This Maple session accompanies the paper "Three quadrant walks by invariants" and considers the case of reverse Kreweras' steps.The proofs are in Section 5.

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

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eqC:=-K*C(x,y)+1-t/x/y*Cm(1/x)-t/x/y*Cm(1/y)-t/x/y*C00;

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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: local i,j: if n=0 then 1 else normal(series(subs(C(x,y)=Cser(n-1),C00=C00ser(n-1),Cm(1/x)=Cmser(n-1),Cm(1/y)=subs(x=y,Cmser(n-1)),eqC+C(x,y)),t,n+1)): fi: end: map(expand,Cser(4));

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C00ser(5); Cmser(7)

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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);

LCgqJiIiJSIiIilJInRHNiIiIiRGJUYlKigiIiNGJSlGJ0YrRiVJInlHRihGJUYlRiVGJQ==

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

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

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Vertical moves in the companion model Vpy:=coeff(tSteps, x,1);Vmy:=coeff(tSteps, x,-1);V0y:=coeff(tSteps, x,0);

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The polynomial Hminus for the original model Hmx:=coeff(Steps, y,-1);

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

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

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eqUfinal:=collect(subs(isolate(eqD,U(0,y)),2*eqU*y),[U,DD],factor);

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

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

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

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The companion kernel Kt:=1-t*(1/x+1/y+x*y);

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

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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)),eqQ+Q(x,y)),t,n+1)): fi: end: map(expand,Qser(4));

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Two pairs of invariants (Table 3) I0x:=1/x^2-1/t/x-x; J0y:=subs(x=y,I0x);

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ratio0:=factor((I0x-J0y)/Kt);

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The quadrant invariants fx:=1/2/t-1/x; gy:=subs(x=y,fx);

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

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

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

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ratio1:=expand(factor(%/Kt));

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

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

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

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The functions F and G from Table 5, which solve the new decoupling problem Fx:=-1/x^2+1/t/x-x; Gy:=1/y/t;

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factor(y-Gy*co-Fx);

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Hxy:=factor(%/(1-t*tSteps));

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

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

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZEZWx0YUdGKCwmKiQpLCYqJkkieUdGKCEiIkkidEdGKCIiIkY0RjZGNiIiI0Y2RjYqKCIiJUY2RjNGNilGNUY3RjZGNDcjRi4=

We now check the identity of Proposition 3.2 factor(Rx^2-Delta*Sy^2);

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

Divisibility by Kt factor(subs(isolate(eqUfinal,U(x,0)),%)/Kt);

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

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

LCQqLCwoKiwiIiMiIiIpSSJ4RzYiRiZGJylJInlHRipGJkYnLUkiVUdGKjYkRilGLEYnSSJ0R0YqRidGJ0YpISIiRixGJ0YnLDZGJUYnKixGJkYnLUkjRERHRio2I0YsRidGMEYnRilGJ0YrRidGJyoqRiZGJ0YtRidGMEYnRilGJ0YnKipGJkYnRi1GJ0YwRidGLEYnRicqKkYmRidGKUYnRixGJ0YtRidGMSoqRiZGJ0YpRidGMEYnLUYuNiRGKSIiIUYnRjEqJkY0RidGMEYnRicqJkYsRidGNEYnRjEqJkYwRictRjU2I0Y9RidGMUYsRidGJ0YwRjFGKUYxRixGMUYx

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

IiIh

Sy;

LCYqJkkieUc2IiIiIi1JI0RER0YlNiNGJEYmRiYqJkkidEdGJSEiIkYkRixGJg==

Rx;Delta;

LCwqKiIiIyIiIkkieEc2IkYlSSJ0R0YnRiUtSSJVR0YnNiRGJiIiIUYlRiUqJClGJkYkISIiRiUqJkYoRi9GJkYvRi9GJkYlKiZGKEYlLUkjRERHRic2I0YsRiVGJQ==

LCYqJCksJiomSSJ5RzYiISIiSSJ0R0YoIiIiRilGK0YrIiIjRitGKyooIiIlRitGJ0YrKUYqRixGK0Yp

Ix:=Rx^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeEdGKCokKSwsKioiIiMiIiJJInhHRihGM0kidEdGKEYzLUkiVUdGKDYkRjQiIiFGM0YzKiQpRjRGMiEiIkYzKiZGNUY8RjRGPEY8RjRGMyomRjVGMy1JI0RER0YoNiNGOUYzRjNGMkYzNyNGLg==

Jy:=Delta*Sy^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKeUdGKComLCYqJCksJiomSSJ5R0YoISIiSSJ0R0YoIiIiRjVGN0Y3IiIjRjdGNyooIiIlRjdGNEY3KUY2RjhGN0Y1RjcpLCYqJkY0RjctSSNEREdGKDYjRjRGN0Y3KiZGNkY1RjRGNUY3RjhGNzcjRi4=

The ratio for I(x), J(y) ratio :=-4*t*x*Vpy*Sy^2+(Hxy-2*x*y*U(x,y))*((t*V0y+2*t*x*Vpy-1)*Sy+Rx);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyYXRpb0dGKCwmKiwiIiUiIiJJInRHRihGMUkieEdGKEYxSSJ5R0YoRjEpLCYqJkY0RjEtSSNEREdGKDYjRjRGMUYxKiZGMiEiIkY0RjxGMSIiI0YxRjwqJiwmKipGNEY8RjNGPCwmRjNGMUY0RjxGMUYyRjxGMSoqRj1GMUYzRjFGNEYxLUkiVUdGKDYkRjNGNEYxRjxGMSwuKiYsKCoqRj1GMUYyRjFGM0YxRjRGMUYxKiZGNEY8RjJGMUYxRjFGPEYxRjZGMUYxKipGPUYxRjNGMUYyRjEtRkQ2JEYzIiIhRjFGMSokKUYzRj1GPEYxKiZGMkY8RjNGPEY8RjNGMSomRjJGMS1GOTYjRk5GMUYxRjFGMTcjRi4=

