This Maple session accompanies the paper "Three quadrant walks by invariants" and considers the model with NSEW steps, plus a diagonal NE step.The proofs are in Section 7. restart:

Functional equation. The series Cm(1/x) is negative in x K:=1-t*(x+y+1/x+1/y+x*y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCYiIiJGLyomSSJ0R0YoRi8sLEkieEdGKEYvSSJ5R0YoRi8qJEYzISIiRi8qJEY0RjZGLyomRjNGL0Y0Ri9GL0YvRjY3I0Yu

eqC:=-K*C(x,y)+1-t/y*Cm(1/x)-t/x*Cm(1/y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcUNHRigsKiomLCYiIiJGMSomSSJ0R0YoRjEsLEkieEdGKEYxSSJ5R0YoRjEqJEY1ISIiRjEqJEY2RjhGMSomRjVGMUY2RjFGMUYxRjhGMS1JIkNHRig2JEY1RjZGMUY4RjFGMSooRjNGMUY2RjgtSSNDbUdGKDYjRjdGMUY4KihGM0YxRjVGOC1GQDYjRjlGMUY4NyNGLg==

Cmser:=proc(n) local i: local res:res:=coeff(convert(Cser(n),polynom),y,0): add(coeff(res,x,-i)/x^i,i=1..n+2): end: C00ser:=proc(n) coeff(coeff(convert(Cser(n),polynom),y,0),x,0): end: Cser:=proc(n) option remember: if n=0 then 1 else convert(normal(series(subs(C(x,y)=Cser(n-1),Cm(1/x)=Cmser(n-1),Cm(1/y)=subs(x=y,Cmser(n-1)),eqC+C(x,y)),t,n+1)),polynom): fi: end: map(expand,Cser(4));

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

We separate in C the contribution of two regions U denotes the part of C above the diagonal, DD the diagonal itself User:=proc(n) local i,j: collect(add(add(coeff(coeff(convert(Cser(n),polynom),x,i),y,j)*x^(j-i-1)*y^(j),i=-n..j-1),j=0..n+4),t,normal): end: User(5);

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

Dser:=proc(n) local i: collect(add(coeff(coeff(convert(Cser(n),polynom),x,i),y,i)*y^(i),i=0..n+4),t,normal):end: Dser(3);

LCoiIiJGIyomLCoqJClJInlHNiIiIiRGI0YjKiYiIidGIylGKCIiI0YjRiMqJiIjN0YjRihGI0YjIiIlRiNGIylJInRHRilGKkYjRiMqJiwoKiRGLUYjRiMqJkYuRiNGKEYjRiNGMUYjRiMpRjNGLkYjRiMqJkYoRiNGM0YjRiM=

Cexpr:=DD(x*y)+1/x*U(1/x,x*y)+1/y*U(1/y,x*y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZDZXhwckdGKCwoLUkjRERHRig2IyomSSJ4R0YoIiIiSSJ5R0YoRjRGNComRjMhIiItSSJVR0YoNiQqJEYzRjdGMkY0RjQqJkY1RjctRjk2JCokRjVGN0YyRjRGNDcjRi4=

Steps:=x*y+1/x+1/y+x+y; tSteps:=expand(subs(y=y/x,x=1/x,Steps));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZTdGVwc0dGKCwsSSJ4R0YoIiIiSSJ5R0YoRjAqJEYvISIiRjAqJEYxRjNGMComRi9GMEYxRjBGMDcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSd0U3RlcHNHRigsLCokSSJ4R0YoISIiIiIiKiZGMEYySSJ5R0YoRjJGMkYwRjIqJkY0RjFGMEYxRjJGNEYyNyNGLg==

The vertical moves in the companion set of steps Vpy:=coeff(tSteps, x,1);Vmy:=coeff(tSteps, x,-1);V0y:=coeff(tSteps, x,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRWcHlHRigsJkkieUdGKCIiIkYwRjA3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRWbXlHRigsJiIiIkYvKiRJInlHRighIiJGLzcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRWMHlHRihJInlHRig3I0Yu

Horizontal moves in the original model Hmx:=coeff(Steps, x,-1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRIbXhHRigiIiI3I0Yu

eqD:=-(1-t*V0y)*DD(y)+1+2*t*Vmy*(U(0,y))-2*t/y*U(0,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcURHRigsKiomLCYqJkkidEdGKCIiIkkieUdGKEYzISIiRjNGM0YzLUkjRERHRig2I0Y0RjNGNUYzRjMqKiIiI0YzRjJGMywmRjNGMyokRjRGNUYzRjMtSSJVR0YoNiQiIiFGNEYzRjMqKkY6RjNGMkYzRjRGNS1GPjYkRkBGQEYzRjU3I0Yu

eqU:=-x*U(x,y)+t*x*Vpy*DD(y)+t*tSteps*x*U(x,y)-t/y*subs(x=1/x,Hmx)*U(x,0)-t*Vmy*U(0,y)+t/y*U(0,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcVVHRigsLiomSSJ4R0YoIiIiLUkiVUdGKDYkRjBJInlHRihGMSEiIioqSSJ0R0YoRjFGMEYxLCZGNUYxRjFGMUYxLUkjRERHRig2I0Y1RjFGMSoqRjhGMSwsKiRGMEY2RjEqJkYwRjFGNUYxRjFGMEYxKiZGNUY2RjBGNkYxRjVGMUYxRjBGMUYyRjFGMSooRjhGMUY1RjYtRjM2JEYwIiIhRjFGNiooRjhGMSwmRjFGMSokRjVGNkYxRjEtRjM2JEZFRjVGMUY2KihGOEYxRjVGNi1GMzYkRkVGRUYxRjE3I0Yu

eqUfinal:=collect(subs(isolate(eqD,U(0,y)),eqU*y),[U,DD],factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSllcVVmaW5hbEdGKCwqKiZJInRHRigiIiItSSJVR0YoNiRJInhHRigiIiFGMSEiIiomLC4qKEYwRjEpRjUiIiNGMSlJInlHRihGPEYxRjEqKEYwRjFGO0YxRj5GMUYxKihGMEYxRjVGMUY9RjFGMSomRjBGMUY+RjFGMSomRjVGMUY+RjFGN0YwRjFGMS1GMzYkRjVGPkYxRjEqKiNGMUY8RjEsKioqRjxGMUYwRjFGNUYxRj5GMUYxKihGPEYxRjBGMUY1RjFGMUZBRjFGMUY3RjFGPkYxLUkjRERHRig2I0Y+RjFGMSomRkZGMUY+RjFGMTcjRi4=

n:=10: normal(series(subs( U(x,y)=User(n), U(x,0)=subs(y=0,User(n)), DD(y)=Dser(n),eqUfinal),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Kt:=1-t*tSteps

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNLdEdGKCwmKiZJInRHRigiIiIsLCokSSJ4R0YoISIiRjEqJkY0RjFJInlHRihGMUYxRjRGMSomRjdGNUY0RjVGMUY3RjFGMUY1RjFGMTcjRi4=

eqQ:=-Kt*Q(x,y)+1-t/y/x*Q(x,0)-t/x/y*Q(0,y)+t/x/y*Q(0,0)-t/x*Q(0,y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcVFHRigsLiomLCYqJkkidEdGKCIiIiwsKiRJInhHRighIiJGMyomRjZGM0kieUdGKEYzRjNGNkYzKiZGOUY3RjZGN0YzRjlGM0YzRjdGM0YzRjMtSSJRR0YoNiRGNkY5RjNGN0YzRjMqKkYyRjNGOUY3RjZGNy1GPDYkRjYiIiFGM0Y3KipGMkYzRjZGN0Y5RjctRjw2JEZBRjlGM0Y3KipGMkYzRjZGN0Y5RjctRjw2JEZBRkFGM0YzKihGMkYzRjZGN0ZDRjNGNzcjRi4=

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

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

A pair of invariants quadrant decoupling, as in Table 4 fx:=-1/x; gy:=y*(1-t*y)/t/(1+y)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmeEdGKCwkKiRJInhHRighIiJGMTcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNneUdGKCoqSSJ5R0YoIiIiLCYqJkkidEdGKEYwRi9GMCEiIkYwRjBGMEYzRjQsJkYvRjBGMEYwRjQ3I0Yu

factor(x*y-fx-gy);factor(%/Kt);

KiosLiooSSJ0RzYiIiIiKUkieEdGJiIiI0YnKUkieUdGJkYqRidGJyooRiVGJ0YoRidGLEYnRicqKEYlRidGKUYnRitGJ0YnKiZGJUYnRixGJ0YnKiZGKUYnRixGJyEiIkYlRidGJ0YpRjEsJkYsRidGJ0YnRjFGJUYx

LCQqKEkieUc2IiIiIiwmRiRGJkYmRiYhIiJJInRHRiVGKEYo

I1x:=t*Q(x,0)-fx; J1y:=-t*y*Vmy*Q(0,y)+t*Q(0,0)+gy

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRJMXhHRigsJiomSSJ0R0YoIiIiLUkiUUdGKDYkSSJ4R0YoIiIhRjFGMSokRjUhIiJGMTcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRKMXlHRigsKCoqSSJ0R0YoIiIiSSJ5R0YoRjEsJkYxRjEqJEYyISIiRjFGMS1JIlFHRig2JCIiIUYyRjFGNSomRjBGMS1GNzYkRjlGOUYxRjEqKkYyRjEsJiomRjBGMUYyRjFGNUYxRjFGMUYwRjUsJkYyRjFGMUYxRjVGMTcjRi4=

factor(I1x-J1y+x*y*eqQ);

KiwsLiooSSJ0RzYiIiIiKUkieEdGJiIiI0YnKUkieUdGJkYqRidGJyooRiVGJ0YoRidGLEYnRicqKEYlRidGKUYnRitGJ0YnKiZGJUYnRixGJ0YnKiZGKUYnRixGJyEiIkYlRidGJywoKiotSSJRR0YmNiRGKUYsRidGJUYnRilGJ0YsRidGJyooRjRGJ0YlRidGKUYnRidGJ0YnRidGKUYxLCZGLEYnRidGJ0YxRiVGMQ==

ratio1:=collect(factor(%/Kt),Q,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdyYXRpbzFHRigsJiooSSJ4R0YoIiIiSSJ5R0YoRjEtSSJRR0YoNiRGMEYyRjEhIiIqKEYyRjEsJkYyRjFGMUYxRjZJInRHRihGNkY2NyNGLg==

Steps; tSteps;

LCxJInhHNiIiIiJJInlHRiRGJSokRiMhIiJGJSokRiZGKEYlKiZGI0YlRiZGJUYl

LCwqJEkieEc2IiEiIiIiIiomRiRGJ0kieUdGJUYnRidGJEYnKiZGKUYmRiRGJkYnRilGJw==

Vpy:=coeff(tSteps, x,1);Vmy:=coeff(tSteps, x,-1);V0y:=coeff(tSteps, x,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRWcHlHRigsJkkieUdGKCIiIkYwRjA3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRWbXlHRigsJiIiIkYvKiRJInlHRighIiJGLzcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRWMHlHRihJInlHRig3I0Yu

co:=t*V0y+2*t*x*Vpy-1;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjb0dGKCwoKioiIiMiIiJJInhHRihGMSwmRjFGMUkieUdGKEYxRjFJInRHRihGMUYxKiZGNEYxRjVGMUYxRjEhIiI3I0Yu

Ysol:=factor([solve(1-t*tSteps,y)])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCosIyIiIiIiI0YyLCoqJkkidEdGKEYyKUkieEdGKEYzRjIhIiJGNkY5RjhGMiokKSwwKiYpRjZGM0YyKUY4IiIlRjJGMiooRjNGMkY+RjJGN0YyRjkqKEYzRjJGNkYyKUY4IiIkRjJGOSooRkBGMkY+RjJGOEYyRjkqJEY+RjJGMiooRjNGMkY2RjJGOEYyRjkqJEY3RjJGMkYxRjJGMkYyRjhGOUY2RjksJkY4RjJGMkYyRjlGMiwkKixGMUYyLCpGNUYyRjpGMkY2RjJGOEY5RjJGOEY5RjZGOUZJRjlGOTcjRi4=

Gform:=proc(d,e,f) local i: add(a[i]*y^i,i=0..d)+add(b[i]/(y-beta[i]),i=1..e) +add(c[i]/(y-gamma[i])^2,i=1..f): end: Gform(2,3,4)

LDYqJiZJImFHNiI2IyIiIyIiIilJInlHRiZGKEYpRikqJiZGJTYjRilGKUYrRilGKSZGJTYjIiIhRikqJiZJImJHRiZGLkYpLCZGK0YpJkklYmV0YUdGJkYuISIiRjhGKSomJkY0RidGKSwmRitGKSZGN0YnRjhGOEYpKiYmRjQ2IyIiJEYpLCZGK0YpJkY3Rj9GOEY4RikqJiZJImNHRiZGLkYpKSwmRitGKSZJJmdhbW1hRyUqcHJvdGVjdGVkR0YuRjhGKEY4RikqJiZGRUYnRikpLCZGK0YpJkZJRidGOEYoRjhGKSomJkZFRj9GKSksJkYrRikmRklGP0Y4RihGOEYpKiYmRkU2IyIiJUYpKSwmRitGKSZGSUZXRjhGKEY4Rik=

d:=1: e:=1: f:=0:collect(numer(factor(expand(rationalize(subs(y=Ysol[1],co*Gform(d,e,f)-y)-subs(y=Ysol[2],co*Gform(d,e,f)-y))))/sqrt(t^2*x^4-2*t^2*x^2-2*t*x^3-4*t^2*x+t^2-2*t*x+x^2)),x,factor)

