YoIiIkRjFGMSomIiIlRjFGNkYxISIiRjlGMUYxRjZGMUkidEdGKEY6RjE3I0Yu

For A2 res:=numer(factor(subs(t=tZ,algA2)));nops(%);

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

IiIj

subs(Z=0,op(2,res));

IiIl

A2Z:=factor(solve(op(1,res),A2));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBMlpHRigsJCouIyIiIiIkYyNGMUkiWkdGKEYxLCZGM0YxIiIjRjFGMSwqKiQpRjMiIiRGMUYxKiYiIiVGMSlGM0Y1RjFGMSomRjVGMUYzRjFGMUY1RjFGMSwsKiQpRjMiIiZGMUYxKiYiI0dGMSlGM0Y7RjFGMSomRjtGMUY4RjFGMSomIiNjRjFGPEYxISIiIiNLRjFGMSksJkYzRjFGMUYxRjVGSEYxNyNGLg==

A2Zt:=factor(A2Z*tZ/t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVBMlp0R0YoLCQqLiMiIiIiJGMjRjEpSSJaR0YoIiIjRjEsJkY0RjFGNUYxRjEsLCokKUY0IiImRjFGMSomIiNHRjEpRjQiIiVGMUYxKiZGPkYxKUY0IiIkRjFGMSomIiNjRjFGM0YxISIiIiNLRjFGMSksJkY0RjFGMUYxRjVGREkidEdGKEZERjE3I0Yu

Back to Q(0,y) eqcat;

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

eqQZ:=collect(numer(factor(subs(A1=A1Z,A2=A2Z,t=tZ,eqcat))),Q,factor);

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

nops(factor(%));

IiQ2Ig==

factor(discrim(eqQZ,Q));

LCQqLCImJVE7IiIiKSwqKiQpSSJaRzYiIiIkRiVGJSomIiIlRiUpRioiIiNGJUYlKiZGMEYlRipGJUYlRjBGJUYwRiUsMiomKUYqRi5GJUkieUdGK0YlRiUqKCIiJ0YlRilGJUY1RiVGJSomRi9GJSlGNUYwRiUhIiIqKCIiKUYlRi9GJUY1RiVGJUYtRjoqKEYuRiVGKkYlRjVGJUYlKiZGPEYlRipGJUY6Ri5GOkYlKSwuKiZGL0YlRjVGJUYlKiZGKkYlRjlGJUYlKiRGL0YlRjoqKEYwRiVGKkYlRjVGJUYlKiZGLEYlRipGJUY6RjBGOkYwRiUpLCpGQUYlRj1GJSomRjBGJUY5RiVGOkYqRiVGMEYlRjo=

QYsolens:=simplify(factor([solve(eqQZ,Q)]),symbolic): map(normal,map(series,%,Z,2));

NyQrJUkiWkc2Ii1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyIiIiEiIispRiQsJCooIiIpRixJInlHRiVGLCksJkYyRixGLEYsIiIlRi1GLSEiJCwkKipGNUYsLCwqJClGMkY1RixGLComIiIkRiwpRjJGPUYsRiwqJiIiKEYsKUYyIiIjRixGLSomRjVGLEYyRixGLEYsRixGLEYyRi1GM0YtRiwhIiNGJkYt

indets(QYsolens);

PCVJIlpHNiJJInlHRiQqJCksKComKUYjIiIjIiIiKUYlRitGLEYsKiYsKiokKUYjIiIlRiwhIiIqJiIiJ0YsKUYjIiIkRixGMyomIiIpRixGKkYsRjMqJkYyRixGI0YsRjNGLEYlRixGLComRjJGLCksJkYjRixGLEYsRitGLEYsI0YsRitGLA==

QZ:=subs(sqrt(delta)=sqrt(-Z^4*y-6*Z^3*y+Z^2*y^2-8*Z^2*y+4*Z^2-4*Z*y+8*Z+4),collect(subs(sqrt(-Z^4*y-6*Z^3*y+Z^2*y^2-8*Z^2*y+4*Z^2-4*Z*y+8*Z+4)=sqrt(delta),op(1,QYsolens)),delta,factor));

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

n:=8: normal(series(subs(Z=Zser,QZ),t,n+1));

Ky9JInRHNiIiIiIiIiFJInlHRiRGJSwoKiQpRiciIiNGJUYlRidGJSIiJUYlRissKiokKUYnIiIkRiVGJSomRjBGJUYqRiVGJSomIiM2RiVGJ0YlRiUiIidGJUYwLCwqJClGJ0YsRiVGJSomRjRGJUYvRiVGJSomIiNCRiVGKkYlRiUqJiIjTUYlRidGJUYlIiNbRiVGLC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIm

subs(x=0,Qser(3));

LCoiIiJGIyomSSJ0RzYiRiNJInlHRiZGI0YjKiYsKCokKUYnIiIjRiNGI0YnRiMiIiVGI0YjKUYlRixGI0YjKiYsKiokKUYnIiIkRiNGIyomRjNGI0YrRiNGIyomIiM2RiNGJ0YjRiMiIidGI0YjKUYlRjNGI0Yj

collect((-Z^4*y-6*Z^3*y+Z^2*y^2-8*Z^2*y+4*Z^2-4*Z*y+8*Z+4),Z,factor);

LCwiIiUiIiIqJilJIlpHNiJGI0YkSSJ5R0YoRiQhIiIqKCIiJ0YkRilGJClGJyIiJEYkRioqJiwoKiQpRikiIiNGJEYkKiYiIilGJEYpRiRGKkYjRiRGJClGJ0YzRiRGJComLCYqJkYjRiRGKUYkRipGNUYkRiRGJ0YkRiQ=

liste:=x+x*y+2*y+y/x+2/x+1/x/y+1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwwSSJ4R0YoIiIiKiZGL0YwSSJ5R0YoRjBGMEYyIiIjKiZGMkYwRi8hIiJGMCokRi9GNUYzKiZGL0Y1RjJGNUYwKiRGMkY1RjA3I0Yu

K:=collect((t*liste-1)*x*y,t,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCYqJiwwKiZJInhHRigiIiNJInlHRihGMyIiIiomRjJGM0Y0RjVGNSomRjJGNUY0RjNGMyokRjRGM0Y1RjJGNUY0RjNGNUY1RjVJInRHRihGNUY1KiZGMkY1RjRGNSEiIjcjRi4=

The functional equation eqQ:=K*Q(x,y)+x*y-subs(x=0,K)*Q(0,y) -subs(y=0,K)*Q(x,0)+subs(x=0,y=0,K)*Q(0,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcVFHRigsLComLCYqJiwwKiZJInhHRigiIiNJInlHRihGNSIiIiomRjRGNUY2RjdGNyomRjRGN0Y2RjVGNSokRjZGNUY3RjRGN0Y2RjVGN0Y3RjdJInRHRihGN0Y3KiZGNEY3RjZGNyEiIkY3LUkiUUdGKDYkRjRGNkY3RjdGPEY3KigsKEY6RjdGNkY1RjdGN0Y3RjtGNy1GPzYkIiIhRjZGN0Y9KigsJkY0RjdGN0Y3RjdGO0Y3LUY/NiRGNEZFRjdGPSomRjtGNy1GPzYkRkVGRUY3Rjc3I0Yu

Qser:=proc(n) option remember: if n=0 then 1 else convert(normal(series(subs(Q(x,y)=Qser(n-1),Q(x,0)=subs(y=0,Qser(n-1)), Q(0,y)=subs(x=0,Qser(n-1)), Q(0,0)=subs(x=0,y=0,Qser(n-1)), expand(eqQ/x/y+Q(x,y))),t,n+1)),polynom): fi: end: Qser(4);