Ix;series(Ix,x,1);

KiQpLCwqKiIiIyIiIkkieEc2IkYnSSJ0R0YpRictSSJVR0YpNiRGKCIiIUYnRicqJClGKEYmISIiRicqJkYqRjFGKEYxRjFGKEYnKiZGKkYnLUkjRERHRik2I0YuRidGJ0YmRic=

KytJInhHNiIiIiIhIiUsJComIiIjRiVJInRHRiQhIiJGKyEiJCwmKihGKUYlRipGJS1JI0RER0YkNiMiIiFGJUYlKiQpRipGKUYrRiUhIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJUYr

I0x; I1x;

LCgqJClJInhHNiIiIiMhIiIiIiIqJkkidEdGJkYoRiVGKEYoRiVGKA==

LCgqKEkidEc2IiIiIkkieEdGJUYmLUkiUUdGJTYkRiciIiFGJkYmKiYjRiYiIiNGJkYkISIiRi8qJEYnRi9GJg==

What do we need to subtract from Ix to hope to have a constant ? series(Ix-I0x^2,x,1);

KydJInhHNiIsJCooIiIjIiIiSSJ0R0YkRigtSSNEREdGJDYjIiIhRihGKCEiIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoISIi

A2:=coeff(%,x,-2); factor(series(Ix-I0x^2-A2*I0x,x,2));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNBMkdGKCwkKigiIiMiIiJJInRHRihGMS1JI0RER0YoNiMiIiFGMUYxNyNGLg==

KydJInhHNiIsJiIiJSIiIiooRiZGJ0kidEdGJEYnLUkiVUdGJDYkIiIhRi1GJ0YnISIiLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRidGLQ==

A1:=coeff(%,x,-1); factor(series(Ix-I0x^2-A2*I0x-A1*I1x,x,3));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNBMUdGKCwmIiIlIiIiKihGL0YwSSJ0R0YoRjAtSSJVR0YoNiQiIiFGNkYwRjA3I0Yu

KydJInhHNiIqJiwqKiYpSSJ0R0YkIiIkIiIiKS1JI0RER0YkNiMiIiEiIiNGK0YrKigiIiVGKylGKUYxRistLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YrNiNJIlVHRiQ2JEYwRjBGK0YrKihGMUYrRilGKy1GPkY/RishIiJGMUZCRitGKUZCRjAtSSJPR0Y5RjxGKw==

What happens to J(y) when we modify it accordingly? An encouraging sign is that there is no pole at y=0 map(factor,series(Jy-J0y^2-A2*J0y-A1*J1y,y,3));

KydJInlHNiIsKiIiIyIiIiooIiIlRidJInRHRiRGJy1JIlVHRiQ2JCIiIUYuRidGJyomRiZGJy1JI0RER0YkNiNGLkYnISIiKihGJkYnRipGJy0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGMUYyRidGJ0YzLUkiT0dGODYjRidGLg==

The coefficient of 1/y is 0, as can be seen by looking at the last step of a walk ending at the origin factor(subs(DD(0)=1+2*t*U(0,0)+t*(D(DD))(0),%));

KyVJInlHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiE=

Now let us consider the corresponding ratio and prove that it is a multiple of x and y ser:=map(factor,series(ratio-ratio0*(I0x+J0y)-A2*ratio0-A1*ratio1,x,3));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzZXJHRigrKUkieEdGKCwkKiZJInRHRighIiIsLCoqIiIjIiIiLUkiVUdGKDYkIiIhSSJ5R0YoRjdGMkY3RjxGN0Y3KiYtSSNEREdGKDYjRjxGN0YyRjdGNyomRjxGN0Y+RjdGMyomRjJGNy1GPzYjRjtGN0YzRjxGN0Y3RjNGMywkKihGMkYzRjxGMywyKipGNkY3Rj5GN0YyRjcpRjwiIiRGN0Y3KipGNkY3KUY8RjZGNy0tJkkiREdGJTYjRjc2I0Y5RjpGN0YyRjdGNyooRjZGN0ZMRjdGOEY3RjMqKkY2RjctRjk2JEY7RjtGN0YyRjdGPEY3RjNGPUYzRkFGN0ZCRjdGPEYzRjdGM0Y7LUkiT0dGJUZRRjc3I0Yu

Let us check that the coefficient of 1/x vanishes eqD;

LCoqJiwmKiZJInlHNiIhIiJJInRHRiciIiJGKEYqRipGKi1JI0RER0YnNiNGJkYqRihGKkYqKihGKUYqRiZGKC1GLDYjIiIhRipGKCooIiIjRipGKUYqLUkiVUdGJzYkRjFGJkYqRio=

factor(subs(isolate(eqD,U(0,y)),coeff(ser,x,-1)));

IiIh

Now we consider the coefficient of x^0 in ser, and want to prove that it is in fact zero eqU;series(eqU,x,2);

LCoqKCwmKiZJInRHNiIiIiIsKCokSSJ4R0YnISIiRigqJkYrRihJInlHRidGKEYoKiRGLkYsRihGKEYsRihGKEYoRitGKC1JIlVHRic2JEYrRi5GKEYsKipGJkYoRitGKEYuRigtSSNEREdGJzYjRi5GKEYoKipGJkYoRi5GLEYrRigtRjE2JEYrIiIhRihGLComRiZGKC1GMTYkRjpGLkYoRiw=

KydJInhHNiIsKiomSSJ0R0YkIiIiLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKDYjSSJVR0YkNiQiIiFJInlHRiRGKEYoKiYsJiomRjUhIiJGJ0YoRjlGKEYoRigtRjJGM0YoRjkqKC1JI0RER0YkNiNGNUYoRidGKEY1RihGKCooRjVGOUYnRigtRjI2JEY0RjRGKEY5RigtSSJPR0YtRjAiIiM=

factor(subs(isolate(convert(%,polynom),(D[1](U))(0, y)),isolate(eqD,U(0,y)),ser));

KyVJInhHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGKw==

Now let us expand the ratio in y map(factor,series(ratio-ratio0*(I0x+J0y)-A2*ratio0-A1*ratio1,y,3));

KydJInlHNiIsJComSSJ0R0YkISIiLDAqKiIiIyIiIkkieEdGJEYsLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKzYjSSJVR0YkNiRGLSIiIUYsRidGLEYsKihGK0YsRidGLC1GN0Y4RixGLCooRitGLEYtRixGO0YsRigqKCIiJUYsRidGLC1GNzYkRjlGOUYsRigqJkYnRiwtLUYxNiNJI0RER0YkNiNGOUYsRigtRkVGRkYsRixGKEYsRihGOS1JIk9HRjI2I0YsRiw=

series(eqU,y,2);

KydJInlHNiIsKCooSSJ4R0YkIiIiLS0mSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiM2I0kiVUdGJDYkRiciIiFGKEkidEdGJEYoRigqKCwmKiZGNkYoRichIiJGOkYoRihGKEYnRigtRjNGNEYoRjoqJkY2RigtRjM2JEY1RjVGKEY6RjUtSSJPR0YtNiNGKEYo

factor(subs(isolate(convert(%,polynom),(D[2](U))(x,0)),%%));

KydJInlHNiIsJComLCoqKCIiIyIiIkkidEdGJEYqLUkiVUdGJDYkIiIhRi9GKiEiIiomRitGKi0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJI0RER0YkNiNGL0YqRjAtRjlGOkYqRipGMEYqRitGMEYwRi8tSSJPR0Y1NiNGKkYq

series(eqD,y,2)

KydJInlHNiIsKiomSSJ0R0YkIiIiLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kjRERHRiQ2IyIiIUYoRigtRjBGMSEiIkYoRigqKCIiI0YoRidGKC1JIlVHRiQ2JEYyRjJGKEYoRjItSSJPR0YsNiNGKEYo

factor(subs(isolate(convert(%,polynom),U(0,0)),%%));

KyVJInlHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGKw==

A2

LCQqKCIiIyIiIkkidEc2IkYlLUkjRERHRic2IyIiIUYlRiU=

We conclude that the following series are independent of x and y (and equal) exprx:=Ix-I0x^2-A2*I0x-A1*I1x; expry:=Jy-J0y^2-A2*J0y-A1*J1y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZleHByeEdGKCwqKiYsJiIiJSIiIiooRjFGMkkidEdGKEYyLUkiVUdGKDYkIiIhRjhGMkYyRjIsKCooRjRGMkkieEdGKEYyLUkiUUdGKDYkRjtGOEYyRjIqJiNGMiIiI0YyRjQhIiJGQiokRjtGQkYyRjJGQioqRkFGMkY0RjItSSNEREdGKDYjRjhGMiwoKiQpRjtGQUZCRjIqJkY0RkJGO0ZCRkJGO0ZCRjJGQiokKUZIRkFGMkZCKiQpLCwqKkZBRjJGO0YyRjRGMi1GNkY+RjJGMkZJRjJGS0ZCRjtGMiomRjRGMkZFRjJGMkZBRjJGMjcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZleHByeUdGKCwqKiYsJiIiJSIiIiooRjFGMkkidEdGKEYyLUkiVUdGKDYkIiIhRjhGMkYyRjIsKCooRjRGMkkieUdGKEYyLUkiUUdGKDYkRjhGO0YyISIiKiYjRjIiIiNGMkY0Rj9GMiokRjtGP0Y/RjJGPyoqRkJGMkY0RjItSSNEREdGKDYjRjhGMiwoKiQpRjtGQkY/RjIqJkY0Rj9GO0Y/Rj9GO0Y/RjJGPyokKUZIRkJGMkY/KiYsJiokKSwmKiZGO0Y/RjRGMkY/RjJGMkZCRjJGMiooRjFGMkY7RjIpRjRGQkYyRj9GMiksJiomRjtGMi1GRjYjRjtGMkYyRktGMkZCRjJGMjcjRi4=

Let us check n:=20: normal(series(subs(U(x,0)=subs(y=0,User(n)),Q(x,0)=subs(y=0,Qser(n)),DD(0)=subs(y=0,Dser(n)),U(0,0)=subs(x=0,y=0,User(n)), exprx),t,n));

KzNJInRHNiIhIiMhIiJGJiIiIyEiJyIiJiEjdSIiKSElIT0iIiM2ISZLOSMiIzkhJ0c0VSIjPC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIjPg==

n:=20: normal(series(subs(DD(y)=Dser(n), Q(0,y)=subs(x=0,Qser(n)),Q(0,0)=subs(x=0,y=0,Qser(n)),DD(0)=subs(y=0,Dser(n)),U(0,0)=subs(x=0,y=0,User(n)), expry),t,n));

KzNJInRHNiIhIiMhIiJGJiIiIyEiJyIiJiEjdSIiKSElIT0iIiM2ISZLOSMiIzkhJ0c0VSIjPC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIjPg==

The solution of quadrant walks with Kreweras steps, taken from earlier papers algV:=-V+t*(2+V^3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdWR0YoLCZJIlZHRighIiIqJkkidEdGKCIiIiwmKiQpRi8iIiRGM0YzIiIjRjNGM0YzNyNGLg==

tV:=solve(algV,t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN0VkdGKComSSJWR0YoIiIiLCYqJClGLyIiJEYwRjAiIiNGMCEiIjcjRi4=

with(gfun): Vser:=proc(n) op(algeqtoseries(algV,t,V,n,true)): end: Vser(4); Vser(5);

KylJInRHNiIiIiMiIiIiIikiIiUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJiIiJg==

KylJInRHNiIiIiMiIiIiIikiIiUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJiIiJw==

Deltapx:=1-x*V^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShEZWx0YXB4R0YoLCYqJilJIlZHRigiIiMiIiJJInhHRihGMyEiIkYzRjM3I0Yu