LC4qLCIiIyIiIilJInRHNiJGJEYlLCYmSSViZXRhR0YoNiNGJUYlRiVGJUYlLCYqJiZJImFHRig2IyIiIUYlRipGJUYlJkkiYkdGKEYsISIiRiUpSSJ4R0YoIiImRiVGJSooRidGJSw6KipGOEYlRidGJUYvRiUpRipGJEYlRiUqKEYnRiUmRjBGLEYlRjxGJUYlKioiIiRGJUYnRiVGL0YlRipGJUYlKihGJ0YlRj5GJUYqRiVGJSoqRjhGJUYnRiVGM0YlRipGJUY1KiZGPkYlRjxGJUYlKigiIiVGJUYnRiVGM0YlRjUqKEYkRiVGL0YlRipGJUY1KiZGPkYlRipGJUYlKiRGPEYlRjVGM0YlRipGNUYlKUY3RkVGJUYlKiYsQCoqRkVGJUYmRiVGL0YlRjxGJUYlKihGJkYlRj5GJUY8RiVGNSoqRkBGJUYmRiVGL0YlRipGJUYlKipGJEYlRiZGJUY+RiVGKkYlRjUqKkZFRiVGJkYlRjNGJUYqRiVGNUY9RiUqKEYkRiVGJkYlRi9GJUYlKihGJEYlRiZGJUYzRiVGNUY/RjVGQUY1KihGJEYlRidGJUY8RiVGNSooRiRGJUYnRiVGM0YlRiUqJkYnRiVGKkYlRjVGR0Y1RipGJUYlKUY3RkBGJUYlKiYsPiooRiZGJUYvRiVGPEYlRiUqKkZARiVGJkYlRj5GJUY8RiVGNUZORiUqKEYmRiVGM0YlRipGJUY1KihGQEYlRiZGJUYvRiVGJSomRiZGJUY+RiVGJSooRidGJUYvRiVGKkYlRjUqKkZARiVGJ0YlRj5GJUYqRiVGJSomRidGJUY8RiVGNSomRidGJUY+RiVGJSomRidGJUYzRiVGJUZVRjVGJ0Y1RipGJUYlKUY3RiRGJUYlKihGJ0YlLDJGPUY1RmhuRiUqKkYkRiVGJ0YlRj5GJUYqRiVGNSomRidGJUYvRiVGJSooRiRGJUYnRiVGPkYlRjVGR0YlRipGNUYlRjVGJUY3RiVGJSooRiZGJUY+RiVGKUYlRjU=

coeffs(%,x)

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

sol:=solve({%}, [seq(a[i], i=0..d), seq(b[i], i=1..e), seq(beta[i], i=1..e), seq(c[i], i=1..f), seq(gamma[i], i=1..f)])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzb2xHRig3IzcmLyZJImFHRig2IyIiISwkKiRJInRHRighIiJGOC8mRjI2IyIiIkY0LyZJImJHRihGOywkKiYiIiNGPEY3RjhGPC8mSSViZXRhR0YoRjtGODcjRi4=

Gy:=subs(op(sol),Gform(d,e,f))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwmKiRJInRHRighIiJGMSooIiIjIiIiRjBGMSwmRjRGNEkieUdGKEY0RjFGNDcjRi4=

The function F(x) Fx:=factor(expand(rationalize(subs(y=Ysol[1], op(sol),-y+co*Gform(d,e,f))))); expand(%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCooLCgqKCIiIyIiIkkidEdGKEYyKUkieEdGKEYxRjJGMiomRjFGMkYzRjJGMkY1ISIiRjJGNUY3RjNGNzcjRi4=

LCgqJiIiIyIiIkkieEc2IkYlRiUqJkYkRiVGJiEiIkYlKiRJInRHRidGKUYp

Decoupling for y (Table 5) Fx:= 1/t-2*x-2/x; Gy:= (1-y)/t/(1+y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwoKiRJInRHRighIiIiIiIqJiIiI0YySSJ4R0YoRjJGMSomRjRGMkY1RjFGMTcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCooLCYiIiJGMEkieUdGKCEiIkYwSSJ0R0YoRjIsJkYwRjBGMUYwRjI3I0Yu

Hxy:=factor((y-Gy*co-Fx)/(1-t*tSteps))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRIeHlHRigsJCoqIiIjIiIiSSJ5R0YoRjFJInRHRighIiIsJkYxRjFGMkYxRjRGNDcjRi4=

Sy:=y*DD(y)+Gy;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNTeUdGKCwmKiZJInlHRigiIiItSSNEREdGKDYjRjBGMUYxKigsJkYxRjFGMCEiIkYxSSJ0R0YoRjcsJkYxRjFGMEYxRjdGMTcjRi4=

Rx:=2*t*subs(x=1/x,Hmx)*U(x,0)-Fx;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNSeEdGKCwqKigiIiMiIiJJInRHRihGMS1JIlVHRig2JEkieEdGKCIiIUYxRjEqJEYyISIiRjkqJkYwRjFGNkYxRjEqJkYwRjFGNkY5RjE3I0Yu

Delta:=(1-t*V0y)^2-4*t^2*Vpy*Vmy;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZEZWx0YUdGKCwmKiQpLCYqJkkidEdGKCIiIkkieUdGKEY0ISIiRjRGNCIiI0Y0RjQqKiIiJUY0KUYzRjdGNCwmRjRGNEY1RjRGNCwmRjRGNCokRjVGNkY0RjRGNjcjRi4=

factor(Rx^2-Delta*Sy^2);

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

factor(subs(isolate(eqUfinal,U(x,0)),%)/(1-t*tSteps));

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

factor(%- (-4*t*x*Vpy*Sy^2+(Hxy-2*x*y*U(x,y))*((t*V0y+2*t*x*Vpy-1)*Sy+Rx)))

LCQqLiIiIyIiIiwoKiotSSJVRzYiNiRJInhHRipJInlHRipGJUYsRiVJInRHRipGJUYtRiVGJSooRihGJUYuRiVGLEYlRiVGJUYlRiUsOiosRiRGJUYoRiVGLkYlKUYsRiRGJSlGLUYkRiVGJSosRiRGJS1JI0RER0YqNiNGLUYlRi5GJUYsRiVGM0YlRiUqLEYkRiVGKEYlRi5GJUYyRiVGLUYlRiUqLEYkRiVGKEYlRi5GJUYsRiVGM0YlRiUqLEYkRiVGNUYlRi5GJUYsRiVGLUYlRiUqKEYzRiVGNUYlRi5GJUYlKipGJEYlRihGJUYuRiVGLUYlRiUqKkYkRiVGLEYlRi1GJUYoRiUhIiIqJkYtRiVGNUYlRj4qKEYkRiVGLkYlLUYpNiRGLCIiIUYlRj4qKEYkRiVGKEYlRi5GJUYlRi1GJUYlRi1GJSwmRiVGJUYtRiVGPkYuRj5GPg==

factor(subs(isolate(eqUfinal,U(x,0)),%));

IiIh

Sy;

LCYqJkkieUc2IiIiIi1JI0RER0YlNiNGJEYmRiYqKCwmRiZGJkYkISIiRiZJInRHRiVGLCwmRiZGJkYkRiZGLEYm

expand(Rx);Delta;

LCoqKCIiIyIiIkkidEc2IkYlLUkiVUdGJzYkSSJ4R0YnIiIhRiVGJSokRiYhIiJGLiomRiRGJUYrRiVGJSomRiRGJUYrRi5GJQ==

LCYqJCksJiomSSJ0RzYiIiIiSSJ5R0YoRikhIiJGKUYpIiIjRilGKSoqIiIlRikpRidGLEYpLCZGKUYpRipGKUYpLCZGKUYpKiRGKkYrRilGKUYr

Ix:=Rx^2; series(Ix,x,1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeEdGKCokKSwqKigiIiMiIiJJInRHRihGMy1JIlVHRig2JEkieEdGKCIiIUYzRjMqJEY0ISIiRjsqJkYyRjNGOEYzRjMqJkYyRjNGOEY7RjNGMkYzNyNGLg==

KylJInhHNiIiIiUhIiMsJiooIiIpIiIiSSJ0R0YkRiotSSJVR0YkNiQiIiFGL0YqRioqJkYlRipGKyEiIkYxRjEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKkYv

Jy:=Delta*Sy^2; series(Jy,y,1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKeUdGKComLCYqJCksJiomSSJ0R0YoIiIiSSJ5R0YoRjUhIiJGNUY1IiIjRjVGNSoqIiIlRjUpRjRGOEY1LCZGNUY1RjZGNUY1LCZGNUY1KiRGNkY3RjVGNUY3RjUpLCYqJkY2RjUtSSNEREdGKDYjRjZGNUY1KigsJkY1RjVGNkY3RjVGNEY3RjxGN0Y1RjhGNTcjRi4=

KydJInlHNiIhIiUhIiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiE=

The quadrant invariants I1x;J1y;

LCYqJkkidEc2IiIiIi1JIlFHRiU2JEkieEdGJSIiIUYmRiYqJEYqISIiRiY=

LCgqKkkidEc2IiIiIkkieUdGJUYmLCZGJkYmKiRGJyEiIkYmRiYtSSJRR0YlNiQiIiFGJ0YmRioqJkYkRiYtRiw2JEYuRi5GJkYmKipGJ0YmLCYqJkYkRiZGJ0YmRipGJkYmRiZGJEYqLCZGJkYmRidGJkYqRiY=

series(I1x,x,1); series(J1y,y,2)

KylJInhHNiIiIiIhIiIqJkkidEdGJEYlLUkiUUdGJDYkIiIhRixGJUYsLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiVGJQ==

KydJInlHNiIsKComSSJ0R0YkIiIiLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiM2I0kiUUdGJDYkIiIhRjVGKCEiIiomRidGKC1GM0Y0RihGNiokRidGNkYoRigtSSJPR0YtNiNGKEYx

series(Jy,y,1)

KydJInlHNiIhIiUhIiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiE=

Delete the pole at 0 by multiplying by J1 series(Jy*J1y,y,2)

KydJInlHNiIsKCooIiIlIiIiSSJ0R0YkRigtLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIzYjSSJRR0YkNiQiIiFGNkYoRigqKEYnRihGKUYoLUY0RjVGKEYoKiZGJ0YoRikhIiJGOkY2LUkiT0dGLjYjRihGKA==

Now consider the pole at -1 series(Jy*J1y,y=-1,1)

KycsJiIiIkYkSSJ5RzYiRiQsJCoqIiIlRiQpLCZGJEYkSSJ0R0YmRiQiIiNGJClGLCIiJCEiIiwmRixGMEYkRjBGJEYkISIkLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjRiQhIiM=

serJ1:=convert(series(J1y,y=-1,0),polynom);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZzZXJKMUdGKCooLCZJInRHRighIiIiIiJGMUYyRjBGMSwmRjJGMkkieUdGKEYyRjE3I0Yu

factor(series(Jy*J1y-4*J1y^3,y=-1,2))

KycsJiIiIkYkSSJ5RzYiRiQsJCoqIiIlRiQpLCZGJEYkSSJ0R0YmRiQiIiNGJCwsKihGLUYkKUYsRi1GJC1JIlFHRiY2JCIiIUY0RiRGJComLUkjRERHRiY2IyEiIkYkRjBGJEY5KiZGNkYkRixGJEY5RixGJEYkRiRGJClGLCIiJEY5RjkhIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJEY5

A2:=convert(%,polynom)/serJ1^2

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNBMkdGKCwkKiwiIiUiIiIpLCZGMUYxSSJ0R0YoRjEiIiNGMSwsKihGNUYxKUY0RjVGMS1JIlFHRig2JCIiIUY8RjFGMSomLUkjRERHRig2IyEiIkYxRjhGMUZBKiZGPkYxRjRGMUZBRjRGMUYxRjFGMUY0RkEpLCZGNEZBRjFGQUY1RkFGQTcjRi4=

collect(A2,[DD,Q],factor)

LCgqJiwmIiIlIiIiKiZGJUYmSSJ0RzYiRiZGJkYmLUkjRERHRik2IyEiIkYmRiYqKCIiKUYmRihGJi1JIlFHRik2JCIiIUYzRiZGLSooRiVGJkYoRi0sJkYmRiZGKEYmRiZGLQ==

Next order factor(series(Jy*J1y-4*J1y^3 -A2*J1y^2,y=-1,3))

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

A1:=convert(%,polynom)/serJ1;

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

collect(A1,[DD,Q],distributed,factor)

LDIqJiksJiIiIkYmSSJ0RzYiRiYiIiNGJiktSSNEREdGKDYjISIiRilGJkYmKiwiIiVGJkYlRiZGJ0YmRitGJi1JIlFHRig2JCIiIUY0RiZGLiosRilGJkYlRiYsJiomIiIkRiZGJ0YmRiZGJkYmRiZGJ0YuRitGJkYmKipGMEYmRiRGJkYnRi4tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKDYjRixGLUYmRi4qJiwmIiIpRi4qJkZDRiZGJ0YmRi5GJi1GMjYkRjRGLkYmRiYqKEYwRiYpRidGKUYmKUYxRilGJkYmKiYsJkYwRiYqJkYwRiZGJ0YmRiZGJkYxRiZGJiomLCgqJiIjOEYmRkhGJkYmKiZGKUYmRidGJkYuRiZGJkYmRkhGLkYm

Now we have eliminated all poles in the y-invariant factor(series(Jy*J1y-4*J1y^3 -A2*J1y^2-A1*J1y,y=-1,4))

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

What happened to the x invariant meanwhile? Does it still has a pole at 0? ser:=factor(series(Ix*I1x-4*I1x^3 -A2*I1x^2-A1*I1x,x,2))

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

We can express series evaluated at y=-1 in terms of more combinatorial series thanks to the functional equation(s) Dm1sol:=isolate(subs(y=-1,eqD),DD(-1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdEbTFzb2xHRigvLUkjRERHRig2IyEiIiomLCYiIiJGMiooIiIjRjVJInRHRihGNS1JIlVHRig2JCIiIUY8RjVGMkY1LCZGOEYyRjVGMkYyNyNGLg==

Hence there is no double pole at x=0 factor(subs(%,ser))