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

Rser:=proc(n) convert(normal(series(subs(y=0,K)*subs(y=0,Qser(n-1)),t,n+1)),polynom):end: Sser:=proc(n) convert(normal(series(subs(x=0,K)*subs(x=0,Qser(n-1)),t,n+1)),polynom):end: Rser(4);Sser(4);

LCoqJiwmSSJ4RzYiIiIiRidGJ0YnSSJ0R0YmRidGJyooRiRGJ0YlRidGKCIiI0YnKihGJEYnLCgqJEYlRipGJ0YlRiciIiZGJ0YnRigiIiRGJyooRiRGJywqKiRGJUYvRidGLUYvRiUiIzgiIipGJ0YnRigiIiVGJw==

LCoqJiwoKiRJInlHNiIiIiMiIiJGJkYoRilGKUYpSSJ0R0YnRilGKSooRiRGKUYmRilGKkYoRigqKEYkRiksKEYlIiImRiYiIiRGLkYpRilGKkYvRikqKEYkRiksKiokRiZGLyIjOUYlIiM9RiYiI0ciIipGKUYpRioiIiVGKQ==

Checking the functional equation n:=10: normal(series(Qser(n)*K+x*y-Rser(n)-Sser(n)+subs(x=0,Rser(n)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Invariants can be constructed from those of the first model with group order 10 (symmetry in a vertical line) Jy:=t^2/y^2-(1+2*t)*t/y-t*y*(1+3*t)-(1+3*t)*(1+4*t)/(1+y)+(1+3*t)^2/(1+y)^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKeUdGKCwsKiZJInRHRigiIiNJInlHRighIiMiIiIqKCwmRjRGNEYwRjFGNEYwRjRGMiEiIkY3KihGMEY0RjJGNCwmRjRGNEYwIiIkRjRGNyooRjlGNCwmRjRGNEYwIiIlRjQsJkYyRjRGNEY0RjdGNyomRjlGMUY+RjNGNDcjRi4=

A decoupling function Gy:=-y-1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwmSSJ5R0YoISIiKiRGL0YwRjA3I0Yu

Jy; Ly:=S(y)-Gy;

LCwqJkkidEc2IiIiI0kieUdGJSEiIyIiIiooLCZGKUYpRiRGJkYpRiRGKUYnISIiRiwqKEYkRilGJ0YpLCZGKUYpRiQiIiRGKUYsKihGLkYpLCZGKUYpRiQiIiVGKSwmRidGKUYpRilGLEYsKiZGLkYmRjNGKEYp

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNMeUdGKCwoLUkiU0dGKDYjSSJ5R0YoIiIiRjJGMyokRjIhIiJGMzcjRi4=

We construct an expression with no pole at y=0 nor y=-1. We begin by getting rid of the pole at y=-1 factor(subs(x=0,K));

KiYsJkkieUc2IiIiIkYmRiYiIiNJInRHRiVGJg==

Again, L(y)-L(-1) has a double root at y=-1 collect(subs(S(y)=subs(x=0,K)*Q(0,y),Ly)-subs(y=-1,S(-1)=0,Ly),Q,factor);

LCYqKCwmSSJ5RzYiIiIiRidGJyIiI0kidEdGJkYnLUkiUUdGJjYkIiIhRiVGJ0YnKiZGJEYoRiUhIiJGJw==

expr:=(Ly-subs(y=-1,S(-1)=0,Ly))*Jy;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVleHByR0YoKiYsKi1JIlNHRig2I0kieUdGKCIiIkYzRjQqJEYzISIiRjQiIiNGNEY0LCwqJkkidEdGKEY3RjMhIiNGNCooLCZGNEY0RjpGN0Y0RjpGNEYzRjZGNiooRjpGNEYzRjQsJkY0RjRGOiIiJEY0RjYqKEY/RjQsJkY0RjRGOiIiJUY0LCZGM0Y0RjRGNEY2RjYqJkY/RjdGREY7RjRGNDcjRi4=

We now have a pole at y=0, of order 3 d:='d': eqcat:=expr-a*Ly^3-b*Ly^2-c*Ly-d;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZlcWNhdEdGKCwsKiYsKC1JIlNHRig2I0kieUdGKCIiIkY0RjUqJEY0ISIiRjUiIiRJImFHRihGNUY3KiZGMCIiI0kiYkdGKEY1RjcqJkYwRjVJImNHRihGNUY3SSJkR0YoRjcqJiwqRjFGNUY0RjVGNkY1RjtGNUY1LCwqJkkidEdGKEY7RjQhIiNGNSooLCZGNUY1RkRGO0Y1RkRGNUY0RjdGNyooRkRGNUY0RjUsJkY1RjVGREY4RjVGNyooRklGNSwmRjVGNUZEIiIlRjUsJkY0RjVGNUY1RjdGNyomRklGO0ZNRkVGNUY1RjU3I0Yu

normal(series(eqcat,y,1));

KylJInlHNiIsJiokSSJ0R0YkIiIjIiIiSSJhR0YkISIiISIkLCoqJkYnRigtSSJTR0YkNiMiIiFGKUYpKiZGL0YpRipGKUYsSSJiR0YkRitGJ0YrISIjLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRilGKw==

asol:=solve(op(1,%),a);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhc29sR0YoKiRJInRHRigiIiM3I0Yu

normal(series(subs(a=asol,eqcat),y,1));

KydJInlHNiIsKComSSJ0R0YkIiIjLUkiU0dGJDYjIiIhIiIiISIjSSJiR0YkISIiRidGMEYuLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRi1GMA==

bsol:=solve(op(1,%),b);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVic29sR0YoLCYqJkkidEdGKCIiIy1JIlNHRig2IyIiISIiIiEiI0YwISIiNyNGLg==

subs(y=0,Sser(3));

LCYqJEkidEc2IiIiJCIiJkYkIiIi

normal(series(subs(a=asol,b=bsol,eqcat),y,2));

KydJInlHNiIsMComLUkiU0dGJDYjIiIhIiIjSSJ0R0YkRisiIiIqJkYsRistLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGKUYtISIjKiZGLEYrRidGLUY2KiZGJ0YtRixGLUYtKiRGLEYrISIqSSJjR0YkISIiRiwhIiRGPC1JIk9HRjI2I0YtRio=

csol:=solve(coeff(%,y,-1),c);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVjc29sR0YoLC4qJi1JIlNHRig2IyIiISIiI0kidEdGKEY0IiIiKiZGNUY0LS1JIkRHRiU2I0YxRjJGNiEiIyomRjVGNEYwRjZGPComRjBGNkY1RjZGNiokRjVGNCEiKkY1ISIkNyNGLg==

collect(csol,S,factor);

LCoqJi1JIlNHNiI2IyIiISIiI0kidEdGJkYpIiIiKihGKkYrLCZGKkYpISIiRitGK0YkRitGLiomRipGKS0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJUYnRishIiMqJkYqRissJkYrRitGKiIiJEYrISIk

normal(series(subs(a=asol,b=bsol,c=csol,eqcat),y,3));