Qx0:=(1/2/t-1/x-(1/V-1/x)*sqrt(Deltapx))/t/x;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRReDBHRigqKCwoKiYjIiIiIiIjRjJJInRHRighIiJGMiokSSJ4R0YoRjVGNSomLCYqJEkiVkdGKEY1RjJGNkY1RjIpLCYqJilGO0YzRjJGN0YyRjVGMkYyRjFGMkY1RjJGNEY1RjdGNTcjRi4=

Let us check n:=12: factor(series(subs(V=Vser(n),Qx0),t,n));

KzdJInRHNiIiIiIiIiFJInhHRiQiIiNGKCIiJCwkKiZGKEYlKUYnRihGJUYlIiIlLCQqJiIiKUYlRidGJUYlIiImLCYqJkYxRiUpRidGKUYlRiUiIztGJSIiJywkKiYiI0lGJUYsRiVGJSIiKCwkKihGKEYlRidGJSwmKiZGOkYlRjRGJUYlIiNbRiVGJUYlRjAsJiomIiQ3IkYlRjRGJUYlIiQjPkYlIiIqLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIzU=

map(factor,subs(y=0,Qser(9)));

KzVJInRHNiIiIiIiIiFJInhHRiQiIiNGKCIiJCwkKiZGKEYlKUYnRihGJUYlIiIlLCQqJiIiKUYlRidGJUYlIiImLCYqJkYxRiUpRidGKUYlRiUiIztGJSIiJywkKiYiI0lGJUYsRiVGJSIiKCwkKihGKEYlRidGJSwmKiZGOkYlRjRGJUYlIiNbRiVGJUYlRjAsJiomIiQ3IkYlRjRGJUYlIiQjPkYlIiIq

normal(series(%%-%,t,n));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

factor(subs(t=tV,series(Qx0,x,3)));

KydJInhHNiIsJCooIyIiIiIiKUYoLCYqJClJIlZHRiQiIiRGKEYoIiIlISIiRigsJkYrRigiIiNGKEYoRjAiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYo

Q00:=convert(%,polynom);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRRMDBHRigsJCooIyIiIiIiKUYxLCYqJClJIlZHRigiIiRGMUYxIiIlISIiRjEsJkY0RjEiIiNGMUYxRjk3I0Yu

I1xalg:=subs(Q(x,0)=Qx0,I1x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdJMXhhbGdHRigsJComLCYqJEkiVkdGKCEiIiIiIiokSSJ4R0YoRjNGM0Y0KSwmKiYpRjIiIiNGNEY2RjRGM0Y0RjQjRjRGO0Y0RjM3I0Yu

Ixform:=I0x^2+AA2*I0x+AA1*%+AA0; Jyform:=J0y^2+AA2*J0y+AA1*subs(x=y,-%%)+AA0;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdJeGZvcm1HRigsKiooLCYqJEkiVkdGKCEiIiIiIiokSSJ4R0YoRjNGM0Y0KSwmKiYpRjIiIiNGNEY2RjRGM0Y0RjQjRjRGO0Y0SSRBQTFHRihGNEYzKiZJJEFBMkdGKEY0LCgqJClGNkY7RjNGNComSSJ0R0YoRjNGNkYzRjNGNkYzRjRGNCokKUZARjtGNEY0SSRBQTBHRihGNDcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdKeWZvcm1HRigsKiokKSwoKiQpSSJ5R0YoIiIjISIiIiIiKiZJInRHRihGNkY0RjZGNkY0RjZGNUY3RjcqJkkkQUEyR0YoRjdGMUY3RjcqKEkkQUExR0YoRjcsJiokSSJWR0YoRjZGNyokRjRGNkY2RjcpLCYqJilGQEY1RjdGNEY3RjZGN0Y3I0Y3RjVGN0Y3SSRBQTBHRihGNzcjRi4=

Relation between I0 and I1 series(I1xalg^2,x,1); series (I0x,x,1)

KydJInhHNiIiIiIhIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSEiIg==

KylJInhHNiIiIiIhIiMsJCokSSJ0R0YkISIiRipGKi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlRiU=

expand(factor(subs(t=tV,I1xalg^2-I0x)));

LCYqJiIiIyIiIkkiVkc2IkYlRiUqJClGJkYkISIiRiU=

cV:=%;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjVkdGKCwmKiYiIiMiIiJJIlZHRihGMUYxKiQpRjJGMCEiIkYxNyNGLg==

Equation for I1x algI1x:=subs(t=tV,II1x^2-I0x-cV)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdJMXhHRigsLiokKUklSUkxeEdGKCIiIyIiIkYzKiQpSSJ4R0YoRjIhIiJGNyooSSJWR0YoRjcsJiokKUY5IiIkRjNGM0YyRjNGM0Y2RjdGM0Y2RjMqJkYyRjNGOUYzRjcqJClGOUYyRjdGNzcjRi4=

algJ1y:=subs(II1x=JJ1y,x=y,algI1x)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdKMXlHRigsLiokKUklSkoxeUdGKCIiIyIiIkYzKiQpSSJ5R0YoRjIhIiJGNyooSSJWR0YoRjcsJiokKUY5IiIkRjNGM0YyRjNGM0Y2RjdGM0Y2RjMqJkYyRjNGOUYzRjcqJClGOUYyRjdGNzcjRi4=

Hence Ix is a quartic polynomial in I1x, and Jy is the same quartic polynomial in J1y P4:=(u^2-cV)^2+AA2*(u^2-cV)+AA1*u+AA0;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNQNEdGKCwqKiQpLCgqJClJInVHRigiIiMiIiJGNiomRjVGNkkiVkdGKEY2ISIiKiQpRjhGNUY5RjlGNUY2RjYqJkkkQUEyR0YoRjZGMUY2RjYqJkkkQUExR0YoRjZGNEY2RjZJJEFBMEdGKEY2NyNGLg==

This polynomial of degree 4 vanishes when u=J1(Y), where Y is any of the roots of Delta well defined at t=0 factor(Delta)