KydJInhHNiIsJCooIiIlIiIiLDYqKEYnRigpSSJ0R0YkIiIjRigtSSJVR0YkNiQiIiFGMUYoRigqKEYtRigtSSJRR0YkNiRGMSEiIkYoRitGKEY2KiYtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSNEREdGJDYjRjZGKEYrRihGNiooRi1GKC0tJkY6NiNGKDYjRjRGMEYoRitGKEYoKihGLUYoLS1GRDYjRi9GMEYoRitGKEY2KihGLUYoRixGKEYuRihGKCooRi1GKEYsRihGM0YoRjYqKEYtRihGLEYoRjhGKEY2RjhGNiomIiIkRihGLEYoRihGKEYsRjZGNkY2LUkiT0dGO0ZFRjE=

What about the simple pole? Dm1diff:=isolate(limit(diff(eqD,y),y=-1),(D(DD))(-1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShEbTFkaWZmR0YoLy0tSSJER0YlNiNJI0RER0YoNiMhIiIqJiwoKiYtRjNGNCIiIkkidEdGKEY6RjUqKCIiI0Y6RjtGOi1JIlVHRig2JCIiIUY1RjpGOiooRj1GOkY7RjotRj82JEZBRkFGOkY1RjosJkY7RjVGOkY1RjU3I0Yu

factor(subs(Dm1diff,Dm1sol,series(Ix*I1x-4*I1x^3 -A2*I1x^2-A1*I1x,x,1)))

KydJInhHNiIsMCooIiIpIiIiSSJ0R0YkRigtSSJVR0YkNiQiIiEhIiJGKEYuKihGJ0YoRilGKC1JIlFHRiRGLEYoRigqKEYnRihGKUYoLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKDYjRjE2JEYtRi1GKEYuKihGJ0YoRilGKC0tRjU2I0YrRjxGKEYoKiZGJ0YoRipGKEYuKiZGJ0YoRjBGKEYoRidGLkYuLUkiT0dGN0Y6Ri0=

One more series to express Q0m1sol:=isolate(limit(subs(y=-1,eqQ),x=0),Q(0, -1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShRMG0xc29sR0YoLy1JIlFHRig2JCIiISEiIiomLCYqJkkidEdGKCIiIi0tJkkiREdGJTYjRjg2I0YwNiRGMkYyRjhGM0Y4RjNGOCwmRjdGM0Y4RjNGMzcjRi4=

factor(subs(Dm1diff,Dm1sol,Q0m1sol,series(Ix*I1x-4*I1x^3 -A2*I1x^2-A1*I1x,x,1)))

KydJInhHNiIsKCooIiIpIiIiSSJ0R0YkRigtSSJVR0YkNiQiIiEhIiJGKEYuKihGJ0YoRilGKC0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRig2I0YrNiRGLUYtRihGKComRidGKEYqRihGLkYuLUkiT0dGNEY3Ri0=

And the final one series(subs(y=-1,eqU),x,2)

KydJInhHNiIsKC1JIlVHRiQ2JCIiISEiIkYqKiZJInRHRiQiIiJGJkYtRioqJkYsRi0tLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YtNiNGJzYkRilGKUYtRi1GLS1JIk9HRjNGNiIiIw==

U0m1sol:=isolate(op(1,%), U(0,-1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShVMG0xc29sR0YoLy1JIlVHRig2JCIiISEiIiwkKihJInRHRigiIiItLSZJIkRHRiU2I0Y3NiNGMDYkRjJGMkY3LCZGNkYzRjdGM0YzRjM3I0Yu

There we are: the new x-invariant has no pole at 0 factor(subs(Dm1diff,Dm1sol,Q0m1sol,U0m1sol,series(Ix*I1x-4*I1x^3 -A2*I1x^2-A1*I1x,x,3)))

KydJInhHNiIsJCooIiIlIiIiLDIqKiIiI0YoLUkiVUdGJDYkIiIhRi9GKC0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRig2I0kiUUdGJEYuRigpSSJ0R0YkIiIkRighIiIqKkYrRihGLEYoLS1GMjYjRi1GLkYoRjpGKEYoKiYpRjtGK0YoLS0mRjM2JEYoRihGQUYuRihGKComRkNGKC0tRkZGOEYuRihGPSooRitGKEZDRihGLEYoRigqJkY7RihGMEYoRigqJkY7RihGP0YoRj1GKEY9RihGO0Y9RihGLy1JIk9HRjRGN0Yo

Have we found trivial invariants? YES, at least experimentally n:=10: factor(series(subs((D(DD))(-1)=subs(y=-1,diff(Dser(n),y)) ,DD(-1)=subs(y=-1,Dser(n)), U(x,0)=subs(y=0,User(n)),Q(x,0)=subs(y=0,Qser(n)), Q(0,0)=subs(x=0,y=0,Qser(n)), Q(0,-1)=subs(x=0,y=-1,Qser(n)),Ix*(I1x)-4*I1x^3 -A2*I1x^2-A1*I1x),t,n))

KzVJInRHNiIhIiUhIiIiIiUiIiJGJyIiIyIjNyIiJCIjS0YnIiMnKiIiJiIkTyQiIiciJSc0IiIiKC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIp

factor(series(subs((D(DD))(-1)=subs(y=-1,diff(Dser(n),y)) ,DD(-1)=subs(y=-1,Dser(n)), DD(y)=Dser(n), Q(0,y)=subs(x=0,Qser(n)),Q(0,0)=subs(x=0,y=0,Qser(n)), Q(0,-1)=subs(x=0,y=-1,Qser(n)),Jy*(J1y)-4*J1y^3 -A2*J1y^2-A1*J1y),t,n))

KzNJInRHNiIhIiUhIiIiIiUiIiJGJyIiIyIjNyIiJCIjS0YnIiMnKiIiJiIkTyQiIictSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIiKA==

Let us form the ratio associated with the new invariants, and check that it is a multiple of xy Ifinal:=Ix*(I1x)-4*I1x^3 -A2*I1x^2-A1*I1x; Jfinal:=Jy*(J1y)-4*J1y^3 -A2*J1y^2-A1*J1y

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdKZmluYWxHRigsKiouIiIlIiIiKSwmRjFGMUkidEdGKEYxIiIjRjEsLCooRjVGMSlGNEY1RjEtSSJRR0YoNiQiIiFGPEYxRjEqJi1JI0RER0YoNiMhIiJGMUY4RjFGQSomRj5GMUY0RjFGQUY0RjFGMUYxRjFGNEZBKSwmRjRGQUYxRkFGNUZBKSwoKipGNEYxSSJ5R0YoRjEsJkYxRjEqJEZIRkFGMUYxLUY6NiRGPEZIRjFGQSomRjRGMUY5RjFGMSoqRkhGMSwmKiZGNEYxRkhGMUZBRjFGMUYxRjRGQSwmRjFGMUZIRjFGQUYxRjVGMUYxKiZGMEYxKUZGIiIkRjFGQSosLEgqKEYwRjEpRjlGNUYxKUY0RjBGMUZBKipGMEYxRjlGMUY+RjFGWUYxRjEqJilGPkY1RjFGWUYxRkEqKkYwRjFGOUYxRj5GMSlGNEZURjFGMSooRjVGMUZmbkYxRmhuRjFGQSooIiIpRjEtRjo2JEY8RkFGMUZobkYxRjEqKEYwRjEtLUkiREdGJTYjRj9GQEYxRmhuRjFGMSooRjBGMUY5RjFGaG5GMUZBKiZGZm5GMUY4RjFGQSooIiInRjFGPkYxRmhuRjFGQSooRltvRjFGXG9GMUY4RjFGMSooRltvRjFGX29GMUY4RjFGMSooRjBGMUY4RjFGOUYxRkEqKEZbb0YxRj5GMUY4RjFGQSooRjBGMUY0RjFGX29GMUYxKihGNUYxRj5GMUY0RjFGQSomIiM4RjFGOEYxRkEqJkY1RjFGNEYxRjFGMUZBRjFGM0YxRjhGQUZERkFGRkYxRkEqKEZGRjEsJiokKUZPRjVGMUYxKipGMEYxRjhGMUZRRjFGSUYxRkFGMSksJiomRkhGMS1GPzYjRkhGMUYxKigsJkYxRjFGSEZBRjFGNEZBRlFGQUYxRjVGMUYxNyNGLg==

Here is the ratio. We first observe that tI(x)-tJ(y) is indeed divisible by the kernel. ratio:=factor(subs(isolate(eqUfinal,U(x,0)), isolate(eqQ,Q(x,0)),(Ifinal-Jfinal)/(1-t*tSteps))): denom(%)

KihJInhHNiIiIiIpLCZGJUYlSSJ5R0YkRiUiIiNGJSlJInRHRiRGKUYl

This ratio should be a multiple of xy. Let us expand it in it first. factor(subs(isolate(eqD,U(0,y)),Dm1diff,Dm1sol, Q0m1sol,U0m1sol,series(ratio,x,2)))

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

We want to prove that this is zero co0:=numer(op(1,%))

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

factor(series(eqU,x,2))

KydJInhHNiIsJComLDAqKClJInlHRiQiIiMiIiItSSNEREdGJDYjRipGLEkidEdGJEYsISIiKigtSSJVR0YkNiQiIiFGKkYsRjBGLEYpRixGMSooRjBGLEYtRixGKkYsRjEqKEYwRiwtLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YsNiNGNEY1RixGKkYsRjEqJkYwRiwtRjo2JEY2RjZGLEYsKiZGM0YsRipGLEYsKiZGMEYsRjlGLEYxRixGKkYxRjFGLC1JIk9HRj1GQEYr

d1U0y:=isolate(op(1,%),(D[1](U))(0, y))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZkMVUweUdGKC8tLSZJIkRHRiU2IyIiIjYjSSJVR0YoNiQiIiFJInlHRigqJiwsKigpRjkiIiNGNC1JI0RER0YoNiNGOUY0SSJ0R0YoRjRGNCooLUY2RjdGNEZCRjRGPUY0RjQqKEZCRjRGP0Y0RjlGNEY0KiZGQkY0LUYwNiRGOEY4RjQhIiIqJkZERjRGOUY0RklGNCwmKiZGQkY0RjlGNEZJRkJGSUZJNyNGLg==

co0:=factor(subs(%,co0))

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

U0yDsol:=isolate(eqD,U(0,y))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShVMHlEc29sR0YoLy1JIlVHRig2JCIiIUkieUdGKCwkKiojIiIiIiIjRjcsKComLCYqJkkidEdGKEY3RjNGNyEiIkY3RjdGNy1JI0RER0YoNiNGM0Y3RjdGN0Y+KipGOEY3Rj1GN0YzRj4tRjA2JEYyRjJGN0Y3RjdGPUY+LCZGN0Y3KiRGM0Y+RjdGPkY3NyNGLg==

co0:=factor(subs(U0yDsol,co0))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRjbzBHRigsJCosIiIpIiIiLCZGMUYxSSJ5R0YoRjFGMSwuKigtSSJRR0YoNiQiIiFGM0YxSSJ0R0YoRjEpRjMiIiNGMSEiIiooLS0mSSJER0YlNiNGMTYjRjdGOEYxRjpGMUYzRjFGPSomRjpGMS1GQDYkRjlGOUYxRjEqJkY2RjFGM0YxRjEqJkY/RjFGOkYxRj1GM0Y9RjFGOkYxRjNGMUY9NyNGLg==

factor(series(eqQ,x,2));

KydJInhHNiIsJComLC4qKC1JIlFHRiQ2JCIiIUkieUdGJCIiIkkidEdGJEYuKUYtIiIjRi4hIiIqKC0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRi42I0YqRitGLkYvRi5GLUYuRjIqJkYvRi4tRjU2JEYsRixGLkYuKiZGKUYuRi1GLkYuKiZGNEYuRi9GLkYyRi1GMkYuRi1GMkYyRiwtSSJPR0Y4RjtGLg==

d1Q0y:=isolate(op(1,%),(D[1](Q))(0, y))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZkMVEweUdGKC8tLSZJIkRHRiU2IyIiIjYjSSJRR0YoNiQiIiFJInlHRigqJiwqKigtRjZGN0Y0SSJ0R0YoRjQpRjkiIiNGNEY0KiZGPkY0LUYwNiRGOEY4RjQhIiIqJkY9RjRGOUY0RkRGOUY0RjQsJiomRj5GNEY5RjRGREY+RkRGRDcjRi4=

co0:=factor(subs(%,co0))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRjbzBHRigiIiE3I0Yu

OK Now in y: immediate series(ratio,y,1)

KyVJInlHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGKw==

So we have indeed a trivial pair of invariants We need to determine the roots of the polynomial denoted P in the paper. We first exploit the root Y of Delta with(gfun): Yser:=op(algeqtoseries(numer(Delta),t,y,14,true))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc2VyR0YoKz9JInRHRigiIiUiIiMiI0tGMEYyIiImIiQ/JCIiJyIkbygiIigiJUtTIiIpIiZPViIiIioiJlM5JyIjNSInU01EIiM2Iihfdi8iIiM3IigrY10lIiM4IilHKCopKT0iIzkiKT8+KyMpIiM6LUkiT0dGJTYjIiIiIiM7NyNGLg==

One root must be J1(Y) simplerootser:=series(subs(Q(0, y)=subs(x=0,Qser(n)),Q(0, 0)=subs(x=0,y=0,Qser(n)), y=Yser,J1y),t,n)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS5zaW1wbGVyb290c2VyR0YoKzFJInRHRigiIiUiIiIiIzciIiRGMkYwIiMlKSIiJiIkMyMiIiciJCcqKSIiKC1JIk9HRiU2I0YxIiIpNyNGLg==

We then exploit the double root of J(y) collect(numer(op(1,op(2,Jy))),DD,factor)