KydJInlHNiIsNiomLUkiU0dGJDYjIiIhIiIjSSJ0R0YkRitGKyomRixGKy0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYpIiIiISIjKiYtLS1JI0BAR0YxNiRGMEYrRjRGKUY1RixGKyEiIiomRixGK0YnRjUiIiUqJkYuRjVGLEY1RjUqJkYnRjVGLEY1RisqJEYsRishIzxJImRHRiRGPUYsISIoRj1GNUYqLUkiT0dGMTYjRjVGNQ==

dsol:=solve(coeff(%,y,0),d);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkc29sR0YoLDQqJi1JIlNHRig2IyIiISIiI0kidEdGKEY0RjQqJkY1RjQtLUkiREdGJTYjRjFGMiIiIiEiIyomLS0tSSNAQEdGJTYkRjlGNEY6RjJGO0Y1RjQhIiIqJkY1RjRGMEY7IiIlKiZGN0Y7RjVGO0Y7KiZGMEY7RjVGO0Y0KiRGNUY0ISM8RjUhIihGQ0Y7NyNGLg==

collect(dsol,S,factor);

LDAqJi1JIlNHNiI2IyIiISIiI0kidEdGJkYpRikqKCwmIiIiRi1GKkYpRi1GKkYtRiRGLUYpKihGKkYtLCZGKkYpISIiRi1GLS0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJUYnRi1GMComLS0tSSNAQEdGNDYkRjNGKUY3RidGLUYqRilGMCokRipGKSEjPEYqISIoRjBGLQ==

eqcat1:=numer(factor(subs(a=asol,b=bsol,c=csol,d=dsol,eqcat)));

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

Can this be true ? n:=20: normal(series(subs(S(y)=Sser(n), S(0)=subs(y=0,Sser(n)), D(S)(0)=subs(y=0,diff(Sser(n),y)),((D@@2)(S))(0)=subs(y=0,diff(Sser(n),y$2)), eqcat1),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0A=

Back to Q(0,y) (denoted here Q(y) to lighten notation) Sy:=subs(x=0,K)*Q(y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNTeUdGKCooLCgqJEkieUdGKCIiIyIiIkYxRjJGM0YzRjNJInRHRihGMy1JIlFHRig2I0YxRjM3I0Yu

factor(subs(S(y)=Sy,S(0)=subs(y=0,Sy),(D(S))(0)=limit(diff(Sy,y),y=0),((D@@2)(S))(0)=limit(diff(Sy,y$2),y=0),eqcat1));

KihJInRHNiIiIiIsJkkieUdGJEYlRiVGJSIiIyxocyoqLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kiUUdGJDYjIiIhRiUtRjI2I0YnRiVGIyIiJEYnIiIlRigqKi1GMkYzRiVGNUYlRiNGKEYnRjghIiIqKkYrRiVGNUYlRiNGN0YnRjdGOCoqRjpGJUY1RiVGI0YoRidGNyEiIyoqRiNGN0Y1RiVGOkYlRidGNyIjPyoqRiNGN0Y1RiVGOkYlRidGKCIjOSoqRjpGJUY1RiVGI0Y3RiciIiZGOCoqRiNGN0Y1RiVGOkYlRidGOEZCKihGI0YoRjpGJUYnRjciIicqKEYjRihGOkYlRidGKEZHKihGI0YoRjpGJUYnRiVGRyooRiNGKEY6RiVGJ0Y4RigqKEYrRiVGI0YlRidGKEY7KihGOkYlRiNGJUYnRiVGOyooRjVGN0YjRjhGJyIiKUY7KihGNUY3RiNGOEYnIiIoISInKihGNUY3RiNGOEYnRkchIzoqKEY1RjdGI0Y4RidGRCEjPyooRjVGN0YjRjhGJ0Y4RlMqKEY1RjdGI0Y4RidGN0ZRKihGNUYoRiNGKEYnRkdGJSooRjVGN0YjRjhGJ0YoRjsqKEY1RihGI0YoRidGREY4KihGNUYoRiNGKEYnRjhGRyooRjVGKEYjRihGJ0Y3RjgqKEY1RihGI0Y3RidGJSEiJCooRjVGKEYjRihGJ0YoRiUqKEY1RiVGI0YlRidGKEY4KihGNUYlRiNGJUYnRiVGJSokRidGN0YlKiZGI0YlRidGREY7KiZGI0YlRidGOEZobiooRjpGKEYjRjdGJ0Y3RjsqKEY6RihGI0Y3RidGKEY+KihGOkYoRiNGN0YnRiVGOyooLS0tSSNAQEdGLjYkRi1GKEYxRjNGJUYjRihGJ0YoRiUqKEY6RiVGI0YlRidGN0Y7KihGOkYlRiNGJUYnRighIiUqKEYrRiVGI0YoRidGJUYoKipGK0YlRjVGJUYjRjdGJ0YoRigqKkY6RiVGNUYlRiNGN0YnRiVGOCoqRjpGJUY1RiVGI0YoRidGKEY7KipGOkYlRjVGKEYjRjhGJ0ZHRigqKkY6RiVGNUYoRiNGOEYnRkRGTioqRjpGKEY1RiVGI0Y4RidGOEY7KipGOkYlRjVGKEYjRjhGJ0Y4IiM3KipGOkYoRjVGJUYjRjhGJ0Y3Rj4qKkY6RiVGNUYoRiNGOEYnRjdGTioqRjpGKEY1RiVGI0Y4RidGKEY7KipGOkYlRjVGKEYjRjhGJ0YoRigqKEY1RihGI0Y3RidGUEZobiooRjVGKEYjRjdGJ0ZHISM3KihGNUYoRiNGN0YnRkQhI0AqKEYjRjdGNUYoRidGOCEjQyooRjVGJUYjRihGJ0ZHRmhuKihGI0Y3RjVGKEYnRjdGXHEqKEYjRihGNUYlRidGRCEiKiooRiNGN0Y1RihGJ0YoRmpwKihGI0YoRjVGJUYnRjhGUSooRjVGJUYjRiVGJ0ZERiUqKEYrRiVGI0YoRidGN0YoKihGI0YoRjVGJUYnRjchIzYqKEY1RiVGI0YlRidGOEZEKihGK0YlRiNGKEYnRihGRyooRiNGKEY1RiVGJ0YoRlEqKEYjRiVGNUYlRidGN0YlKihGI0YoRjVGJUYnRiVGUSomRjpGJUYjRihGKComRjVGJUYnRjdGOyomRjVGJUYjRihGPkYl

eqcatQ:=collect(op(3,%),Q(y),factor);

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

n:=30: normal(series(subs(Q(y)=subs(x=0,Qser(n)),Q(0)=subs(x=0,y=0,Qser(n)), (D(Q))(0)= subs(y=0,diff(subs(x=0,Qser(n)),y)), ((D@@2)(Q))(0)= subs(y=0,diff(subs(x=0,Qser(n)),y$2)),eqcatQ),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0o=

eqcat:=subs(Q(y)=Q,Q(0)=A1,(D(Q))(0)=A2,((D@@2)(Q))(0)=A3,eqcatQ);

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

Three series cancel the following derivative edQ:=factor(diff(eqcat,Q));subs(t=0,%);

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

LCQqJEkieUc2IiIiJCEiIg==

Let us compute their expansions. y0:='y0': Y0ser:=proc(n) add(y0[i]*t^i,i=1..n): end: n:=2: normal(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)),A2=subs(x=0,y=0,diff(Qser(n),y)),A3=subs(x=0,y=0,diff(Qser(n),y$2)), y=Y0ser(n), edQ),t,n+1));

KydJInRHNiIsJiZJI3kwR0YkNiMiIiJGKSEiI0YpIiIjLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJEYoIiIk

y0[n-1]:=solve(coeff(%,t,n),y0[n-1]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTBHRig2IyIiIiIiIzcjRjE=