LCQqJiwqKigiIiUiIiIpSSJ0RzYiIiIjRicpSSJ5R0YqIiIkRidGJyokRihGJyEiIiooRitGJ0YpRidGLUYnRicqJClGLUYrRidGMEYnRjNGMEYw

Two roots are finite at t=0 map(allvalues,algeqtoseries(numer(Delta),t,y,10, true));

NyQsLEkidEc2IiIiIiomIiIjRiYpRiQjIiImRihGJkYmKiYiIidGJilGJCIiJUYmRiYqJiIjQEYmKUYkIyIjNkYoRiZGJi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyokKUYkIiIoRiZGJiwsRiRGJkYnISIiRixGJkYwRj9GNUYm

We can determine the minimal polynomial of Y+ and Y- in terms of V numer(factor(subs(t=tV,Delta)));

KiYsJiomKUkiVkc2IiIiIyIiIkkieUdGJ0YpRilGKSEiIkYpLCoqJilGJiIiJUYpRipGKUYpKiRGJUYpRisqKEYvRilGJkYpRipGKUYpKiZGL0YpKUYqRihGKUYrRik=

algY:=collect(-op(2,%),y,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdZR0YoLCgqJiIiJSIiIilJInlHRigiIiNGMUYxKihJIlZHRihGMSwmKiQpRjYiIiRGMUYxRjBGMUYxRjNGMSEiIiokKUY2RjRGMUYxNyNGLg==

Can we express Y using the other two series W and Z? We prove below that these roots of Delta are not in the extension generated by W and Z

solve(algY,y)

NiQqJiwoKiYjIiIiIiIpRicpSSJWRzYiIiIkRidGJyNGJyIiI0YnKiZGJkYnKSwmKiQpRioiIidGJ0YnKiZGKEYnRilGJ0YnRi1GJ0YnRidGKkYnKiYsKEYlRidGLUYnRi8hIiJGJ0YqRic=

We re-introduce the series used in the solution of Kreweras' model algWVt:=Vt-4*W*(1-W);solve(%,W)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdXVnRHRigsJkkjVnRHRigiIiIqKCIiJUYwSSJXR0YoRjAsJkYwRjBGMyEiIkYwRjU3I0Yu

NiQsJiMiIiIiIiNGJSomRiRGJSksJkkjVnRHNiIhIiJGJUYlRiRGJUYlLCZGJEYlRidGLA==

VtW:=solve(%%,Vt);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRWdFdHRigsJCooIiIlIiIiSSJXR0YoRjEsJkYxISIiRjJGMUYxRjQ3I0Yu

Wser:=proc(n) op(2,algeqtoseries(algWVt,Vt,W,n,true)): end: Wser(4);

Ky1JI1Z0RzYiIyIiIiIiJUYmI0YmIiM7IiIjI0YmIiNLIiIkIyIiJiIkYyNGJy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YmRi8=

algZW:=numer(W-2*Z/(1+Z^2));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdaV0dGKCwoKiZJIldHRigiIiIpSSJaR0YoIiIjRjFGMUYwRjEqJkY0RjFGM0YxISIiNyNGLg==

WZ:=factor(solve(%,W))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNXWkdGKCwkKigiIiMiIiJJIlpHRihGMSwmKiQpRjJGMEYxRjFGMUYxISIiRjE3I0Yu

ZWser:=proc(n) op(algeqtoseries(algZW,W,Z,n,true)): end: ZWser(5);

KytJIldHNiIjIiIiIiIjRiYjRiYiIikiIiQjRiYiIzsiIiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJiIiJw==

op(3,factor(subs(y=V*z, algY)))

LCoqJilJIlZHNiIiIiQiIiJJInpHRiZGKEYoKiYiIiVGKClGKSIiI0YoISIiKiZGK0YoRilGKEYoRihGLg==

subs(V=Vt^(1/3),%)

LCoqJkkjVnRHNiIiIiJJInpHRiVGJkYmKiYiIiVGJilGJyIiI0YmISIiKiZGKUYmRidGJkYmRiZGLA==

factor(subs(Vt=VtW,%))

LCwqKCIiJSIiIilJIldHNiIiIiNGJUkiekdGKEYlISIiKihGJEYlRidGJUYqRiVGJSomRiRGJSlGKkYpRiVGKyomRiRGJUYqRiVGJUYlRis=

factor(subs(W=WZ,%))

LCQqJiw4KigiIiUiIiIpSSJaRzYiRiZGJylJInpHRioiIiNGJ0YnKihGJkYnRihGJ0YsRichIiIqJEYoRidGJyooIiIpRicpRikiIiRGJ0YsRidGLyooRjJGJylGKUYtRidGK0YnRicqKEYyRidGNkYnRixGJ0YnKiZGLUYnRjZGJ0YnKihGMkYnRilGJ0YsRidGLyomRiZGJ0YrRidGJyomRiZGJ0YsRidGL0YnRidGJyksJiokRjZGJ0YnRidGJ0YtRi9GLw==

Not one of the known extensions. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv

Anyway, we are interested in J1(Y) rather than Y itself factor(resultant(numer(algJ1y),algY,y))

KiYpSSJWRzYiIiIlIiIiLDIqJilJJUpKMXlHRiUiIiNGJylGJCIiKUYnRicqJkYmRicpRiQiIipGJyEiIiooRiZGJylGK0YmRidGI0YnRicqKCIjP0YnRipGJylGJCIiJkYnRjIqJiIjN0YnKUYkIiInRidGJyooRi5GJ0YqRicpRiRGLEYnRjIqJkY6RicpRiQiIiRGJ0YyRiZGJ0Yn

collect(op(2,factor(resultant(numer(algJ1y),algY,y))),JJ1y,factor)