LCgqKkkieUc2IiIiIkkidEdGJUYmLCZGJkYmRiRGJkYmLUkjRERHRiU2I0YkRiZGJkYmRiZGJCEiIg==

n:=20: ser:=subs(DD(y)=Dser(n),%%): Y0ser:=op(algeqtoseries(ser,t,y,15,true))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZZMHNlckdGKCtDSSJ0R0YoIiIiIiIhIiIjRjAiIilGMiIjXyIiJCIkWyQiIiUiJSNcIyIiJiImcyU9IiInIicleVMiIiIoIihLazQiRjMiKCllKG8pIiIqIikhR0YpcCIjNSIqZywoemMiIzYiKztMJW9tJSIjNyIsdyJwL29RIiM4Ii1XTjo/SUsiIzktSSJPR0YlNiNGMCIjOjcjRi4=

Y0 cannot be a root of this (because of the term in 1/t) collect(diff(J1y,y),Q,factor)

LCgqJkkidEc2IiIiIi1JIlFHRiU2JCIiIUkieUdGJUYmISIiKigsJkYmRiZGK0YmRiZGJEYmLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMiIiM2I0YoRilGJkYsKigsKComRiRGJilGK0Y3RiZGJiooRjdGJkYkRiZGK0YmRiZGJkYsRiZGJEYsKUYuRjdGLEYs

Value of J1(y) at Y0 doublerootser:=series(subs(Q(0, y)=subs(x=0,Qser(n)),Q(0, 0)=subs(x=0,y=0,Qser(n)), y=Y0ser,J1y),t,n)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS5kb3VibGVyb290c2VyR0YoK0FJInRHRigjIiIiIiIjISIiRjNGMUYzRjIhIiUiIiQhIzYiIiUhI1EiIiYhJFUiIiInISQtJiIiKCElTT8iIikhJSZvKCIiKiEmIXBKIiM1ISdwbDciIzYhJytuXyIjNyEodCw9IyIjOC1JIk9HRiU2I0YxIiM5NyNGLg==

Alternatively, we could have exploited the double root of I(x) numer(op(1,Ix))

LCoqKiIiIyIiIilJInRHNiJGJEYlLUkiVUdGKDYkSSJ4R0YoIiIhRiVGLEYlRiUqKEYkRiVGJ0YlKUYsRiRGJUYlKiZGJEYlRidGJUYlRiwhIiI=

n:=20: ser:=subs(U(x,0)=subs(y=0,User(n)), numer(op(1,Ix))): Xser:=op(algeqtoseries(ser,t,x,15,true))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc2VyR0YoK0FJInRHRigiIiMiIiIiIikiIiQiIiVGNCIjayIiJiIjJSkiIiciJHcnIiIoIiVvOUYyIiVrJikiIioiJldaIyIjNSInd0o3IiM2IidHIj4lIiM3Iig/QyQ+IiM4IihvP0IoIiM5IilbXDxLIiM6LUkiT0dGJTYjRjEiIzs3I0Yu

X cannot be a root of this: diff(I1x,x)

LCYqJkkidEc2IiIiIi0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiY2I0kiUUdGJTYkSSJ4R0YlIiIhRiZGJiokKUYyIiIjISIiRjc=

Value of Ix at X (same as the double root above, as expected) series(subs(Q(x,0)=subs(y=0,Qser(n)), x=Xser,I1x),t,n)

K0FJInRHNiIjIiIiIiIjISIiRihGJkYoRichIiUiIiQhIzYiIiUhI1EiIiYhJFUiIiInISQtJiIiKCElTT8iIikhJSZvKCIiKiEmIXBKIiM1ISdwbDciIzYhJytuXyIjNyEodCw9IyIjOC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YmIiM5

The two following invariant vanish Iformnul:=Ix*I1-4*(I1-A)^2*(I1-B); Jformnul:=Jy*J1-4*(J1-A)^2*(J1-B);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlJZm9ybW51bEdGKCwmKiYpLCoqKCIiIyIiIkkidEdGKEY0LUkiVUdGKDYkSSJ4R0YoIiIhRjRGNCokRjUhIiJGPComRjNGNEY5RjRGNComRjNGNEY5RjxGNEYzRjRJI0kxR0YoRjRGNCooIiIlRjQpLCZGP0Y0SSJBR0YoRjxGM0Y0LCZGP0Y0SSJCR0YoRjxGNEY8NyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlKZm9ybW51bEdGKCwmKigsJiokKSwmKiZJInRHRigiIiJJInlHRihGNiEiIkY2RjYiIiNGNkY2KioiIiVGNilGNUY5RjYsJkY2RjZGN0Y2RjYsJkY2RjYqJEY3RjhGNkY2RjhGNiksJiomRjdGNi1JI0RER0YoNiNGN0Y2RjYqKCwmRjZGNkY3RjhGNkY1RjhGPUY4RjZGOUY2SSNKMUdGKEY2RjYqKEY7RjYpLCZGSEY2SSJBR0YoRjhGOUY2LCZGSEY2SSJCR0YoRjhGNkY4NyNGLg==

We have determined B= J1(Y). Now we need a relation between the double root A and the simple root B factor(series(subs(J1=J1y,Jformnul),y,2))

KydJInlHNiIsJCooIiIlIiIiLCoqKClJIkFHRiQiIiNGKEkiQkdGJEYoSSJ0R0YkRihGKComKUYvRi1GKC0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRi02I0kiUUdGJDYkIiIhRj1GKEYoKiZGMUYoLUY7RjxGKEYoRighIiJGKEYvRkBGKEY9LUkiT0dGNjYjRihGKA==

doublerootsquared:=(op(2,subs(Q(0, 0)=Q00,(D[2](Q))(0, 0)=Q01,isolate(op(1,%),A^2))))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+STJkb3VibGVyb290c3F1YXJlZEdGKCooLCgqJkkkUTAwR0YoIiIiKUkidEdGKCIiI0YyISIiKiZJJFEwMUdGKEYyRjNGMkY2RjJGMkYySSJCR0YoRjZGNEY2NyNGLg==

Let us check, and choose the sign of the double root n:=10: series(subs(Q00=coeff(coeff(Qser(n),x,0),y,0),Q01=coeff(coeff(Qser(n),x,0),y,1),B=simplerootser,doublerootsquared),t,n)

KzFJInRHNiIjIiIiIiIlISIjISIiIiIhRilGJiEiJCIiIyEiKiIiJCEjSEYnLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiYiIiY=

subs(csgn(1/t)=1,series(sqrt(%),t))

KzFJInRHNiIjIiIiIiIjISIiRihGJkYoRichIiUiIiQhIzYiIiUhI1EiIiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJiIiJw==

series(doublerootser,t,7)

KzNJInRHNiIjIiIiIiIjISIiRihGJkYoRichIiUiIiQhIzYiIiUhI1EiIiYhJFUiIiInLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiYiIig=

Other relations between the roots A and B ? They involve unknown series like D(-1) or U(0,0) factor(series(subs(J1=J1y,Jformnul),y=-1,2))

KycsJiIiIkYkSSJ5RzYiRiQsJCoqIiIlRiQpLCZGJEYkSSJ0R0YmRiQiIiNGJCwwKiYtSSNEREdGJjYjISIiRiQpRixGLUYkRiQqKEYtRiRGNEYkLUkiUUdGJjYkIiIhRjlGJEYzKihGLUYkSSJBR0YmRiRGLEYkRiQqJkkiQkdGJkYkRixGJEYkKiZGMEYkRixGJEYkRixGM0YkRjNGJClGLCIiJEYzRiQhIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJEYz

factor(series(subs(I1=I1x,Iformnul),x,1))

KydJInhHNiIsJCooIiIlIiIiLCwqKCIiI0YoKUkidEdGJEYrRigtSSJVR0YkNiQiIiFGMUYoRigqKEYrRihGLEYoLUkiUUdGJEYwRighIiIqKEYrRihJIkFHRiRGKEYtRihGKComSSJCR0YkRihGLUYoRihGKEY1RihGLUY1RighIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEY1

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv

The definition of the kernel in [BeBMRa] is a bit different Kpol:=-numer(1-t*tSteps)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVLcG9sR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGMkYxRjZGMUYxKihGMEYxRjNGMUY1RjFGMSomRjBGMUY2RjFGMSomRjNGMUY2RjEhIiJGMEYxNyNGLg==

Let us examine the 4 roots of the discriminant of Kpol in x disc:=factor(discrim(Kpol,x))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkaXNjR0YoKiYsLiomKUkidEdGKCIiIyIiIilJInlHRigiIiRGNEY0KigiIiVGNEYxRjQpRjZGM0Y0ISIiKigiIilGNEYxRjRGNkY0RjsqKEYzRjRGMkY0RjpGNEY7KiZGOUY0RjFGNEY7RjZGNEY0RjZGNDcjRi4=

factor(%/Delta)

KiQpSSJ5RzYiIiIjIiIi

One root, in fact y2, is very close to 0 fsolve(subs(t=1/10,disc))

NiYkIiIhRiQkIitsbD4pUiUhIzYkIitRUF1JWiEiKiQiK0k5YkE+ISIp

plot(subs(t=1/10,disc),y=-2..20,v=-3..3)

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Here we zoom in to see y1=0 and y2 plot(subs(t=1/10,disc),y=-0.1..0.1,v=-0.01..0.01)

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What are the series y3 and y4? They diverge in 1/t, while y2=O(t^2) is our series Y with(gfun):map(allvalues,algeqtoseries(disc,t,y,3))

NyUsKiokSSJ0RzYiISIiIiIiKiYiIiNGKClGJSNGKEYqRidGKEYqRigtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiMqJEYrRihGKCwqRiRGKEYpRidGKkYoRi1GKCspRiUiIiVGKiIjS0Y2LUYuNiNGKCIiJg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv

Y0ser

K0NJInRHNiIiIiIiIiEiIiNGJSIiKUYnIiNfIiIkIiRbJCIiJSIlI1wjIiImIiZzJT0iIiciJyV5UyIiIigiKEtrNCJGKCIoKWUobykiIioiKSFHRilwIiM1IipnLCh6YyIjNiIrO0wlb20lIiM3Iix3InAvb1EiIzgiLVdOOj9JSyIjOS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiM6

The curve L intersects the real axis at Y(x2). Is this larger than Y0? Kpol:=-numer(1-t*tSteps)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVLcG9sR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGMkYxRjZGMUYxKihGMEYxRjNGMUY1RjFGMSomRjBGMUY2RjFGMSomRjNGMUY2RjEhIiJGMEYxNyNGLg==

discx:=factor(discrim(Kpol,y))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZkaXNjeEdGKCwwKiYpSSJ0R0YoIiIjIiIiKUkieEdGKCIiJUYzRjMqKEYyRjNGMEYzKUY1RjJGMyEiIiooRjJGM0YxRjMpRjUiIiRGM0Y5KihGNkYzRjBGM0Y1RjNGOSokRjBGM0YzKihGMkYzRjFGM0Y1RjNGOSokRjhGM0YzNyNGLg==

plot(subs(t=1/10,discx),x=-2..13,v=-3..3)

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Here we zoom to see x1 and x2 plot(subs(t=1/10,discx),x=-0.1..0.5,v=-0.01..0.01)

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map(allvalues,algeqtoseries(discx,t,x,6))

NyYsKiokSSJ0RzYiISIiIiIiIiIjRigqJiNGKCIiJUYoKUYlRilGKEYnLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjKiQpRiUjIiImRilGKEYoLCxGJEYoRilGJyomRilGKEYlRihGJyomIyIjOkYsRihGLUYoRidGLkYoLDBGJUYoKiZGKUYoKUYlIyIiJEYpRihGKComRilGKEYtRihGKComRilGKEY1RihGKComRjdGKClGJUZBRihGKComIiM3RigpRiUjIiIoRilGKEYoLUYvNiMqJClGJUYsRihGKCwwRiVGKEY+RidGQkYoRkNGJ0ZERihGRkYnRktGKA==

x2ser:=op(3,%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZ4MnNlckdGKCwwSSJ0R0YoIiIiKiYiIiNGMClGLyMiIiRGMkYwRjAqJkYyRjApRi9GMkYwRjAqJkYyRjApRi8jIiImRjJGMEYwKiZGO0YwKUYvRjVGMEYwKiYiIzdGMClGLyMiIihGMkYwRjAtSSJPR0YlNiMqJClGLyIiJUYwRjA3I0Yu

Now what is Y(x2) ? simplify(subs(x=convert(x2ser,polynom),t=s^2,Kpol)) assuming s>0

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

the expansion of Y(x2) is much larger thn Y0 algeqtoseries(%,s,y,5)

NyMsLCokSSJzRzYiISIiIiIiIiIiISIiKiYtSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmKiQpSSNfWkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IiIiIyIiIiIiIiIjPyIiIiIiIilJInNHNiIjIiIkIiIjIiIiIiIiKiQpSSJzRzYiIiIjIiIiISIiLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiQpSSJzRzYiIyIiKiIiJSIiIiIiIg==

Y0ser

K0NJInRHNiIiIiIiIiEiIiNGJSIiKUYnIiNfIiIkIiRbJCIiJSIlI1wjIiImIiZzJT0iIiciJyV5UyIiIigiKEtrNCJGKCIoKWUobykiIioiKSFHRilwIiM1IipnLCh6YyIjNiIrO0wlb20lIiM3Iix3InAvb1EiIzgiLVdOOj9JSyIjOS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiM6

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv

With the weak invariant Jyweak:=alpha*(w(y)-a)^2/(w(y)-w(0))/(w(y)-w(-1))^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdKeXdlYWtHRigqKkkmYWxwaGFHRigiIiIpLCYtSSJ3R0YoNiNJInlHRihGMEkiYUdGKCEiIiIiI0YwLCZGM0YwLUY0NiMiIiFGOEY4KSwmRjNGMC1GNDYjRjhGOEY5Rjg3I0Yu

We can express the constants alpha and a in terms of w Let us expand first around 0 series(Jyweak-Jy,y,2)