From now on we execute the next two lines several times to obtain coefficients of Y0 n:=n+1: normal(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)),A2=subs(x=0,y=0,diff(Qser(n),y)),A3=subs(x=0,y=0,diff(Qser(n),y$2)), y=Y0ser(n), edQ),t,n+1));

KydJInRHNiIsJiEiJSIiIiZJI3kwR0YkNiMiIiNGJyIiJC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YnIiIl

y0[n-1]:=solve(coeff(%,t,n),y0[n-1]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTBHRig2IyIiIyIiJTcjRjE=

Y0ser(n);

LDQqJilJInRHNiIiIioiIiImSSN5MEdGJjYjRidGKEYoKiYiJ0N5PkYoKUYlIiIpRihGKComIiZrJ0hGKClGJSIiKEYoRigqJiIlV1hGKClGJSIiJ0YoRigqJiIkTyhGKClGJSIiJkYoRigqJiIkPyJGKClGJSIiJUYoRigqJiIjQ0YoKUYlIiIkRihGKComRj9GKClGJSIiI0YoRigqJkZGRihGJUYoRig=

For the other two roots, we need series in s=sqrt(t) instead y1:='y1': Y1ser:=proc(n) add(y1[i]*s^i,i=1..n): end: n:=3: factor(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)),A2=subs(x=0,y=0,diff(Qser(n),y)),A3=subs(x=0,y=0,diff(Qser(n),y$2)), y=Y1ser(n), t=s^2,edQ),s,n+1));

KydJInNHNiIsJCooJkkjeTFHRiQ2IyIiIkYqLCZGJ0YqRiohIiJGKiwmRidGKkYqRipGKkYsIiIkLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJEYpIiIl

y1[1]:=1: From now on we execute the next two lines several times to obtain coefficients of Y1 n:=n+1: normal(series(subs(Q=subs(x=0,Qser(n)), A1=subs(x=0,y=0,Qser(n)),A2=subs(x=0,y=0,diff(Qser(n),y)),A3=subs(x=0,y=0,diff(Qser(n),y$2)), y=Y1ser(n), t=s^2,edQ),s,n+1));

KydJInNHNiIsJiIjUyIiIiomIiIjRicmSSN5MUdGJDYjIiIpRichIiIiIzUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJyIjNg==

y1[n-2]:=solve(coeff(%,s,n),y1[n-2]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeTFHRig2IyIiKSIjPzcjRjE=

Y1ser(n);

LDJJInNHNiIiIiIqJClGIyIiI0YlRiUqJiMiIihGKEYlKUYjIiImRiVGJSomIiIpRiUpRiMiIidGJUYlKiYjIiM+RihGJSlGI0YrRiVGJSomIiM/RiUpRiNGL0YlRiUqJiZJI3kxR0YkNiMiIipGJSlGI0Y9RiVGJSomJkY7NiMiIzVGJSlGI0ZCRiVGJQ==

We can determine a minimal polynomial for each Y_i by eliminating A2 and A3 from our three equations edy:=factor(diff(eqcat,y));

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

indets(edy);indets(edQ);

PChJI0ExRzYiSSNBMkdGJEkjQTNHRiRJIlFHRiRJInRHRiRJInlHRiQ=

PCdJI0ExRzYiSSNBMkdGJEkiUUdGJEkidEdGJEkieUdGJA==

res1:=op(4,factor(resultant(eqcat,edy,A3)));

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

algcandY:=factor(resultant(res1,edQ,A2));nops(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlhbGdjYW5kWUdGKCwkKjAiIiMiIiIpSSJ0R0YoRjBGMUkieUdGKEYxLCZGNEYxRjFGMUYxLC4qKCIiJEYxRjNGMSlGNEY4RjFGMSooRjhGMUYzRjEpRjRGMEYxISIiKiRGOUYxRjEqJkYzRjFGNEYxRjEqJEY7RjFGPEYzRjFGMSwsKiZGM0YxRjlGMUYxKigiIiVGMUYzRjFGO0YxRjEqKEYwRjFGM0YxRjRGMUYxKiZGMEYxRjNGMUYxRjRGPEYxLCYqKEkiUUdGKEYxRjNGMUY0RjFGMUYxRjFGMUY8NyNGLg==

IiIo

Since Y0 starts with t and Y1 with sqrt(t), we can identify which factor vanishes at Y0 and Y1 algY0:=t*y^3+4*t*y^2+2*t*y+2*t-y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdZMEdGKCwsKiZJInRHRigiIiIpSSJ5R0YoIiIkRjFGMSooIiIlRjFGMEYxKUYzIiIjRjFGMSooRjhGMUYwRjFGM0YxRjEqJkY4RjFGMEYxRjFGMyEiIjcjRi4=

algY1:=3*t*y^3-3*t*y^2+y^3+t*y-y^2+t;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdZMUdGKCwuKigiIiQiIiJJInRHRihGMSlJInlHRihGMEYxRjEqKEYwRjFGMkYxKUY0IiIjRjEhIiIqJEYzRjFGMSomRjJGMUY0RjFGMSokRjZGMUY4RjJGMTcjRi4=

series(subs(y=Y0ser(n),algY0),t); series(subs(y=Y1ser(n),t=s^2,algY1),s);

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIic=

KyVJInNHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIic=

disc:=factor(discrim(eqcat,Q)): nops(disc);

IiIl

seq(indets(op(i,disc)),i=1..nops(disc));

NiY8I0kieUc2IjwnSSNBMUdGJUkjQTJHRiVJI0EzR0YlSSJ0R0YlRiQ8I0YqRiM=

nops(op(2,disc));

IiRfJg==

discs:=op(2,disc):degree(%,y);

IiM6

We check that Y0 and Y1 are indeed double roots of discs n:=17:normal(series(subs(y=Y0ser(n),A1=subs(x=0,y=0,Qser(n)),A2=subs(x=0,y=0,diff(Qser(n),y)),A3=subs(x=0,y=0,diff(Qser(n),y$2)),discs),t,n));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzw=

n:=14:normal(series(subs(y=Y0ser(n),A1=subs(x=0,y=0,Qser(n)),A2=subs(x=0,y=0,diff(Qser(n),y)),A3=subs(x=0,y=0,diff(Qser(n),y$2)),diff(discs,y)),t,n));

KydJInRHNiIsJiIoZ0xxIyEiIiomIiIjIiIiJkkjeTBHRiQ2IyIiKkYqRioiIzgtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKiIjOQ==

n:=19:normal(series(subs(y=Y1ser(n),A1=subs(x=0,y=0,Qser(n)),A2=subs(x=0,y=0,diff(Qser(n),y)),A3=subs(x=0,y=0,diff(Qser(n),y$2)),t=s^2,discs),s,n));

KyVJInNHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIz4=

n:=19:normal(series(subs(y=Y1ser(n),A1=subs(x=0,y=0,Qser(n)),A2=subs(x=0,y=0,diff(Qser(n),y)),A3=subs(x=0,y=0,diff(Qser(n),y$2)),t=s^2,diff(discs,y)),s,n));

Ky1JInNHNiIsJiomIiIpIiIiJkkjeTFHRiQ2IyIiKkYoRigiJGImISIiIiM6LCgqJkYnRigmRio2IyIjNUYoRigqJiIjS0YoRilGKEYuIiQ3KUYoIiM7LCoiJTc6RigqJkYnRigmRio2IyIjNkYoRigqJkY2RihGMkYoRi4qJkY4RihGKUYoRigiIzwsLCImPT4iRigqJiIkVyJGKEYpRihGLiomRjZGKEY8RihGLiomRidGKCZGKjYjIiM3RihGKComRjhGKEYyRihGKCIjPS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiM+