LCgqKCIiJSIiIilJJUpKMXlHNiJGJEYlKUkiVkdGKEYkRiVGJSoqKUYqIiIjRiUsKCokRixGJUYlKiZGLUYlRipGJSEiIkYtRjFGJSwsKiRGKUYlRiUqJkYtRiUpRioiIiRGJUYlKiYiIidGJUYsRiVGJSomRiRGJUYqRiVGMUYkRiVGJSlGJ0YtRiVGJSooRiRGJSksJkYqRiVGJUYxRjZGJSksKEYvRiVGKkYlRiVGJUY2RiVGMQ==

A biquadratic equation algJ1Y:=subs(JJ1y=J1Y,%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdKMVlHRigsKCooIiIlIiIiKUkkSjFZR0YoRjBGMSlJIlZHRihGMEYxRjEqKilGNSIiI0YxLCgqJEY3RjFGMSomRjhGMUY1RjEhIiJGOEY8RjEsLCokRjRGMUYxKiZGOEYxKUY1IiIkRjFGMSomIiInRjFGN0YxRjEqJkYwRjFGNUYxRjxGMEYxRjEpRjNGOEYxRjEqKEYwRjEpLCZGNUYxRjFGPEZBRjEpLChGOkYxRjVGMUYxRjFGQUYxRjw3I0Yu

The expression is smaller when we do not factor coefficients collect(%,J1Y,expand)

LC4qKCIiJSIiIilJJEoxWUc2IkYkRiUpSSJWR0YoRiRGJUYlKiYsKCokKUYqIiIpRiVGJSomIiM/RiUpRioiIiZGJSEiIiomRi9GJSlGKiIiI0YlRjRGJSlGJ0Y3RiVGJSomRiRGJSlGKiIiKkYlRjQqJiIjN0YlKUYqIiInRiVGJSomRj1GJSlGKiIiJEYlRjRGJEYl

The values J1(Y) map(allvalues,algeqtoseries(algJ1Y,V,J1Y,5))

NyYsKiokSSJWRzYiISIiIiIiKiYpIiIjI0YoRitGKClGJUYsRihGKComI0YoIiIlRigpRiVGK0YoRigtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiMqJClGJSMiIiZGK0YoRigsKkYkRihGKUYnRi5GKEYyRigsKkYkRidGKUYoRi5GJ0YyRigsKkYkRidGKUYnRi5GJ0YyRig=

J1Yvalens:=map(series,subs(V=Vser(10),%),t,4)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpKMVl2YWxlbnNHRig3JiwoKiYjIiIiIiIjRjJJInRHRighIiJGMiomRjNGMilGNEYxRjJGMi1JIk9HRiU2IyokKUY0RjNGMkYyLChGMEYyRjZGNUY4RjIsKEYwRjVGNkYyRjhGMiwoRjBGNUY2RjVGOEYyNyNGLg==

Is there a solution, or at least a factorization, in terms of W and Z ? No, as we shall see in the next few lines. First, we observe that J1(Y)/V is a series in t^3 factor(subs(J1Y=sqrt(U)/V,algJ1Y));

LDIqJiIiJSIiIilJIlZHNiIiIipGJSEiIiomSSJVR0YoRiUpRiciIidGJUYlKiYiIzdGJUYtRiVGJSooIiM/RiVGLEYlKUYnIiIkRiVGKiomRjBGJUYzRiVGKiomRiRGJSlGLCIiI0YlRiUqJiIiKUYlRixGJUYqRiRGJQ==

subs(V=Vt^(1/3),%)

LDIqJkkiVUc2IiIiIilJI1Z0R0YlIiIjRiZGJiomIiIlRiYpRigiIiRGJiEiIiomRitGJilGJEYpRiZGJiooIiM/RiZGJEYmRihGJkYuKiYiIzdGJkYnRiZGJiomIiIpRiZGJEYmRi4qJkY0RiZGKEYmRi5GK0Ym

factor(subs(Vt=VtW,%))

LDwqJiIkYyMiIiIpSSJXRzYiIiInRiVGJSooIiM7RiVJIlVHRihGJSlGJyIiJUYlRiUqJiIkbyhGJSlGJyIiJkYlISIiKigiI0tGJUYsRiUpRiciIiRGJUYzKiYiJGcqRiVGLUYlRiUqKCIjJypGJUYsRiUpRiciIiNGJUYlKiYiJFMnRiVGNkYlRjMqJkYuRiUpRixGPUYlRiUqKCIjISlGJUYsRiVGJ0YlRjMqJiIkUyNGJUY8RiVGJSomIiIpRiVGLEYlRjMqJiIjW0YlRidGJUYzRi5GJQ==

Not better numer(factor(subs(W=WZ,%)))

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

The following two polynomials have two roots in common, namely J1(Yplus) and J1(Yminus). One is biquadratic, while the other has no term in u^3, but a linear term in u subs(J1Y=u,algJ1Y);P4

LCgqKCIiJSIiIilJInVHNiJGJEYlKUkiVkdGKEYkRiVGJSoqKUYqIiIjRiUsKCokRixGJUYlKiZGLUYlRipGJSEiIkYtRjFGJSwsKiRGKUYlRiUqJkYtRiUpRioiIiRGJUYlKiYiIidGJUYsRiVGJSomRiRGJUYqRiVGMUYkRiVGJSlGJ0YtRiVGJSooRiRGJSksJkYqRiVGJUYxRjZGJSksKEYvRiVGKkYlRiVGJUY2RiVGMQ==

LCoqJCksKCokKUkidUc2IiIiIyIiIkYrKiZGKkYrSSJWR0YpRishIiIqJClGLUYqRi5GLkYqRitGKyomSSRBQTJHRilGK0YlRitGKyomSSRBQTFHRilGK0YoRitGK0kkQUEwR0YpRis=