KydJInlHNiIsJioqSSZhbHBoYUdGJCIiIiksJi1JIndHRiQ2IyIiIUYoSSJhR0YkISIiIiIjRigtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRixGLUYwKSwmRitGKC1GLDYjRjBGMEYxRjBGKCIiJUYoRjAtSSJPR0Y1NiNGKEYu

alphaw1:=factor(solve(convert(%,polynom),alpha))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShhbHBoYXcxR0YoLCQqKiIiJSIiIi0tSSJER0YlNiNJIndHRig2IyIiIUYxKSwmLUY2RjdGMS1GNjYjISIiRj4iIiNGMSksJkY7RjFJImFHRihGPkY/Rj5GPjcjRi4=

Or conversely dw0:=factor(solve(alphaw1-alpha, D(w)(0)))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRkdzBHRigsJCoqIyIiIiIiJUYxSSZhbHBoYUdGKEYxKSwmLUkid0dGKDYjIiIhRjFJImFHRighIiIiIiNGMSksJkY2RjEtRjc2I0Y7RjtGPEY7Rjs3I0Yu

Expansion at -1 series(Jy,y=-1,1); factor(series(subs(alpha=alphaw1,Jyweak),y=-1,1))

KycsJiIiIkYkSSJ5RzYiRiQsJCooIiIlRiQpLCZGJEYkSSJ0R0YmRiQiIiNGJClGLEYtISIiRiQhIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJEYv

KycsJiIiIkYkSSJ5RzYiRiQsJCouIiIlRiQpLCYtSSJ3R0YmNiMhIiJGJEkiYUdGJkYvIiIjRiQsJi1GLTYjIiIhRiRGLEYvRiQtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjRi1GNEYkKS1GN0YuRjFGLyksJkYzRiRGMEYvRjFGL0YkISIjLUkiT0dGOTYjRiRGLw==

factor(op(1,%)/op(1,%%))-1

LCYqMCksJi1JIndHNiI2IyEiIiIiIkkiYUdGKEYqIiIjRissJi1GJzYjIiIhRitGJkYqRistLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKDYjRidGMEYrKS1GM0YpRi1GKiksJkYvRitGLEYqRi1GKiksJkYrRitJInRHRihGK0YtRiopRj9GLUYrRitGK0Yq

Quadratic equation for a expraw1:=map(c->factor(c*D(w)(-1)^2*(1 + t)^2/t^2/((w(0) - w(-1))*D(w)(0))),%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShleHByYXcxR0YoLCYqJiksJi1JIndHRig2IyEiIiIiIkkiYUdGKEY1IiIjRjYpLCYtRjM2IyIiIUY2RjdGNUY4RjVGNiosKS0tSSJER0YlNiNGM0Y0RjhGNiksJkY2RjZJInRHRihGNkY4RjYpRkZGOEY1LCZGO0Y2RjJGNUY1LUZBRjxGNUY1NyNGLg==

The values w(0, w(1), D(w)(0), D(w(-1)) are related to g2 and g3 using the differential equation for w(y) eqw:=4*dy*(diff(w(y), y))^2-(4*w(y)^3-g2*w(y)-g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcXdHRigsKioqIiIlIiIiLC4qJilJInRHRigiIiNGMSlJInlHRigiIiRGMUYxKihGMEYxRjRGMSlGOEY2RjEhIiIqKCIiKUYxRjRGMUY4RjFGPCooRjZGMUY1RjFGO0YxRjwqJkYwRjFGNEYxRjxGOEYxRjFGOEYxKS1JJWRpZmZHRiY2JC1JIndHRig2I0Y4RjhGNkYxRjEqJkYwRjEpRkVGOUYxRjwqJkkjZzJHRihGMUZFRjFGMUkjZzNHRihGMTcjRi4=

g3w0:=factor(solve(limit(eqw,y=0),g3))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVnM3cwR0YoKiYtSSJ3R0YoNiMiIiEiIiIsJiomIiIlRjMpRi8iIiNGM0YzSSNnMkdGKCEiIkYzNyNGLg==

eqw1:=collect(subs(g3=g3w0,eqw),[w,g2])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcXcxR0YoLCwqKiIiJSIiIiwuKiYpSSJ0R0YoIiIjRjEpSSJ5R0YoIiIkRjFGMSooRjBGMUY0RjEpRjhGNkYxISIiKigiIilGMUY0RjFGOEYxRjwqKEY2RjFGNUYxRjtGMUY8KiZGMEYxRjRGMUY8RjhGMUYxRjhGMSktSSVkaWZmR0YmNiQtSSJ3R0YoNiNGOEY4RjZGMUYxKiZGMEYxKUZFRjlGMUY8KiZJI2cyR0YoRjFGRUYxRjEqJkYwRjEpLUZGNiMiIiFGOUYxRjEqJkZLRjFGTkYxRjw3I0Yu

series(eqw1,y=0,2)

KydJInlHNiIsKCooIiM7IiIiKUkidEdGJCIiI0YoKS0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJIndHRiQ2IyIiIUYrRighIiIqKCIjN0YoKS1GNEY1RitGKEYtRihGNyomSSNnMkdGJEYoRi1GKEYoRigtSSJPR0YwNiNGKEYr

g2w0:=factor(solve(op(1,%),g2))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVnMncwR0YoLCYqKCIjOyIiIilJInRHRigiIiNGMS0tSSJER0YlNiNJIndHRig2IyIiIUYxRjEqJiIjN0YxKS1GOUY6RjRGMUYxNyNGLg==

g3w0:=factor(subs(g2=%,g3w0))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVnM3cwR0YoLCQqKCIiKSIiIi1JIndHRig2IyIiIUYxLCYqKCIiI0YxKUkidEdGKEY4RjEtLUkiREdGJTYjRjNGNEYxRjEqJClGMkY4RjFGMUYxISIiNyNGLg==

eqwfinal:=collect(subs(g3=g3w0,g2=g2w0,eqw1),w(y))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSllcXdmaW5hbEdGKCwsKioiIiUiIiIsLiomKUkidEdGKCIiI0YxKUkieUdGKCIiJEYxRjEqKEYwRjFGNEYxKUY4RjZGMSEiIiooIiIpRjFGNEYxRjhGMUY8KihGNkYxRjVGMUY7RjFGPComRjBGMUY0RjFGPEY4RjFGMUY4RjEpLUklZGlmZkdGJjYkLUkid0dGKDYjRjhGOEY2RjFGMSomRjBGMSlGRUY5RjFGPComLCYqKCIjO0YxRjRGMS0tSSJER0YlNiNGRjYjIiIhRjFGMSomIiM3RjEpLUZGRlJGNkYxRjFGMUZFRjFGMSomRjBGMSlGV0Y5RjFGMSomRktGMUZXRjFGPDcjRi4=

rel_derw:=collect(convert(factor(series(eqwfinal,y=-1,1)),polynom),[D(w)(-1),D(w)(0)],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlyZWxfZGVyd0dGKCwoKigiIiUiIiIpLCZGMUYxSSJ0R0YoRjEiIiNGMSktLUkiREdGJTYjSSJ3R0YoNiMhIiJGNUYxRjEqKiIjO0YxKUY0RjVGMSwmLUY7NiMiIiFGMS1GO0Y8Rj1GMS1GOEZDRjFGPSooRjBGMSwmRkVGMSomRjVGMUZCRjFGMUYxKUZBRjVGMUY9NyNGLg==

expraw2:=subs(isolate(rel_derw,D(w)(0)),expraw1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShleHByYXcyR0YoLCYqJiksJi1JIndHRig2IyEiIiIiIkkiYUdGKEY1IiIjRjYpLCYtRjM2IyIiIUY2RjdGNUY4RjVGNioqIiM7RjYpLS1JIkRHRiU2I0YzRjRGOEY2KSwmRjZGNkkidEdGKEY2RjhGNiwmKigiIiVGNkZFRjZGQEY2RjUqKEZKRjYsJkYyRjYqJkY4RjZGO0Y2RjZGNiksJkY7RjZGMkY1RjhGNkY2RjVGNjcjRi4=

expraw3:=map(factor,subs(isolate(rel_derw,D(w)(-1)^2),expraw1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShleHByYXczR0YoLCYqJiksJi1JIndHRig2IyEiIiIiIkkiYUdGKEY1IiIjRjYpLCYtRjM2IyIiIUY2RjdGNUY4RjVGNiooLCoqKCIiJUY2KUkidEdGKEY4RjYtLUkiREdGJTYjRjNGPEY2RjYqJkY4RjYpRjtGOEY2RjYqJkY7RjZGMkY2RjUqJClGMkY4RjZGNUY2RkJGNUZERjVGNTcjRi4=

dw0sol:=factor(solve(%,D(w)(0)))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdkdzBzb2xHRigsJCouKSwmLUkid0dGKDYjIiIhIiIiSSJhR0YoISIiIiIjRjYsJi1GMzYjRjhGNiomRjlGNkYyRjZGNkY2LCZGMkY2RjtGOEY2KUkidEdGKEY5RjgsKEY9RjZGO0Y2KiYiIiRGNkY3RjZGOEY4LChGPUY2RjtGOEY3RjhGOEY4NyNGLg==

alphaw1

LCQqKiIiJSIiIi0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJIndHRiw2IyIiIUYlKSwmLUYuRi9GJS1GLjYjISIiRjYiIiNGJSksJkYzRiVJImFHRixGNkY3RjZGNg==

Expression of Q(0,y) in terms of w, from the earlier paper [BeBMRa] Kpol:=factor(x*y*(t*tSteps-1)); subs(x=0,%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVLcG9sR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGMkYxRjZGMUYxKihGMEYxRjNGMUY1RjFGMSomRjBGMUY2RjFGMSomRjNGMUY2RjEhIiJGMEYxNyNGLg==

LCYqJkkidEc2IiIiIkkieUdGJUYmRiZGJEYm

Gy_early:=-y-(1+t)/t/(1+y); factor(%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlHeV9lYXJseUdGKCwmSSJ5R0YoISIiKigsJiIiIkYzSSJ0R0YoRjNGM0Y0RjAsJkYzRjNGL0YzRjBGMDcjRi4=

LCQqKCwqKiZJInRHNiIiIiIpSSJ5R0YnIiIjRihGKComRiZGKEYqRihGKEYmRihGKEYoRigsJkYoRihGKkYoISIiRiZGLkYu

residue r_early:=-(1+t)/t

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShyX2Vhcmx5R0YoLCQqJkkidEdGKCEiIiwmIiIiRjNGMEYzRjNGMTcjRi4=

limit(diff(w(y),y$2),y=-1)

LS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSJER0YmIiIjNiNJIndHRik2IyEiIg==

id_early:=-subs(x=0,Kpol)*Q(0,y)+Gy_early-1-r_early *(D(w)(-1)/(w(y)-w(-1))+1/2*(D@@2)(w)(-1)/D(w)(-1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlpZF9lYXJseUdGKCwsKiYsJiomSSJ0R0YoIiIiSSJ5R0YoRjNGM0YyRjNGMy1JIlFHRig2JCIiIUY0RjMhIiJGNEY5KigsJkYzRjNGMkYzRjNGMkY5LCZGM0YzRjRGM0Y5RjlGM0Y5KihGMkY5RjtGMywmKiYtLUkiREdGJTYjSSJ3R0YoNiNGOUYzLCYtRkQ2I0Y0RjMtRkRGRUY5RjlGMyooI0YzIiIjRjNGQEY5LS0tSSNAQEdGJTYkRkJGTEZDRkVGM0YzRjNGMzcjRi4=

id_early0:=limit(%,y=0)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSppZF9lYXJseTBHRigsKiomSSJ0R0YoIiIiLUkiUUdGKDYkIiIhRjVGMSEiIiomRjBGNiwmRjFGMUYwRjFGMUY2RjFGNiooRjBGNkY4RjEsJiomLS1JIkRHRiU2I0kid0dGKDYjRjZGMSwmLUZANiNGNUYxLUZARkFGNkY2RjEqKCNGMSIiI0YxRjxGNi0tLUkjQEBHRiU2JEY+RkhGP0ZBRjFGMUYxRjE3I0Yu

series(id_early,y=-1,2)

KyUsJiIiIkYkSSJ5RzYiRiQtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJCIiIQ==

J1y

LCgqKkkidEc2IiIiIkkieUdGJUYmLCZGJkYmKiRGJyEiIkYmRiYtSSJRR0YlNiQiIiFGJ0YmRioqJkYkRiYtRiw2JEYuRi5GJkYmKipGJ0YmLCYqJkYkRiZGJ0YmRipGJkYmRiZGJEYqLCZGJkYmRidGJkYqRiY=

J1yweak:=factor(subs(J1=J1y,isolate(id_early,Q(0,y)),isolate(id_early0,Q(0,0)),J1y))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShKMXl3ZWFrR0YoKi4sJiIiIkYwSSJ0R0YoRjBGMCwmLUkid0dGKDYjSSJ5R0YoISIiLUY0NiMiIiFGMEYwLS1JIkRHRiU2I0Y0NiNGN0YwRjFGNywmRjhGMC1GNEY/RjdGNywmRjNGN0ZBRjBGNzcjRi4=

factor(series(J1yweak-J1y,y=-1,2))

KyUsJiIiIkYkSSJ5RzYiRiQtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJCIiIQ==

At Y=y2 Bweak:=factor(limit(subs(w(y)=wy,J1yweak),wy=infinity))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZCd2Vha0dGKCoqLCYiIiJGMEkidEdGKEYwRjAtLUkiREdGJTYjSSJ3R0YoNiMhIiJGMEYxRjgsJi1GNjYjIiIhRjAtRjZGN0Y4Rjg3I0Yu

-op(2,Jformnul)/J1

LCQqKiIiJSIiIiksJkkjSjFHNiJGJUkiQUdGKSEiIiIiI0YlLCZGKEYlSSJCR0YpRitGJUYoRitGJQ==

Jyweak2:=factor(subs(J1=J1yweak,B=Bweak,%))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShKeXdlYWsyR0YoLCQqLiIiJSIiIiksMioqSSJBR0YoRjEtSSJ3R0YoNiMiIiFGMS1GNzYjISIiRjFJInRHRihGMUYxKipGNUYxLUY3NiNJInlHRihGMUY2RjFGPUYxRjwqKEY1RjEpRjoiIiNGMUY9RjFGPCoqRjVGMUY/RjFGOkYxRj1GMUYxKihGPUYxLS1JIkRHRiU2I0Y3RjtGMUY2RjFGPCooRj9GMUZHRjFGPUYxRjEqJkZHRjFGNkYxRjwqJkY/RjFGR0YxRjFGREYxKUY9RkRGPCwmRj9GPEY2RjFGPCksJkY/RjxGOkYxRkRGPCwmRjZGMUY6RjxGPEYxNyNGLg==