Connection between the minimal polynomials of Y0 and Y1 algY0;algY1;

LCwqJkkidEc2IiIiIilJInlHRiUiIiRGJkYmKigiIiVGJkYkRiYpRigiIiNGJkYmKihGLUYmRiRGJkYoRiZGJiomRi1GJkYkRiZGJkYoISIi

LC4qKCIiJCIiIkkidEc2IkYlKUkieUdGJ0YkRiVGJSooRiRGJUYmRiUpRikiIiNGJSEiIiokRihGJUYlKiZGJkYlRilGJUYlKiRGK0YlRi1GJkYl

factor(algY1/subs(y=-1+1/y,algY0));

KiQpSSJ5RzYiIiIkIiIi

factor(algY0/subs(y=1/(1+y),algY1));

KiQpLCZJInlHNiIiIiJGJ0YnIiIkRic=

factor(coeff(discs,y,15));

LCQqKCIiJSIiIilJInRHNiIiIiRGJSksJkYlRiUqJkYpRiVGJ0YlRiVGKUYlISIi

di:=factor((y-1)^6*discs-y^18*(y+1)^6*subs(y=-1+1/y,discs));

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

algY1;

LC4qKCIiJCIiIkkidEc2IkYlKUkieUdGJ0YkRiVGJSooRiRGJUYmRiUpRikiIiNGJSEiIiokRihGJUYlKiZGJkYlRilGJUYlKiRGK0YlRi1GJkYl

res:=rem(discs,algY0^2*algY1^2,y): degree(%,y);

IiM2

eqens:=[seq(numer(coeff(res,y,i)),i=0..11)]: map(nops, eqens);

Ny4iIiRGI0YjRiNGI0YjRiNGIyIiI0YjRiNGIw==

alg1:=op(3,coeff(res,y,11));

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

alg2:=op(3,coeff(res,y,0));

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

alg3:=op(3,coeff(res,y,1));

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

res1:=op(3,factor(resultant(alg1,alg2,A3))): res2:=op(3,factor(resultant(alg1,alg3,A3))): Equation for A1 res3:=factor(resultant(res1,res2,A2)): nops(res3);

IiIo

seq(indets(op(i,res3)),i=1..nops(res3)); seq(nops(op(i,res3)),i=1..nops(res3));

Nik8IjwjSSJ0RzYiRiRGJEYkPCRJI0ExR0YmRiVGJw==

NikiIiIiIiNGJEYkRiQiI1ZGJA==

op(7,res3);

KiQpLCgqKCIiIyIiIkkjQTFHNiJGJylJInRHRilGJkYnRicqJiIiJ0YnRitGJ0YnRidGJyIiKUYn

algA1:=collect(op(6,res3),A1,factor);

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

algA1 := 27*A1^8*t^14+108*t^12*(1+6*t)*A1^7+189*t^10*(1+6*t)^2*A1^6+189*t^8*(1+6*t)^3*A1^5+9*t^6*(15760*t^4+10215*t^3+2510*t^2+285*t+13)*A1^4+9*t^4*(1+6*t)*(4304*t^4+2286*t^3+484*t^2+66*t+5)*A1^3+t^2*(71749*t^6-29376*t^5-41190*t^4-10575*t^3-717*t^2+63*t+10)*A1^2+(1+6*t)*(4357*t^6-2268*t^5+30*t^4+630*t^3+21*t^2-18*t+1)*A1+28397*t^6+8571*t^5-2769*t^4-768*t^3+72*t^2+12*t-1;

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

n:=10: normal(series(subs(A1=subs(x=0,y=0,Qser(n)),algA1),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Equation for A2 nops(res1); nops(res2);

IiQoPQ==

IiQ2Iw==

res4:=factor(resultant(res1,algA1,A1)): nops(%);

IiIl

seq(indets(op(i,res4)),i=1..nops(res4)); seq(nops(op(i,res4)),i=1..nops(res4));

NiY8IjwjSSJ0RzYiPCRJI0EyR0YmRiVGJw==

NiYiIiIiIiMiI3UiJUQ5

n:=10: normal(series(subs(A2=subs(x=0,y=0,diff(Qser(n),y)),op(3,res4)),t,n+1)); n:=4:normal(series(subs(A2=subs(x=0,y=0,diff(Qser(n),y)),op(4,res4)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Ky9JInRHNiIiJSc0JSIiISIodyRwUyIiIiIrYzk8bkoiIiMiLk92Wml4QyIiIiQiMDhCViNwOyJwJCIiJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIm

algA2:=collect(op(3,res4),A2,factor);

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

algA2 := 531441*t^20*A2^8-354294*t^17*(32*t^2-1)*A2^7+19683*t^14*(2876*t^4-2250*t^3-986*t^2-60*t+5)*A2^6-1458*t^11*(528947*t^6+74250*t^5-35463*t^4+855*t^3+3336*t^2+342*t-10)*A2^5+243*t^8*(71530190*t^8+73964250*t^7+32359145*t^6+7595130*t^5+1040697*t^4+81672*t^3+1457*t^2-330*t+5)*A2^4-54*t^5*(1521384074*t^10+2090962125*t^9+1232529190*t^8+394881240*t^7+72439136*t^6+7179039*t^5+247706*t^4-22788*t^3-2504*t^2+111*t-1)*A2^3-t^2*(92987803547*t^12+137016002250*t^11+71757251274*t^10+13351401810*t^9-1344630111*t^8-867129318*t^7-94643552*t^6+4488534*t^5+1656957*t^4+65808*t^3-8988*t^2+198*t-1)*A2^2+(-30852252238*t^13-56998004250*t^12-29293137062*t^11-698963030*t^10+2898532526*t^9+462771308*t^8-87355006*t^7-22330202*t^6+756820*t^5+446104*t^4-3964*t^3-3940*t^2+168*t-2)*A2-t*(3123122379*t^11+8207933250*t^10+4131881424*t^9+306412220*t^8-192419934*t^7-32679670*t^6+2455353*t^5+786126*t^4-15258*t^3-7432*t^2+330*t-4): Equation for A3 res6:=op(3,factor(resultant(alg1,alg2,A2))): res7:=factor(resultant(res6,algA1,A1)): nops(%);

IiIl

seq(indets(op(i,res7)),i=1..nops(res7)); seq(nops(op(i,res7)),i=1..nops(res7));

NiY8IjwjSSJ0RzYiPCRJI0EzR0YmRiVGJw==

NiYiIiIiIiMiJDQiIiVEQA==

n:=10: normal(series(subs(A3=subs(x=0,y=0,diff(Qser(n),y$2)),op(3,res7)),t,n+1)); n:=4:normal(series(subs(A3=subs(x=0,y=0,diff(Qser(n),y$2)),op(4,res7)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Ky9JInRHNiIiOTMtUio+IWZfNnM3WVIiIiEiPGshKVtxXzIvJDQkPjonZkgiIiIhP2c8XSlveSNcPk5aJDRxKXlUIiIjIUI3OF16ViYpUkE7Pmg+VGc1JEciIiQiRWxDV3BrPFhuRGEiei9RNURBPCMiIiUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIiJg==

algA3:=collect(op(3,res7),A3,factor);