This means that each of these two polynomials is divisible by this quadratic one, which is thus the minimal polynomial of J1(Yplus) and J1(Yminus) over Q(V) P2:=collect(4*V^2*P4-subs(J1Y=u,algJ1Y)/V^2,[u],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNQMkdGKCwyKigpSSJWR0YoIiIjIiIiLCgqJClGMSIiJUYzISIiKiZGN0YzSSRBQTJHRihGM0YzKiZGN0YzRjFGM0YzRjMpSSJ1R0YoRjJGM0YzKipGN0YzRjBGM0kkQUExR0YoRjNGPUYzRjMqJkY3RjMpRjEiIihGM0YzKigiIilGM0Y6RjMpRjEiIiRGM0Y4KiZGN0YzRjZGM0YzKihGN0YzSSRBQTBHRihGM0YwRjNGM0Y5RjgqJiIjR0YzRjFGM0YzNyNGLg==

Another information will come from the (double) root of I(x) algX:=numer(op(1,Ix));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdYR0YoLCwqKiIiIyIiIilJInhHRigiIiRGMSlJInRHRihGMEYxLUkiVUdGKDYkRjMiIiFGMUYxKihGNUYxLUkjRERHRig2I0Y6RjEpRjNGMEYxRjEqJkY2RjFGMkYxRjFGNkYxRjMhIiI3I0Yu

Xser:=proc(n) option remember: if n=0 then 0 else normal(series(subs(DD(0)=subs(y=0,Dser(n)), U(x,0)=subs(y=0,User(n)),x=Xser(n-1),algX+x),t,n+1)): fi: end: Xser(10);

Ky1JInRHNiIiIiJGJSIiIyIiJSIjOyIiKCIkJz4iIzUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjOA==

Let us check that the derivative of I1 does not cancel at X diff(I1xalg,x);

LCYqJilJInhHNiIiIiMhIiIpLCYqJilJIlZHRiZGJyIiIkYlRi5GKEYuRi4jRi5GJ0YuRigqKkYvRi4sJiokRi1GKEYuKiRGJUYoRihGLkYpRihGLEYuRi4=

series(subs(x=Xser(10),V=Vser(n),%),t,10);

Ky1JInRHNiIhIiIhIiMiIiYiIiIiI0MiIiUiJHUjIiIoLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIzU=

Here is a double root of P4 I1Xser:=series(subs(x=Xser(n),V=Vser(n),I1xalg),t,n)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdJMVhzZXJHRigrLUkidEdGKCMiIiIiIiMhIiJGM0YyISIoIiImISMhKiIiKS1JIk9HRiU2I0YxIiM2NyNGLg==

We check that we have not already found this root J1Yvalens

NyYsKComIyIiIiIiI0YmSSJ0RzYiISIiRiYqJkYnRiYpRihGJUYmRiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YpNiMqJClGKEYnRiZGJiwoRiRGJkYrRipGLUYmLChGJEYqRitGJkYtRiYsKEYkRipGK0YqRi1GJg==

Hence P4 has a double root at I1(X). The form of P4 is as follows P4; (u-I1X)^2*P2/lcoeff(P2,u);

LCoqJCksKCokKUkidUc2IiIiIyIiIkYrKiZGKkYrSSJWR0YpRishIiIqJClGLUYqRi5GLkYqRitGKyomSSRBQTJHRilGK0YlRitGKyomSSRBQTFHRilGK0YoRitGK0kkQUEwR0YpRis=

KiopLCZJInVHNiIiIiJJJEkxWEdGJiEiIiIiI0YnLDIqKClJIlZHRiZGKkYnLCgqJClGLiIiJUYnRikqJkYyRidJJEFBMkdGJkYnRicqJkYyRidGLkYnRidGJylGJUYqRidGJyoqRjJGJ0YtRidJJEFBMUdGJkYnRiVGJ0YnKiZGMkYnKUYuIiIoRidGJyooIiIpRidGNEYnKUYuIiIkRidGKSomRjJGJ0YxRidGJyooRjJGJ0kkQUEwR0YmRidGLUYnRidGM0YpKiYiI0dGJ0YuRidGJ0YnRi1GKUYvRik=

We now equate these two polynomials diff1:=collect(P4-%,u,factor);

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

An expression of I1(X) I1Xsol:=solve(coeff(diff1,u,3),I1X);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdJMVhzb2xHRigsJCooIiIjIiIiSSRBQTFHRihGMSwoKiQpSSJWR0YoIiIlRjEhIiIqJkY3RjFJJEFBMkdGKEYxRjEqJkY3RjFGNkYxRjFGOEYxNyNGLg==

The denominator does not vanish (it was also the leading coefficient of P2, which has two roots) n:=8: series(subs(DD(0)=subs(y=0,Dser(n)),V=Vser(n),4*A2-V*(V^3-4)),t,n)

KytJInRHNiIiIzsiIiIiI1siIiUiJCdcIiIoLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiYiIio=

We replace I1(X) by its expression diff2:=map(factor,subs(I1X=I1Xsol,diff1));

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

A0sol:=collect(factor(solve(coeff(%,u,2),AA0)),AA2,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZBMHNvbEdGKCwkKiojIiIiIiIlRjEsKiooIiM7RjEpSSRBQTJHRigiIiRGMSlJIlZHRigiIiNGMUYxKiYsJiomIiIpRjEpRjoiIidGMSEiIkY1RkJGMSlGN0Y7RjFGMSouRjpGMSwmRjpGMUY7RjFGMSwmKiQpRjpGOEYxRjFGO0YxRjEsJkZHRjEiIzVGQkYxLCgqJEY5RjFGMSomRjtGMUY6RjFGQkYyRjFGMUY3RjFGMSooRkFGMUY5RjEsKiomRjhGMUZARjFGMSomRkFGMUZIRjFGQiomRjtGMSlJJEFBMUdGKEY7RjFGMSIjQ0ZCRjFGMUYxRjlGQiwmKiZGMkYxRjdGMUYxKiZGOkYxLCZGR0YxRjJGQkYxRkJGQkYxNyNGLg==

diff3:=map(factor,subs(AA0=A0sol,diff2));