We recover the square at the numerator, and the form of the denominator Jyweak

KipJJmFscGhhRzYiIiIiKSwmLUkid0dGJDYjSSJ5R0YkRiVJImFHRiQhIiIiIiNGJSwmRihGJS1GKTYjIiIhRi1GLSksJkYoRiUtRik2I0YtRi1GLkYt

We can express A, or rather A^2, in terms of w factor(series(Jyweak2-Jy,y=0,2))

KydJInlHNiIsJCooIiIlIiIiLCgqJilJIkFHRiQiIiNGKC1JIndHRiQ2IyIiIUYoRigqJkYrRigtRi82IyEiIkYoRjUtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRi9GMEY1RihGNkY1RjVGNS1JIk9HRjk2I0YoRjE=

Assol:=factor(isolate(op(1,%),A^2))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZBc3NvbEdGKC8qJClJIkFHRigiIiMiIiIqJiwmLUkid0dGKDYjIiIhRjMtRjc2IyEiIkY8RjwtLUkiREdGJTYjRjdGOEYzNyNGLg==

The expression of alpha that we obtain here alphaw2:=factor(limit(subs(w(y)=wy,Jyweak2*w(y)),wy=infinity))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShhbHBoYXcyR0YoLCQqKiIiJSIiIiksKiooSSJBR0YoRjEtSSJ3R0YoNiMiIiFGMUkidEdGKEYxRjEqKEY1RjEtRjc2IyEiIkYxRjpGMUY+KiZGOkYxLS1JIkRHRiU2I0Y3Rj1GMUY+RkBGPiIiI0YxKUY6RkRGPiwmRjZGMUY8Rj5GPkY+NyNGLg==

The expression of a that we obtain here aw2:=op(factor({solve(numer(Jyweak2),w(y))}))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRhdzJHRigqJiwqKipJIkFHRigiIiItSSJ3R0YoNiMiIiFGMi1GNDYjISIiRjJJInRHRihGMkYyKihGMUYyKUY3IiIjRjJGOkYyRjkqKEY6RjItLUkiREdGJTYjRjRGOEYyRjNGMkY5KiZGP0YyRjNGMkY5RjIsKiooRjFGMkYzRjJGOkYyRjIqKEYxRjJGN0YyRjpGMkY5KiZGOkYyRj9GMkY5Rj9GOUY5NyNGLg==

Let us compare with what we obtained by applying directly the invariant lemma to J(y) expraw1;alphaw1

LCYqJiksJi1JIndHNiI2IyEiIiIiIkkiYUdGKEYqIiIjRispLCYtRic2IyIiIUYrRixGKkYtRipGKyosKS0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YoNiNGJ0YpRi1GKyksJkYrRitJInRHRihGK0YtRispRj5GLUYqLCZGMEYrRiZGKkYqLUY2RjFGKkYq

LCQqKiIiJSIiIi0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJIndHRiw2IyIiIUYlKSwmLUYuRi9GJS1GLjYjISIiRjYiIiNGJSksJkYzRiVJImFHRixGNkY3RjZGNg==

numer(factor(subs(a=aw2, expraw1)));isolate(%,A^2)

LCQqKCktLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ3R0YrNiMhIiIiIiMiIiIpLCZGMUYxSSJ0R0YrRjFGMEYxLCgqJilJIkFHRitGMEYxLUYtNiMiIiFGMUYxKiZGN0YxLUYtRi5GMUYvLUYmRjpGL0YxRi8=

LyokKUkiQUc2IiIiIyIiIiomLCYtSSJ3R0YmNiMiIiFGKC1GLDYjISIiRjFGMS0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGLEYtRig=

numer(factor(subs(a=aw2, alphaw2-alphaw1)));isolate(%,A^2)

LCQqKCIiJSIiIiksKiooSSJBRzYiRiUtSSJ3R0YqNiMiIiFGJUkidEdGKkYlRiUqKEYpRiUtRiw2IyEiIkYlRi9GJUYzKiZGL0YlLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRio2I0YsRjJGJUYzRjVGMyIiI0YlLCgqJilGKUY8RiVGK0YlRiUqJkY/RiVGMUYlRjMtRjZGLUYzRiVGMw==

LyokKUkiQUc2IiIiIyIiIiomLCYtSSJ3R0YmNiMiIiFGKC1GLDYjISIiRjFGMS0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGLEYtRig=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv

We expect the growth constant to be the same for the quadrant model with steps in tS and the three quadrant model with steps in S For the quadrant problem, we follow [Bostan et al.] tSteps

LCwqJEkieEc2IiEiIiIiIiomRiRGJ0kieUdGJUYnRidGJEYnKiZGKUYmRiRGJkYnRilGJw==

eqsQ:={diff(tSteps,x),diff(tSteps,y)}

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcXNRR0YoPCQsKEkieEdGKCIiIiomRjAhIiIpSSJ5R0YoIiIjRjNGM0YxRjEsKiokKUYwRjZGM0YzRjVGMUYxRjEqJkY1RjNGOUYzRjM3I0Yu

As predicted by the general theory (and convexity!), we have only one positive solution map(allvalues,solve(eqsQ,[x,y])): nops(%); evalf(%%)

IiIm

Nyc3JC9JInhHNiIkIishKSlSLj0nISM1L0kieUdGJiQhIiIiIiE3JC9GJSQhKykpUi49OyEiKkYqNyQvRiUkIitlenJDOEYzL0YrJCIrJipHUylwJkYpNyQvRiUsJiQiKyd5KmVCbUYpRi0qJiQiK0ReekFjRikiIiJeI0ZDRkNGQy9GKywmJCIrXSYpel1ARilGQyomJCIrIUdUckkiRjNGQ0ZERkNGQzckL0YlLCZGPkYtRkBGLS9GKywmRkdGQ0ZJRi0=

Here are the real values allvalues(op(2,solve(eqsQ,[x,y]))): evalf(%)

NiU3JC9JInhHNiIkIitlenJDOCEiKi9JInlHRiYkIismKkdTKXAmISM1NyQvRiUsJiQiKyd5KmVCbUYuISIiKiYkIitEXnpBY0YuIiIiXiNGOEY4RjgvRissJiQiK10mKXpdQEYuRjgqJiQiKyFHVHJJIkYpRjhGOUY4Rjg3JC9GJSwmRjJGNEY1RjQvRissJkY8RjhGPkY0

critQ:=op(2,solve(eqsQ,[x,y]))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjcml0UUdGKDckL0kieEdGKC1JJ1Jvb3RPZkdGJTYjLCgqJClJI19aR0YlIiIkIiIiRjlGNyEiIkY5RjovSSJ5R0YoLCgqJClGMSIiI0Y5RjpGMUY5RjlGOTcjRi4=

Estimate of mu subs(x = 1.324717958, y = .5698402895,tSteps)

JCIrUTouSFohIio=

Let us determine the minimal polynomial of mu algmuQ:=-factor(resultant(numer(subs(y=-x^2+x+1,muQ-tSteps)),subs(_Z=x,_Z^3-_Z-1),x))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdtdVFHRigsKiokKUkkbXVRR0YoIiIkIiIiRjMqJClGMSIiI0YzRjMqJiIjPUYzRjFGMyEiIiIjVkY5NyNGLg==

muQest:=fsolve(algmuQ)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdtdVFlc3RHRigkIitROi5IWiEiKjcjRi4=

plot(algmuQ,muQ=-5..5)

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Let us check numerically by taking ratios of consecutive terms N:=100:serQ:=subs(x=0,y=0,Qser(N)): plot([seq([1/n,coeff(serQ,t,n)/coeff(serQ,t,n-1)],n=3..N)],abs=0..0.1,ord=4..4.8,style=point)

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Ig==

muQest

JCIrUTouSFohIio=

We expect the same mu for the 3/4 and 1/4 problems. We go on with the 3/4 problem, following [Mustapha] Steps

LCxJInhHNiIiIiJJInlHRiRGJSokRiMhIiJGJSokRiZGKEYlKiZGI0YlRiZGJUYl

eqs:={diff(Steps,x),diff(Steps,y)}

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcXNHRig8JCwoIiIiRjAqJClJInhHRigiIiMhIiJGNUkieUdGKEYwLChGMEYwKiQpRjZGNEY1RjVGM0YwNyNGLg==

solve(eqs,[x,y])

NyQ3JC9JInhHNiIsJi1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjLCgqJClJI19aR0YqIiIjIiIiRjNGMSEiIkYzRjRGNEYzRjMvSSJ5R0YmRig3JC9GJS1GKTYjLCgqJClGMSIiJEYzRjNGL0YzRjNGNC9GNkY5

We look for real positive values map(allvalues,%): evalf(%)

Nyc3JC9JInhHNiIkISshKSlSLj0nISM1L0kieUdGJiQiKykpUi49OyEiKjckL0YlRiwvRitGJzckL0YlJCIrbm14W3ZGKS9GK0Y0NyQvRiUsJiQiK0okKVF1KClGKSEiIiomJCIrcXdoW3VGKSIiIl4jRkBGQEZAL0YrRjk3JC9GJSwmRjpGPEY9RjwvRitGRQ==

fsolve(_Z^3+_Z^2-1)

JCIraW14W3YhIzU=

algmuC:=-factor(resultant(numer(muC-subs(y=x,Steps)),subs(_Z=x,_Z^3+_Z^2-1),x))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdtdUNHRigsKiokKUkkbXVDR0YoIiIkIiIiRjMqJClGMSIiI0YzRjMqJiIjPUYzRjFGMyEiIiIjVkY5NyNGLg==

Same growth constant as before algmuQ

LCoqJClJJG11UUc2IiIiJCIiIkYoKiQpRiUiIiNGKEYoKiYiIz1GKEYlRighIiIiI1ZGLg==

Numerically: N:=100:serC:=subs(y=0,Dser(N)): plot([seq([1/n,coeff(serC,t,n)/coeff(serC,t,n-1)],n=3..N)],abs=0..0.1,ord=4.3..4.8,style=point)

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

muQest

JCIrUTouSFohIio=

We begin with the quadrant problem (cf. [Bostan et al.]) tSteps

LCwqJEkieEc2IiEiIiIiIiomRiRGJ0kieUdGJUYnRidGJEYnKiZGKUYmRiRGJkYnRilGJw==

cQ:=simplify(factor(diff(diff(tSteps,x),y)/sqrt(diff(tSteps,x$2)*diff(tSteps,y$2))),symbolic)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjUUdGKCwkKigjIiIiIiIjRjEsJiomKUkieEdGKEYyRjEpSSJ5R0YoRjJGMUYxRjFGMUYxKSwmRjhGMUYxRjFGMCEiIkYxNyNGLg==

critQ

NyQvSSJ4RzYiLUknUm9vdE9mRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsKCokKUkjX1pHRigiIiQiIiJGMUYvISIiRjFGMi9JInlHRiUsKCokKUYmIiIjRjFGMkYmRjFGMUYx

minimal polynomial of c algcQ:=factor(resultant(numer(subs(y=-x^2+x+1, c^2-cQ^2)),subs(_Z=x,_Z^3-_Z-1),x))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdjUUdGKCwqKiYiJD8kIiIiKUkiY0dGKCIiJ0YxISIiKiZGMEYxKUYzIiIlRjFGMSomIiRTIkYxKUYzIiIjRjFGNSIjREYxNyNGLg==

algcQ:=-algcQ/5

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdjUUdGKCwqKiYiI2siIiIpSSJjR0YoIiInRjFGMSomRjBGMSlGMyIiJUYxISIiKiYiI0dGMSlGMyIiI0YxRjEiIiZGODcjRi4=

We need to choose the sign of c: it is clearly positive, thanks to the above expression of cQ fsolve(%)

NiQkISsiUWpZRSchIzUkIisiUWpZRSdGJQ==

cQest:=op(2,[%])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjUWVzdEdGKCQiKyJRallFJyEjNTcjRi4=

Estimate of the quadrant exponent alpha_Q aQest:=Pi/arccos(-cQest)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhUWVzdEdGKCQiKyI+RHdSIiEiKjcjRi4=

GAMMA(-aQest)

JCIrYTQlKnBFISIq

gQest:=-1-aQest

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZnUWVzdEdGKCQhKyI+RHdSIyEiKjcjRi4=

check N:=100:serQ:=subs(x=0,y=0,Qser(N)): plot([seq([1/n,n^2*(1-coeff(serQ,t,n+1)*coeff(serQ,t,n-1)/coeff(serQ,t,n)^2)],n=3..N)],abs=0..0.05,ord=-2.4..-2,style=point)

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Ig==

Now for the three quadrant problem [Mustapha] Steps

LCxJInhHNiIiIiJJInlHRiRGJSokRiMhIiJGJSokRiZGKEYlKiZGI0YlRiZGJUYl

cC:=subs(y=x,simplify(factor(diff(diff(Steps,x),y)/sqrt(diff(Steps,x$2)*diff(Steps,y$2))),symbolic))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjQ0dGKCwkKiYjIiIiIiIjRjEpSSJ4R0YoIiIkRjFGMTcjRi4=

algx0

LCgqJClJInhHNiIiIiQiIiJGKCokKUYlIiIjRihGKEYoISIi

subs(x=fsolve(algx0),cC)

JCIrVyYpel1AISM1

algcC:=factor(resultant(numer(c^2-cC^2),algx0,x))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdjQ0dGKComLCoqJiIiKSIiIilJImNHRigiIiRGMkYyKiZGMUYyKUY0IiIjRjJGMiomIiInRjJGNEYyRjJGMkYyRjIsKkYwRjJGNiEiIkY5RjJGMkY8RjI3I0Yu

choose the right factor, since c is positive factor(diff(diff(Steps,x),y))