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

algA3 := 7625597484987*t^27*A3^8+5083731656658*t^23*(220*t^3+112*t^2+22*t+1)*A3^7+4649045868*t^19*(13357825*t^6+14346990*t^5+6836592*t^4+1647954*t^3+196293*t^2+10323*t+304)*A3^6+1033121304*t^15*(1867311125*t^9+2876921150*t^8+2133457715*t^7+936868028*t^6+251821101*t^5+40523130*t^4+3775019*t^3+207149*t^2+7961*t+203)*A3^5+944784*t^11*(50615719470625*t^12+80895857938875*t^11+67515027101925*t^10+39491531031870*t^9+16627808094963*t^8+4782239806482*t^7+895760220509*t^6+106596613089*t^5+8350868607*t^4+497807307*t^3+24446706*t^2+778023*t+18544)*A3^4+209952*t^7*(3308513314009375*t^15+5476482075718750*t^14+4324353500575000*t^13+2482641181867750*t^12+1254747750461850*t^11+522924459933534*t^10+157170587799527*t^9+31653271225343*t^8+4182561579119*t^7+384621681017*t^6+30165639717*t^5+2188369707*t^4+122657317*t^3+5814871*t^2+186259*t+3721)*A3^3-64*t^3*(21525297844309550000*t^18+21273050817789243750*t^17+2464095136851292500*t^16-3998120250036942000*t^15-2505292876951539975*t^14-1593995634328657167*t^13-942070407921591420*t^12-303776027154125721*t^11-36287831872847034*t^10+4490854704660204*t^9+1739351597191722*t^8+147927322563795*t^7-2173718397252*t^6-1140086708226*t^5-142156864764*t^4-10110435174*t^3-436914036*t^2-21769830*t-226981)*A3^2+(843126116721046000000*t^20-9706155599886400000*t^19-903042748933783600000*t^18-344130244718187280000*t^17+47749659703421184000*t^16+6032954713542741760*t^15+13871494894966656*t^14+15567735274931770880*t^13+6825645360591677184*t^12+93430814549826304*t^11-412453130484593664*t^10-54250788316648960*t^9+7441160388129792*t^8+1685526881912320*t^7-11692525303296*t^6-15367153924480*t^5-929159245440*t^4-35395843712*t^3+12625523200*t^2-105958784*t+29053568)*A3-256*t^2*(2502646668396171875*t^17+2796360898382750000*t^16+796001317891278125*t^15+86175823865075000*t^14+72250646494285625*t^13-14056109041050925*t^12-35047176678470735*t^11-10247818372603478*t^10-16784556798765*t^9+395533703487701*t^8+43107989234399*t^7-3345599497164*t^6-689970417144*t^5-26540209587*t^4+452471451*t^3+510969310*t^2-53357*t+1134905): Looking for linear relations between the series A_i A1ser:=proc(n) subs(x=0,y=0,Qser(n)): end: A2ser:=proc(n) subs(x=0,y=0,diff(Qser(n),y)): end: A3ser:=proc(n) subs(x=0,y=0,diff(Qser(n),y$2)): end: algA1;

LEAqJkkjQTFHNiIiIilJInRHRiUiIzkiI0YqKEYnIiM3LCYiIiJGLUYnIiInRi1GJCIiKCIkMyIqKEYnIiM1RiwiIiNGJEYuIiQqPSooRidGJkYsIiIkRiQiIiZGNCooRidGLiwsKiRGJyIiJSImZ2QiKiRGJ0Y2IiY6LSIqJEYnRjMiJTVERiciJCZHIiM4Ri1GLUYkRjsiIioqKkYnRjtGLEYtLCxGOiIlL1ZGPSIlJ0cjRj8iJCVbRiciI21GN0YtRi1GJEY2RkMqKEYnRjMsMCokRidGLiImXDwoKiRGJ0Y3ISZ3JEhGOiEmIT5URj0hJnYwIkY/ISQ8KEYnIiNqRjJGLUYtRiRGM0YtKihGLEYtLDBGTCIlZFZGTiElb0FGOiIjSUY9IiRJJ0Y/IiNARichIz1GLUYtRi1GJEYtRi1GTCImKFJHRk4iJXImKUY6ISVwRkY9ISRvKEY/IiNzRidGKyEiIkYt

with(numapprox): Could A2 lie in the same extension as A1 ? n:=30:hermite_pade([seq(A1ser(n)^i,i=0..7),A2ser(n)],t,n);

NysiIiIhIiIsJCokSSJ0RzYiIiIjRiQiIiFGKkYqRipGKiwkRiciIiQ=

Let us check this factor(resultant(algA1, 1-A1-t^2*A1^2+3*t*A2,A1));

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

We have recovered indeed the equation satisfied by A2 factor(%/algA2);

LCQqJEkidEc2IiIjOyIiKg==

A2A1:=-(1-A1-t^2*A1^2)/3/t;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVBMkExR0YoLCQqJiwoKiZJI0ExR0YoIiIjSSJ0R0YoRjMhIiJGMkY1IiIiRjZGNkY0RjUjRjUiIiQ3I0Yu

n:=7: series(subs(A1=A1ser(n),A2A1),t,n); A2ser(n);

Ky9JInRHNiIiIiMiIiIiIiRGJSIjR0YnIiQ0IiIiJSIkLCgiIiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJiIiJw==

LDAqJEkidEc2IiIiKCImXU8jKiRGJCIiJyIlMVEqJEYkIiImIiQsKCokRiQiIiUiJDQiKiRGJCIiJCIjRyokRiQiIiNGMkYkRjU=

n:=120:factor(hermite_pade([seq(A1ser(n)^i,i=0..7),A3ser(n)],t,n));

NyssLiokSSJ0RzYiIiImIicleWEjKiRGJSIiJSImW2knKiRGJSIiJCEmdyNSKiRGJSIiIyEmcVsiRiUhJTk4ISNTIiIiLDAqJEYlIiInIScvYkxGJCEnL2xkRikhJ2dCO0YsIiY1KEdGLyImMVciRiUiJTk4IiNTRjQsJComRiVGMCwuRiQiJjNpI0YpISU1XEYsISVYWEYvIiQrJkYlIiQuJCIjQUY0RjQiIzcsJComRiVGKiwsRikiJ2d0XkYsIidTRktGLyImZ3QoRiUiJWchKiIkaiVGNEY0RjAsJComRiVGNywmRjRGNEYlRjdGLSIlISo9LCQqJkYlIiIpRlNGMCIlb0EsJComRiUiIzVGU0Y0IiU3OiwkKiRGJUZIIiRLJSwkKiZGJUYwLCxGKSImKysiRiwiJSshKkYvIiUrRUYlIiRTI0Y0RjRGNCEjRg==

Let us check factor(resultant(254784*t^5+66248*t^4-39276*t^3-14870*t^2-1314*t-40+A1*( -335504*t^6-576504*t^5-162360*t^4+28710*t^3+14406*t^2+1314*t+40)+A1^2* 12*t^2*(26208*t^5-4910*t^4-4545*t^3+500*t^2+303*t+22)+A1^3* 2*t^4*(517360*t^4+322740*t^3+77360*t^2+9060*t+463)+A1^4*1890*t^6*(6*t+1)^3+A1^5* 2268*t^8*(6*t+1)^2+A1^6* 1512*t^10*(6*t+1)+A1^7* 432*t^12+A3*( -27)*t^2*(10000*t^4+9000*t^3+2600*t^2+240*t+1), algA1,A1)/algA3);

LCQqJiwsKiRJInRHNiIiIiUiJisrIiokRiYiIiQiJSshKiokRiYiIiMiJStFRiYiJFMjIiIiRjEiIilGJiIjKCkiKipbP3VR

A3A1:=(254784*t^5+66248*t^4-39276*t^3-14870*t^2-1314*t-40+A1*( -335504*t^6-576504*t^5-162360*t^4+28710*t^3+14406*t^2+1314*t+40)+A1^2* 12*t^2*(26208*t^5-4910*t^4-4545*t^3+500*t^2+303*t+22)+A1^3* 2*t^4*(517360*t^4+322740*t^3+77360*t^2+9060*t+463)+A1^4*1890*t^6*(6*t+1)^3+A1^5* 2268*t^8*(6*t+1)^2+A1^6* 1512*t^10*(6*t+1)+A1^7* 432*t^12)/( 27*t^2*(10000*t^4+9000*t^3+2600*t^2+240*t+1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVBM0ExR0YoLCQqKCw8KiRJInRHRigiIiYiJyV5YSMqJEYyIiIlIiZbaScqJEYyIiIkISZ3I1IqJEYyIiIjISZxWyJGMiElOTghI1MiIiIqJkkjQTFHRihGQCwwKiRGMiIiJyEnL2JMRjEhJy9sZEY1ISdnQjtGOCImNShHRjsiJjFXIkYyIiU5OCIjU0ZARkBGQCooRkJGPEYyRjwsLkYxIiYzaSNGNSElNVxGOCElWFhGOyIkKyZGMiIkLiQiI0FGQEZAIiM3KihGQkY5RjJGNiwsRjUiJ2d0XkY4IidTRktGOyImZ3QoRjIiJWchKiIkaiVGQEZARjwqKEZCRjZGMkZFLCZGQEZARjJGRUY5IiUhKj0qKEZCRjNGMiIiKUZobkY8IiVvQSooRkJGRUYyIiM1RmhuRkAiJTc6KiZGQiIiKEYyRlUiJEslRkBGMiEiIywsRjUiJisrIkY4IiUrISpGOyIlK0VGMiIkUyNGQEZAISIiI0ZAIiNGNyNGLg==

n:=10:series(subs(A1=A1ser(n),A3A1),t,n);A3ser(n);

KzFJInRHNiIiIzUiIiMiI08iIiQiJClHIiIlIiV3OiIiJiImMy8iIiInIiZHXCciIigtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIik=

LDQqJEkidEc2IiIjNSIpL0h1PSokRiQiIioiKDl3eiMqJEYkIiIpIic/ZlUqJEYkIiIoIiZHXCcqJEYkIiInIiYzLyIqJEYkIiImIiV3OiokRiQiIiUiJClHKiRGJCIiJCIjTyokRiQiIiNGJg==

The genus of A1 with(algcurves): genus(algA1,t,A1);

IiIi

[solve(algA1,A1)];

NyQsJComSSJ0RzYiISIjLCghIiIiIiJGJSEiJyokLCoqJkYlIiIjLUknUm9vdE9mRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiMsOCokRiUiIiciJihSRyokRiUiIiYiJXImKSokRiUiIiUhJXBGKiRGJSIiJCEkbygqJEYlRi8iI3NGJSIjN0YpRioqJiwwRjciJWRWRjohJW9BRj0iI0lGQCIkSSdGQyIjQEYlISM9RipGKkYqSSNfWkdGMkYqRioqJiwsRjciJXM9RjohJXg4Rj0hJFEoRkAhI0ZGQyIiKkYqRk5GL0YqKiYsKEY3IiRzKkY6IiRDJEY9IiNGRipGTkZBRioqJkYlRjhGTkY+RlpGKkY+RkMiI09GJUZFRipGKiNGKkYvRipGKkZnbiwkKiYsKEYqRipGJUY4RixGKkYqRiVGJyNGKUYv

A1 is almost the square root of a series of degree 4 algB1:=collect(numer(factor(subs(A1=(-1-6*t+B1)/2/t^2,algA1))),B1,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdCMUdGKCwqKiRJI0IxR0YoIiIpIiNGKiYsLCokSSJ0R0YoIiIlIScrKz0qJEY2IiIkIScrPzsqJEY2IiIjISYrbyVGNiElP1YhIz0iIiJGQUYwRjdGQSooLChGPCIjNUY2IiInRkFGQUZBLCxGNSInK0Q1RjkiJitOKEY8IiZdWSJGNiIkNSYhIiJGQUZBRjBGPUYxKiQsLEY1IiYrKyJGOSIlKyEqRjwiJStFRjYiJFMjRkFGQUY9Rks3I0Yu

B1ser:=proc(n) series(2*t^2*A1ser(n)+1+6*t,t,n+1): end: B1ser(5);series(%^2,t);

Ky9JInRHNiIiIiIiIiEiIidGJSIiI0YoIiM1IiIlIiM9IiImLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiVGJw==

KzFJInRHNiIiIiIiIiEiIzdGJSIjUyIiIyIjQyIiJEYqIiIlIiRjIiIiJi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIn

algZ:=subs(B1=sqrt(Z),algB1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdaR0YoLCoqJEkiWkdGKCIiJSIjRiomLCwqJEkidEdGKEYxIScrKz0qJEY2IiIkIScrPzsqJEY2IiIjISYrbyVGNiElP1YhIz0iIiJGQEYwRjxGQCooLChGOyIjNUY2IiInRkBGQEZALCxGNSInK0Q1RjgiJitOKEY7IiZdWSJGNiIkNSYhIiJGQEZARjBGQCIiKSokLCxGNSImKysiRjgiJSshKkY7IiUrRUY2IiRTI0ZARkBGPEZKNyNGLg==

map(T->factor(T/(10000*t^4+9000*t^3+2600*t^2+240*t+1)^2),algZ);

LCoqJkkiWkc2IiIiJSwsKiRJInRHRiVGJiImKysiKiRGKSIiJCIlKyEqKiRGKSIiIyIlK0VGKSIkUyMiIiJGMiEiIyIjRiomRiRGL0YnISIiISM9KiosKEYuIiM1RikiIidGMkYyRjIsLEYoIicrRDVGKyImK04oRi4iJl1ZIkYpIiQ1JkY2RjJGMkYkRjJGJ0YzIiIpRjZGMg==

subs(Z=S*sqrt(10000*t^4+9000*t^3+2600*t^2+240*t+1),%);

LCoqJEkiU0c2IiIiJSIjRiokRiQiIiMhIz0qKiwoKiRJInRHRiVGKSIjNUYuIiInIiIiRjFGMSwsKiRGLkYmIicrRDUqJEYuIiIkIiYrTihGLSImXVkiRi4iJDUmISIiRjFGMUYkRjEsLEYzIiYrKyJGNSIlKyEqRi0iJStFRi4iJFMjRjFGMSMhIiRGKSIiKUY6RjE=

factor(discrim(algZ,Z));

LCQqKEkidEc2IiIiIywmIiIiRihGJCIiJEYmLCgqJEYkRiYiI0RGJCIiJiEiIkYoIiM1ITA3SDU3WnRTIw==

genus(algZ,t,Z);

IiIi

factor(-coeff(algZ,Z,2)/18);subs(t=t/10,%);

LCwqJEkidEc2IiIiJSImKysiKiRGJCIiJCIlKyEqKiRGJCIiIyIlK0VGJCIkUyMiIiJGLw==

LCwqJEkidEc2IiIiJSIiIiokRiQiIiQiIioqJEYkIiIjIiNFRiQiI0NGJ0Yn

The Galois group is S4, so there is no extension of degree 2 in Q(t,Z) galois(algZ,Z);