IiIi

algcC:=op(2,algcC)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdjQ0dGKCwqKiYiIikiIiIpSSJjR0YoIiIkRjFGMSomRjBGMSlGMyIiI0YxISIiKiYiIidGMUYzRjFGMUYxRjg3I0Yu

cCest:=fsolve(%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjQ2VzdEdGKCQiK1gmKXpdQCEjNTcjRi4=

The exponent for 3/4 walks gCest:=-(3*Pi-arccos(-cCest))/(2*Pi-arccos(-cCest))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZnQ2VzdEdGKCQhKyZmNykpcCIhIio3I0Yu

aCest:=-1-gCest

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhQ2VzdEdGKCQiKiZmNykpcCEiKjcjRi4=

Note that Gamma (-alpha) is negative GAMMA(-aCest)

JCErXjg/alUhIio=

check N:=100:serC:=subs(y=0,Dser(N)): plot([seq([1/n,n^2*(1-coeff(serC,t,n+1)*coeff(serC,t,n-1)/coeff(serC,t,n)^2)],n=3..N)],abs=0..0.05,ord=-1.7..-1.4,style=point)

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

Connection between the two exponents algcC; algcQ

LCoqJiIiKSIiIilJImNHNiIiIiRGJUYlKiZGJEYlKUYnIiIjRiUhIiIqJiIiJ0YlRidGJUYlRiVGLQ==

LCoqJiIjayIiIilJImNHNiIiIidGJUYlKiZGJEYlKUYnIiIlRiUhIiIqJiIjR0YlKUYnIiIjRiVGJSIiJkYt

res:=subs(RootOf(algcC,c)=cc,factor(algcQ, RootOf(algcC,c)))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyZXNHRigsJCooIiIjIiIiLCwqJiIjO0YxKUkiY0dGKCIiJUYxRjEqKCIiKUYxKUY2RjBGMUkjY2NHRihGMSEiIiomRjlGMUY6RjFGPComRjdGMSlGO0YwRjFGMSIiJEYxRjEsKComRjBGMUY6RjFGMUY7RjFGMUY8RjFGMTcjRi4=

op(3,%)

LCgqJiIiIyIiIilJImNHNiJGJEYlRiVJI2NjR0YoRiVGJSEiIg==

subs(cc=cCest, c=cQest,op(2,res)); subs(cc=cCest, c=cQest,op(3,res))

JCIrKkd0VyQ9ISIq

JCIiIiEjNQ==

Hence cC=1-2 cQ^2. This translates into a simple indentity between the values aC and aQ aQ:=arccos(-cQest);aC:=arccos(-cCest)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNhUUdGKCQiKyNwMnlDIyEiKjcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNhQ0dGKCQiK0MqcHZ5IiEiKjcjRi4=

evalf(-2*aQ+2*Pi-aC)

JCIiIUYj

The simplest relation is between the values of alpha for the two models aQest/aCest

JCIrLCsrKz8hIio=

Some steps in the derivation of the functional equation (still involving sub-diagonal terms) Heq0:=-H(x,y)*(x^2*y+y+x^2*y^2+1+x*y^2-mu*x*y)+Hm(x)+(1+y)*Hd(y)-x*(1+y)*Hsd(y)+x*H0m1

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIZXEwR0YoLCwqJi1JIkhHRig2JEkieEdGKEkieUdGKCIiIiwuKiYpRjMiIiNGNSlGNEY5RjVGNSooSSNtdUdGKEY1RjNGNUY0RjUhIiIqJkY4RjVGNEY1RjUqJkYzRjVGOkY1RjVGNEY1RjVGNUY1Rj0tSSNIbUdGKDYjRjNGNSomLCZGNEY1RjVGNUY1LUkjSGRHRig2I0Y0RjVGNSooRjNGNUZERjUtSSRIc2RHRihGR0Y1Rj0qJkYzRjVJJUgwbTFHRihGNUY1NyNGLg==

ser:=subs(H(0,y)=Hd(y),Hm(0)=0,series(Heq0,x,2))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzZXJHRigrJ0kieEdGKCwsKiYtSSNIZEdGKDYjSSJ5R0YoIiIiLCYqJkkjbXVHRihGNkY1RjYhIiIqJClGNSIiI0Y2RjZGNkY6KiYtLSZJIkRHRiU2I0Y2NiNJIkhHRig2JCIiIUY1RjYsJkY1RjZGNkY2RjZGOi0tRkI2I0kjSG1HRig2I0ZHRjYqJkZIRjYtSSRIc2RHRihGNEY2RjpJJUgwbTFHRihGNkY2LUkiT0dGJUZDRj03I0Yu

Hsdsol:=collect(solve(subs((D[1](H))(0, y)=Hsd(y),coeff(%,x)),Hsd(y)),H,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdIc2Rzb2xHRigsJCooIyIiIiIiI0YxLCoqKC1JI0hkR0YoNiNJInlHRihGMUkjbXVHRihGMUY4RjFGMSomRjVGMSlGOEYyRjEhIiJJJUgwbTFHRihGMS0tSSJER0YlNiNJI0htR0YoNiMiIiFGMUYxLCZGOEYxRjFGMUY8RjE3I0Yu

Heq1:=collect(subs(Hsd(y)=Hsdsol,H0m1=(D(Hm))(0),Heq0),[H,Hm,Hd],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIZXExR0YoLCgqJiwuKiYpSSJ4R0YoIiIjIiIiKUkieUdGKEY0RjUhIiIqKEkjbXVHRihGNUYzRjVGN0Y1RjUqJkYyRjVGN0Y1RjgqJkYzRjVGNkY1RjhGN0Y4RjVGOEY1LUkiSEdGKDYkRjNGN0Y1RjUtSSNIbUdGKDYjRjNGNSomLCpGN0Y1RjVGNSoqI0Y1RjRGNUY6RjVGM0Y1RjdGNUY4KihGRkY1RjNGNUY2RjVGNUY1LUkjSGRHRig2I0Y3RjVGNTcjRi4=

The functional equation for a harmonic function in C, with steps in the original step set S Heq:=-H(x,y)*(x^2*y+y+x^2*y^2+1+x*y^2-mu*x*y)+Hm(x)+(y+1+x*y/2*(y-mu))*Hd(y)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRIZXFHRigsKComLUkiSEdGKDYkSSJ4R0YoSSJ5R0YoIiIiLC4qJilGMyIiI0Y1KUY0RjlGNUY1KihJI211R0YoRjVGM0Y1RjRGNSEiIiomRjhGNUY0RjVGNSomRjNGNUY6RjVGNUY0RjVGNUY1RjVGPS1JI0htR0YoNiNGM0Y1KiYsKEY0RjVGNUY1KiojRjVGOUY1RjNGNUY0RjUsJkY0RjVGPEY9RjVGNUY1LUkjSGRHRig2I0Y0RjVGNTcjRi4=

factor(Heq1-Heq)

IiIh

co:=coeff(Heq,Hd(y))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjb0dGKCwoSSJ5R0YoIiIiRjBGMCoqI0YwIiIjRjBJInhHRihGMEYvRjAsJkYvRjBJI211R0YoISIiRjBGMDcjRi4=

The kernel of the companion model, at t=1/mu ker:=expand(subs(t=1/mu,1-t*tSteps))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRrZXJHRigsLiomSSNtdUdGKCEiIkkieEdGKEYxRjEqKEYwRjFGMiIiIkkieUdGKEY0RjEqJkYwRjFGMkY0RjEqKEYwRjFGNUYxRjJGMUYxKiZGMEYxRjVGNEYxRjRGNDcjRi4=

The polynomial kernel that appears in the paper kerpol:=x^2*y^2-mu*x*y+x^2*y+x*y^2+y+1

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdrZXJwb2xHRigsLiomKUkieEdGKCIiIyIiIilJInlHRihGMkYzRjMqKEkjbXVHRihGM0YxRjNGNUYzISIiKiZGMEYzRjVGM0YzKiZGMUYzRjRGM0YzRjVGM0YzRjM3I0Yu

factor(kerpol/ker)

LCQqKEkjbXVHNiIiIiJJInhHRiVGJkkieUdGJUYmISIi

factor(rem(co^2,kerpol,x))

LCQqKiMiIiIiIiVGJSwuKiYpSSNtdUc2IiIiI0YlSSJ5R0YrRiVGJSooRixGJUYqRiUpRi1GLEYlISIiKiQpRi0iIiRGJUYlKiZGJkYlRi9GJUYwKiYiIilGJUYtRiVGMEYmRjBGJSwqKihGKkYlSSJ4R0YrRiVGLUYlRiUqJkY5RiVGL0YlRjBGLUYwRiVGMEYlLCZGLUYlRiVGJUYwRiU=

We recognize in the factor involving x four of the six terms of kerpol. We also expect to see its discriminant disc:=factor(discrim(factor(-kerpol),x))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVkaXNjR0YoKiYsLiomKUkjbXVHRigiIiMiIiJJInlHRihGNEY0KihGM0Y0RjJGNClGNUYzRjQhIiIqJClGNSIiJEY0RjQqJiIiJUY0RjdGNEY4KiYiIilGNEY1RjRGOEY9RjhGNEY1RjQ3I0Yu

factor(co^2-disc/4*x^2);factor(%/kerpol)

LCQqJiwmSSJ5RzYiIiIiRidGJ0YnLC4qJilJInhHRiYiIiNGJylGJUYsRichIiIqKEkjbXVHRiZGJ0YrRidGJUYnRicqJkYqRidGJUYnRi4qJkYrRidGLUYnRi5GJUYuRidGLkYnRi4=

LCZJInlHNiIiIiJGJUYl

We multiply the basic equation by a well chosen term (Hm(x)-co*Hd(y))/x^2 Heqmod:=-Hd(y)^2*kerpol*(1+y)/x^2+ collect((Hm(x)-co*Hd(y))/x^2*Heq + Hd(y)^2*kerpol*(1+y)/x^2,[H,Hd,Hm],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdIZXFtb2RHRigsKioqKS1JI0hkR0YoNiNJInlHRigiIiMiIiIsLiomKUkieEdGKEY1RjYpRjRGNUY2RjYqKEkjbXVHRihGNkY6RjZGNEY2ISIiKiZGOUY2RjRGNkY2KiZGOkY2RjtGNkY2RjRGNkY2RjZGNiwmRjRGNkY2RjZGNkY5Rj5GPiomLCYqLCNGNkY1RjYsKkY8RjZGQEY+KiZGNUY2RjRGNkY+RjVGPkY2LC5GOEY+RjxGNkY/Rj5GQEY+RjRGPkY2Rj5GNkY5Rj5GMUY2RjYqKC1JI0htR0YoNiNGOkY2RjlGPkZIRjZGNkY2LUkiSEdGKDYkRjpGNEY2RjYqKiNGNiIiJUY2LC4qJilGPUY1RjZGNEY2RjYqKEY1RjZGPUY2RjtGNkY+KiQpRjQiIiRGNkY2KiZGUkY2RjtGNkY+KiYiIilGNkY0RjZGPkZSRj5GNkY0RjZGMEY2Rj4qJilGSkY1RjZGOUY+RjY3I0Yu

factor(Heqmod/Heq)

LCQqKCMiIiIiIiNGJSwsKiotSSNIZEc2IjYjSSJ5R0YrRiVJI211R0YrRiVJInhHRitGJUYtRiVGJSooRilGJUYvRiUpRi1GJkYlISIiKihGJkYlRilGJUYtRiVGMiomRiZGJS1JI0htR0YrNiNGL0YlRiUqJkYmRiVGKUYlRjJGJSlGL0YmRjJGJQ==

Take the factors that contain a factor tK(1/mu) hxy:=-factor((op(1,Heqmod)+op(2,Heqmod))/kerpol)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRoeHlHRigsJCooIyIiIiIiI0YxLDAqLC1JIkhHRig2JEkieEdGKEkieUdGKEYxLUkjSGRHRig2I0Y5RjFJI211R0YoRjFGOEYxRjlGMUYxKipGNUYxRjpGMUY4RjEpRjlGMkYxISIiKipGMkYxRjVGMUY6RjFGOUYxRkAqKEYyRjEpRjpGMkYxRjlGMUYxKihGMkYxRjVGMS1JI0htR0YoNiNGOEYxRjEqKEYyRjFGNUYxRjpGMUZAKiZGMkYxRkNGMUYxRjEpRjhGMkZARjE3I0Yu

hxy:=collect(%,[H,Hd],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRoeHlHRigsJiomLCYqKiMiIiIiIiNGMywqKihJI211R0YoRjNJInhHRihGM0kieUdGKEYzRjMqJkY4RjMpRjlGNEYzISIiKiZGNEYzRjlGM0Y8RjRGPEYzKUY4RjRGPC1JI0hkR0YoNiNGOUYzRjMqJi1JI0htR0YoNiNGOEYzRj5GPEYzRjMtSSJIR0YoNiRGOEY5RjNGMyooLCZGOUYzRjNGM0YzRj5GPClGP0Y0RjNGMzcjRi4=

Let us prove that hxy is a formal power series in x and y serx:=factor(subs(Hm(0)=0,Hd(y)=H(0,y),series(hxy,x,2)))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVzZXJ4R0YoKydJInhHRigsJCooIyIiIiIiI0YzLUkiSEdGKDYkIiIhSSJ5R0YoRjMsLCooRjVGM0kjbXVHRihGM0Y5RjNGMyomRjVGMylGOUY0RjMhIiIqKEY0RjMtLSZJIkRHRiU2I0YzNiNGNkY3RjNGOUYzRj8qJkY0RjNGQUYzRj8qJkY0RjMtLUZENiNJI0htR0YoNiNGOEYzRjNGM0YzRj8tSSJPR0YlRkVGODcjRi4=