NidRJDRUNTYiPCNRJVMoNClGJFEiLUYkIiNDPCVRJigxfjQpRiRRJigyfjQpRiRRJigzfjQpRiQ=

n:='n': DE for A1 deA1:=map(factor,algeqtodiffeq(algA1,A1(t)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkZUExR0YoLCoiIzUiIiIqKEYvRjAsKComIiNYRjApSSJ0R0YoIiIkRjBGMComIiNBRjApRjYiIiNGMEYwRjAhIiJGMC1JI0ExR0YoNiNGNkYwRjAqKkY7RjBGNkYwLCoqJiIkXSVGMEY1RjBGMComIiRJI0YwRjpGMEYwKiYiIidGMEY2RjBGMCIiKEY8RjAtSSVkaWZmR0YmNiRGPUY2RjBGMCosRjdGMCwmRjBGMComRjdGMEY2RjBGMEYwLCgqJiIjREYwRjpGMEYwKiYiIiZGMEY2RjBGMEYwRjxGMEY6RjAtRko2JEY9LUkiJEdGJjYkRjZGO0YwRjA3IywqRi9GMEYxRjBGQEYwKixGN0YwRk1GMEZPRjBGOkYwLUZKNiRGSUY2RjBGMA==

reca1:=diffeqtorec(%,A1(t),a1(n));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyZWNhMUdGKDwmLCoqJiwoKiYiJEQjIiIiKUkibkdGKCIiI0Y0RjQqJiIkdidGNEY2RjRGNCIkXSVGNEY0LUkjYTFHRig2I0Y2RjRGNComLCgqJiIkPyJGNEY1RjRGNComIiQhZUY0RjZGNEY0IiQhb0Y0RjQtRjw2IywmRjZGNEY0RjRGNEY0KiYsKComIiInRjRGNUY0RjQqJiIjSUY0RjZGNEY0IiNPRjRGNC1GPDYjLCZGNkY0RjdGNEY0RjQqJiwoKiYiIiRGNEY1RjQhIiIqJiIjSEY0RjZGNEZWIiNxRlZGNC1GPDYjLCZGNkY0RlVGNEY0RjQvLUY8NiMiIiFGNC8tRjw2I0Y0RmpuLy1GPDYjRjciIiY3I0Yu

reca1:=map(factor,op(1,%));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyZWNhMUdGKCwqKioiJEQjIiIiLCZJIm5HRihGMSIiI0YxRjEsJkYzRjFGMUYxRjEtSSNhMUdGKDYjRjNGMUYxKioiIz9GMSwmKiYiIidGMUYzRjFGMSIjPEYxRjFGMkYxLUY3NiNGNUYxRjEqKkY9RjEsJkYzRjEiIiRGMUYxRjJGMS1GNzYjRjJGMUYxKigsJkYzRjEiIiZGMUYxLCYqJkZDRjFGM0YxRjEiIzlGMUYxLUY3NiNGQkYxISIiNyNGLg==

with(LREtools);

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

hypergeomsols(reca1,a1(n),{});

IiIh

DE for Z algZ;

LCoqJiIjRiIiIilJIlpHNiIiIiVGJUYlKiYsLComIicrKz1GJSlJInRHRihGKUYlISIiKiYiJys/O0YlKUYvIiIkRiVGMComIiYrbyVGJSlGLyIiI0YlRjAqJiIlP1ZGJUYvRiVGMCIjPUYwRiUpRidGOEYlRiUqKiIiKUYlLCgqJiIjNUYlRjdGJUYlKiYiIidGJUYvRiVGJUYlRiVGJSwsKiYiJytENUYlRi5GJUYlKiYiJitOKEYlRjNGJUYlKiYiJl1ZIkYlRjdGJUYlKiYiJDUmRiVGL0YlRiVGJUYwRiVGJ0YlRiUqJCksLComIiYrKyJGJUYuRiVGJSomIiUrISpGJUYzRiVGJSomIiUrRUYlRjdGJUYlKiYiJFMjRiVGL0YlRiVGJUYlRjhGJUYw

map(factor,op(1,algeqtodiffeq(algZ,Z(t))));

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

diffeqtorec(%,Z(t),a(k));

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

Zser:=proc(n) op(1,algeqtoseries(algZ,t,Z,n+1,true)): end: Zser(10);

KztJInRHNiIiIiIiIiEiIzdGJSIjUyIiIyIjQyIiJEYqIiIlIiRjIiIiJiIkXyYiIiciJSUzJCIiKCImR2UiIiIpIiYjKj4qIiIqIidTM2EiIzUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNg==

JSFH

Back to Q(0,y) eqcat;

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

Let us express this equation in terms of the series B1 eqQyB1:=numer(factor(subs(A3=A3A1,A2=A2A1,A1=(-1-6*t+B1)/(2*t^2), eqcat))): degree(%,Q); nops(eqQyB1);

IiIk

IiRrIw==

Compute an equation for Q(0,y) over Q(t,y) res:=factor(resultant(eqQyB1,algB1,B1)): nops(%);

IiIl

seq(indets(op(i,res)),i=1..nops(res));

NiY8IjwjSSJ0RzYiRiQ8JUkiUUdGJkYlSSJ5R0Ym

algQy:=op(4,res): nops(%);

IiZTLSI=

degree(algQy,Q);

IiND

Back to Q(x,0), using R(x)+S(Y)-R(0)=xy if (x,y) cancels the kernel eqxy:=subs(Q(x,y)=0,Q(0,0)=Q00,Q(x,0)=Qx,Q(0,y)=Qy,eqQ);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcXh5R0YoLCoqJkkieEdGKCIiIkkieUdGKEYxRjEqKCwoKiRGMiIiI0YxRjJGNkYxRjFGMUkidEdGKEYxSSNReUdGKEYxISIiKigsJkYwRjFGMUYxRjFGN0YxSSNReEdGKEYxRjkqJkY3RjFJJFEwMEdGKEYxRjE3I0Yu

The expression of Q(0,Y0) (with y=Y0) solve(eqxy,Qy);

KigsKiooSSNReEc2IiIiIkkidEdGJkYnSSJ4R0YmRichIiIqJkkkUTAwR0YmRidGKEYnRicqJkYlRidGKEYnRioqJkYpRidJInlHRiZGJ0YnRicsKCokRi8iIiNGJ0YvRjJGJ0YnRipGKEYq

res1:=numer(factor(subs(Q=%,Q00=(-1-6*t+B1)/2/t^2,eqQyB1))): nops(%);

IiQncA==

indets(res1);

PCdJI0IxRzYiSSNReEdGJEkidEdGJEkieEdGJEkieUdGJA==

Eliminating y=Y0 res2:=factor(resultant(res1,K,y)): nops(res2);

IiIl

map(indets,[seq(op(i,res2),i=1..nops(res2))]);

NyY8I0kidEc2IjwjSSJ4R0YlRiM8JkkjQjFHRiVJI1F4R0YlRiRGJw==

nops(op(4,res2));

IiIj

Q(x,0) has degree 3 over Q(t,x,B1) nops(op(1,op(4,res2)));degree(op(1,op(4,res2)),Qx);

IiRRIw==

IiIk

res4:=factor(resultant(op(1,op(4,res2)),algB1,B1)): nops(res4);

IiIl

map(indets,[seq(op(i,res4),i=1..nops(res4))]);

NyY8IjwjSSJ0RzYiRiQ8JUkjUXhHRiZGJUkieEdGJg==

nops(op(4,res4));

IiVsJSo=

n:=14: normal(series(subs(Qx=subs(y=0,Qser(n)),op(4,res4)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

algQx:=collect(op(4,res4),Qx,factor): degree(%,Qx);

IiND

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=