We get back to the functional equation satisfied by H(x,y) factor(subs(Hm(0)=0,Hd(y)=H(0,y),series(Heq,x,2)))

KydJInhHNiIsLComLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiI2I0kiSEdGJDYkIiIhSSJ5R0YkRi9GNEYvISIiRidGNSoqI0YvIiIjRi8tRjFGMkYvSSNtdUdGJEYvRjRGL0YvKihGN0YvRjlGLylGNEY4Ri9GNS0tRio2I0kjSG1HRiQ2I0YzRi9GLy1JIk9HRitGLkY4

D1H0ysubs:=isolate(coeff(%,x,1),D[1](H)(0,y))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpEMUgweXN1YnNHRigvLS0mSSJER0YlNiMiIiI2I0kiSEdGKDYkIiIhSSJ5R0YoKiYsKCoqI0Y0IiIjRjQtRjZGN0Y0SSNtdUdGKEY0RjlGNCEiIiooRj1GNEY/RjQpRjlGPkY0RjQtLUYyNiNJI0htR0YoNiNGOEZBRjQsJkY5RkFGNEZBRkE3I0Yu

Hence h(x,y) has no negative power of x factor(subs(%,serx))

KyVJInhHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiE=

One step further, in order to determine h(0,y) serx:=factor(subs(Hm(0)=0,Hd(y)=H(0,y),D1H0ysubs,series(hxy,x,3)))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVzZXJ4R0YoKydJInhHRigsJCooIyIiIiIiJUYzLDgqKCktSSJIR0YoNiQiIiFJInlHRigiIiNGMylJI211R0YoRj1GMylGPEY9RjNGMyoqRj1GM0Y3RjNGP0YzKUY8IiIkRjMhIiIqJkY3RjMpRjxGNEYzRjMqLEY0RjNGOEYzLS1JIkRHRiU2I0kjSG1HRig2I0Y7RjNGP0YzRjxGM0YzKipGNEYzRjhGM0ZIRjNGQEYzRkQqKkY9RjNGOEYzLS0mRko2JEYzRjM2I0Y5RjpGM0ZARjNGRCoqRjRGM0Y4RjNGUEYzRjxGM0ZEKipGPUYzLS0tSSNAQEdGJTYkRkpGPUZLRk1GM0Y4RjNGPEYzRjMqKEY9RjNGOEYzRlBGM0ZEKihGPUYzRldGM0Y4RjNGMyomRjRGMylGSEY9RjNGM0YzLCZGPEYzRjNGM0ZERjNGOy1JIk9HRiU2I0YzRjM3I0Yu

Here is the value of h(x,y) at x=0 h0y:=convert(%,polynom)

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

Once again, we get back to the functional equation for H(x,y) coeff(factor(subs(Hm(0)=0,Hd(y)=H(0,y),D1H0ysubs,series(Heq,x,3))),x,2)

LCQqKCMiIiIiIiNGJSw8KigtSSJIRzYiNiQiIiFJInlHRitGJSlJI211R0YrRiZGJSlGLkYmRiVGJSoqRiZGJUYpRiVGMEYlKUYuIiIkRiUhIiIqJkYpRiUpRi4iIiVGJUYlKihGJkYlRilGJUYzRiVGNSooRjhGJUYpRiVGMUYlRjUqKkYmRiUtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKzYjSSNIbUdGKzYjRi1GJUYwRiVGLkYlRiUqKEYmRiVGPEYlRjFGJUY1KiYtLSZGPjYkRiVGJTYjRipGLEYlRjFGJUY1KihGJkYlRi5GJUYpRiVGNSooRiZGJUZHRiVGLkYlRjUqJi0tLUkjQEBHRj82JEY+RiZGQkZERiVGLkYlRiVGR0Y1Rk9GJUYlLCZGLkYlRiVGJUY1RiU=

D111H0y:=isolate(%,(D[1,1](H))(0, y))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShEMTExSDB5R0YoLy0tJkkiREdGJTYkIiIiRjQ2I0kiSEdGKDYkIiIhSSJ5R0YoKiYsNiooLUY2RjdGNClJI211R0YoIiIjRjQpRjlGQEY0ISIiKipGQEY0Rj1GNEY/RjQpRjkiIiRGNEY0KiZGPUY0KUY5IiIlRjRGQiooRkBGNEY9RjRGREY0RjQqKEZIRjRGPUY0RkFGNEY0KipGQEY0LS1GMjYjSSNIbUdGKDYjRjhGNEY/RjRGOUY0RkIqKEZARjRGTEY0RkFGNEY0KihGQEY0RjlGNEY9RjRGNComLS0tSSNAQEdGJTYkRjJGQEZORlBGNEY5RjRGQkZURkJGNCwoKiRGQUY0RkIqJkZARjRGOUY0RkJGNEZCRkI3I0Yu

h0y:=collect(subs(D111H0y,h0y),[H],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRoMHlHRigsJiosIyIiIiIiJUYxLC4qJilJI211R0YoIiIjRjFJInlHRihGMUYxKihGN0YxRjZGMSlGOEY3RjEhIiIqJClGOCIiJEYxRjEqJkYyRjFGOkYxRjsqJiIiKUYxRjhGMUY7RjJGO0YxRjhGMSwmRjhGMUYxRjFGOyktSSJIR0YoNiQiIiFGOEY3RjFGOyomKS0tSSJER0YlNiNJI0htR0YoNiNGR0Y3RjFGQkY7RjE3I0Yu

disc

KiYsLiomKUkjbXVHNiIiIiMiIiJJInlHRidGKUYpKihGKEYpRiZGKSlGKkYoRikhIiIqJClGKiIiJEYpRikqJiIiJUYpRixGKUYtKiYiIilGKUYqRilGLUYyRi1GKUYqRik=

Finally, the value of h(x,0) factor(subs(H(x,0)=Hm(x)+Hd(0),series(hxy,y,1)))

KydJInlHNiIqJiktSSNIbUdGJDYjSSJ4R0YkIiIjIiIiKUYqRishIiIiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGLEYs

Hence the equation for h(x,y), which is Heqmod

LCoqKiktSSNIZEc2IjYjSSJ5R0YnIiIjIiIiLC4qJilJInhHRidGKkYrKUYpRipGK0YrKihJI211R0YnRitGL0YrRilGKyEiIiomRi5GK0YpRitGKyomRi9GK0YwRitGK0YpRitGK0YrRissJkYpRitGK0YrRitGLkYzRjMqJiwmKiwjRitGKkYrLCpGMUYrRjVGMyomRipGK0YpRitGM0YqRjNGKywuRi1GM0YxRitGNEYzRjVGM0YpRjNGK0YzRitGLkYzRiVGK0YrKigtSSNIbUdGJzYjRi9GK0YuRjNGPUYrRitGKy1JIkhHRic2JEYvRilGK0YrKiojRisiIiVGKywuKiYpRjJGKkYrRilGK0YrKihGKkYrRjJGK0YwRitGMyokKUYpIiIkRitGKyomRkdGK0YwRitGMyomIiIpRitGKUYrRjNGR0YzRitGKUYrRiRGK0YzKiYpRj9GKkYrRi5GM0Yr

coincides with the quadrant harmonic equation for the companion model eqh:=kerpol*h(x,y)+(1+y)*h(0,y)+h(x,0)-h(0,0)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcWhHRigsKiomLC4qJilJInhHRigiIiMiIiIpSSJ5R0YoRjRGNUY1KihJI211R0YoRjVGM0Y1RjdGNSEiIiomRjJGNUY3RjVGNSomRjNGNUY2RjVGNUY3RjVGNUY1RjUtSSJoR0YoNiRGM0Y3RjVGNSomLCZGN0Y1RjVGNUY1LUY+NiQiIiFGN0Y1RjUtRj42JEYzRkRGNS1GPjYkRkRGREY6NyNGLg==

-coeff(eqh, h(x,y));

LC4qJilJInhHNiIiIiMiIiIpSSJ5R0YmRidGKCEiIiooSSNtdUdGJkYoRiVGKEYqRihGKComRiRGKEYqRihGKyomRiVGKEYpRihGK0YqRitGKEYr

Hmxsol:=kappa*x*sqrt(h(x,0)); Hdysol:=kappa*sqrt(((y+1)*h(0, y)-h(0,0)))*2/sqrt(-(mu^2*y-2*mu*y^2+y^3-4*y^2-8*y-4)*y)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdIbXhzb2xHRigqKEkma2FwcGFHRigiIiJJInhHRihGMCktSSJoR0YoNiRGMSIiISNGMCIiI0YwNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdIZHlzb2xHRigsJCoqIiIjIiIiSSZrYXBwYUdGKEYxKSwmKiYsJkkieUdGKEYxRjFGMUYxLUkiaEdGKDYkIiIhRjdGMUYxLUY5NiRGO0Y7ISIiI0YxRjBGMSksJComLC4qJilJI211R0YoRjBGMUY3RjFGMSooRjBGMUZGRjEpRjdGMEYxRj4qJClGNyIiJEYxRjEqJiIiJUYxRkhGMUY+KiYiIilGMUY3RjFGPkZNRj5GMUY3RjFGPkY/Rj5GMTcjRi4=

disc

KiYsLiomKUkjbXVHNiIiIiMiIiJJInlHRidGKUYpKihGKEYpRiZGKSlGKkYoRikhIiIqJClGKiIiJEYpRikqJiIiJUYpRixGKUYtKiYiIilGKUYqRilGLUYyRi1GKUYqRik=

Numerical check series(subs(h(x,0)=add(h[i,0] *x^i,i=0..10), Hmxsol),x,2)

KydJInhHNiIqJkkma2FwcGFHRiQiIiIpJkkiaEdGJDYkIiIhRiwjRiciIiNGJ0YnLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRidGLg==

subs(csgn(sqrt(h[0, 1]+h[0, 0])*sqrt(y))=1,simplify(series(subs(h(0, 0)=h[0,0],h(0,y)=add(h[0,i] *y^i,i=0..10), Hdysol),y,2),symbolic))

KylJInlHNiIqJkkma2FwcGFHRiQiIiIpLCYmSSJoR0YkNiQiIiFGJ0YnJkYrNiRGLUYtRicjRiciIiNGJ0YtLCQqKiNGJyIiKUYnRiZGJywsKiYpSSNtdUdGJEYxRidGLkYnRicqJkY4RidGKkYnRicqJkY1RidGLkYnISIiKiYiIiVGJ0YqRidGPComRj5GJyZGKzYkRi1GMUYnRidGJ0YoRjxGJ0YnLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRidGMQ==

A test N:=150: serCm10:=coeff(Cmser(N),x,-1): serC00:= C00ser(N): unassign('n') P1:=plot([seq([1/i,coeff(serC00,t,i)/coeff(serCm10,t,i)],i=3..N)],abs=0..0.1, ord= 1.7..1.95, style = POINT,labels=[1/'n', " "]): serQ00:=coeff(coeff(Qser(N),x,0),y,0): serQ01:=coeff(coeff(Qser(N),x,0),y,1): P2:=plot([seq([1/n,sqrt(1+coeff(serQ01,t,n)/coeff(serQ00,t,n))],n=3..N)], abs=0..0.1, ord= 1.7..1.95,style = POINT,labels=[1/'n',ww]): with(plots): display(P1,P2)

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Checking further with x series(subs(h(x,0)=add(h[i,0] *x^i,i=0..10), Hmxsol),x,3)

KylJInhHNiIqJkkma2FwcGFHRiQiIiIpJkkiaEdGJDYkIiIhRiwjRiciIiNGJ0YnLCQqKkYtRidGJkYnRighIiImRio2JEYnRixGJ0YnRi4tSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJyIiJA==

N:=100: serCm10:=coeff(Cmser(N),x,-1): serCm20:=coeff(Cmser(N),x,-2): P1:=plot([seq([1/n,coeff(serCm10,t,n)/coeff(serCm20,t,n)],n=4..N)],abs=0..0.1, ord= 1.5..2, style = POINT);

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NiI=

serQ00:=coeff(coeff(Qser(N),x,0),y,0): serQ10:=coeff(coeff(Qser(N),x,1),y,0): P2:=plot([seq([1/n,2*coeff(serQ00,t,n)/coeff(serQ10,t,n)],n=4..N)], abs=0..0.1, ord=1.5..2,style = POINT,color=blue);

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NiI=

perfect agreement with(plots): display(P1,P2)

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Checking further with y subs(csgn(sqrt(h[0, 1]+h[0, 0])*sqrt(y))=1,simplify(series(subs(h(0, 0)=h[0,0],h(0,y)=add(h[0,i] *y^i,i=0..10), Hdysol),y,2),symbolic))

KylJInlHNiIqJkkma2FwcGFHRiQiIiIpLCYmSSJoR0YkNiQiIiFGJ0YnJkYrNiRGLUYtRicjRiciIiNGJ0YtLCQqKiNGJyIiKUYnRiZGJywsKiYpSSNtdUdGJEYxRidGLkYnRicqJkY4RidGKkYnRicqJkY1RidGLkYnISIiKiYiIiVGJ0YqRidGPComRj5GJyZGKzYkRi1GMUYnRidGJ0YoRjxGJ0YnLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRidGMQ==

N:=150: serC11:=coeff(coeff(Cser(N),x,1),y,1): serC00:= C00ser(N): P1:=plot([seq([1/n,coeff(serC11,t,n)/coeff(serC00,t,n)],n=3..N)],abs=0..0.1, ord= 2..3, style = POINT): serQ00:=coeff(coeff(Qser(N),x,0),y,0): serQ01:=coeff(coeff(Qser(N),x,0),y,1): serQ02:=coeff(coeff(Qser(N),x,0),y,2): muQest

JCIrUTouSFohIio=

P2:=plot([seq([1/n,muQest^2/8-1+1/2*(coeff(serQ01,t,n)+coeff(serQ02,t,n))/(coeff(serQ00,t,n)+coeff(serQ01,t,n))],n=3..N)], abs=0..0.1, ord= 2..3,style = POINT);

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Ig==

with(plots): display(P1,P2)

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