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

The last two coefficients (or their numerators) co1:=collect(op(1,numer(coeff(diff3,u,1))),[AA1,AA2],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRjbzFHRigsLCooIiNLIiIiKUkkQUExR0YoIiIjRjEpSSJWR0YoRjRGMUYxKigiJEciRjEpSSRBQTJHRigiIiRGMUY1RjFGMSomLCgqJiIjISlGMSlGNiIiJ0YxISIiKiYiI2tGMSlGNkY7RjFGMUY4RkJGMSlGOkY0RjFGMSosIiM7RjFGNkYxLCYqJEZFRjFGMSIiJUZCRjEpLCZGSkYxRjRGMUY0RjFGOkYxRjEqKEY1RjEsKCokRkBGMUYxKiYiIzdGMUZFRjFGMSIiKUYxRjEpRklGNEYxRkI3I0Yu

co0:=collect(numer(coeff(diff3,u,0)),[AA1,AA2],factor)

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

res:=factor(resultant(co1,co0,AA1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyZXNHRigsJCoqIiUnNCUiIiIpSSJWR0YoIiM3RjEpLDIqJiIiKkYxKUYzIiM1RjFGMSooIiMnKkYxSSRBQTJHRihGMSlGMyIiJ0YxISIiKiYiIykpRjEpRjMiIihGMUYxKigiJGMjRjEpRj0iIiNGMSlGM0ZIRjFGMSooIiRHIkYxRj1GMSlGMyIiJEYxRkAqJkZLRjEpRjMiIiVGMUZAKiZGRkYxRj1GMUZAKiZGRkYxRjNGMUYxRkhGMSksKCokRk9GMUZAKiZGUEYxRj1GMUYxKiZGUEYxRjNGMUYxIiIpRjFGMTcjRi4=

We have already seen that the small factor does not vanish algA2V:=collect(op(1,op(3,res)),AA2,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdBMlZHRigsKCooIiRjIyIiIilJJEFBMkdGKCIiI0YxKUkiVkdGKEY0RjFGMSomLCgqJiIjJypGMSlGNiIiJ0YxISIiKiYiJEciRjEpRjYiIiRGMUY9RjBGPUYxRjNGMUYxKiZGNkYxLCoqJiIiKkYxKUY2RkVGMUYxKiYiIykpRjFGO0YxRjFGPkY9RjBGMUYxRjE3I0Yu

Can we solve this using W? We observe that A2/V is a series in t^3 collect(op(2,factor(subs(AA2=V*B2,algA2V))),B2,factor)

LC4qKCIkYyMiIiIpSSJWRzYiIiIkRiUpSSNCMkdGKCIiI0YlRiUqJiwoKiYiIycqRiUpRiciIidGJSEiIiomIiRHIkYlRiZGJUYzRiRGM0YlRitGJUYlKiYiIipGJSlGJ0Y3RiVGJSomIiMpKUYlRjFGJUYlRjRGM0YkRiU=

Here we have a factorization. Hence B2 is in Q(W) res:=factor(subs(V=Vt^(1/3),Vt=VtW,%)); nops(res)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyZXNHRigsJCooIiNrIiIiLC4qJiIiJEYxKUkiV0dGKEY0RjFGMSooIiIlRjFJI0IyR0YoRjFGNkYxRjEqJiIiI0YxKUY2RjtGMSEiIiomRjhGMUY5RjFGPSomRjhGMUY2RjFGPUY4RjFGMSwsRjNGMUY3RjEqJiIiKEYxRjxGMUY9RjZGMUYxRj1GMUY9NyNGLg==

IiIk

series(subs(B2=AA2/V, AA2= 2*t*subs(y=0,Dser(n)),W=Wser(n),Vt=Vser(n)^3,V=Vser(n),op(2,res)),t,n); series(subs(B2=AA2/V, AA2= 2*t*subs(y=0,Dser(n)),W=Wser(n),Vt=Vser(n)^3,V=Vser(n),op(3,res)),t,n);

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIik=

KytJInRHNiIhIiIiIiEiIzUiIiQiJDciIiInLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiIq

B2sol:=factor(solve(op(2,res),B2))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZCMnNvbEdGKCwkKigjIiIiIiIlRjEsKiomIiIkRjEpSSJXR0YoRjVGMUYxKiYiIiNGMSlGN0Y5RjEhIiIqJkYyRjFGN0YxRjtGMkYxRjEsJkYxRjtGN0YxRjtGOzcjRi4=

A2sol:=V*%;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZBMnNvbEdGKCwkKiojIiIiIiIlRjFJIlZHRihGMSwqKiYiIiRGMSlJIldHRihGNkYxRjEqJiIiI0YxKUY4RjpGMSEiIiomRjJGMUY4RjFGPEYyRjFGMSwmRjFGPEY4RjFGPEY8NyNGLg==

Back to A1 factor(subs(V=Vt^(1/3),Vt=VtW,op(2,numer(factor(subs(AA2=A2sol,co1))))))

LCQqKCIiKSIiIiksJkYlISIiSSJXRzYiRiUiIiNGJSw4KiQpRikiIipGJUYlKiZGL0YlKUYpRiRGJUYoKiYiI0ZGJSlGKSIiKEYlRiUqJiIjOkYlKUYpIiInRiVGKComIiNzRiUpRikiIiZGJUYoKiYiJDMiRiUpRikiIiVGJUYlKihGQUYlKUkkQUExR0YqRitGJUYpRiVGJSomIiNbRiUpRikiIiRGJUYlKiZGQUYlRkNGJUYoKiYiJFciRiUpRilGK0YlRigiI2tGJUYlRiU=

algA1W:=collect(op(3,%),AA1,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdBMVdHRigsJiomLCYiIiUhIiIqJkYxIiIiSSJXR0YoRjRGNEY0KUkkQUExR0YoIiIjRjRGNComKSwmRjVGNEY0RjQiIiRGNCksJkY1RjRGOEYyIiInRjRGNDcjRi4=

A1Wsolens:=simplify([solve(algA1W,AA1)]) assuming -1<W assuming W<1;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpBMVdzb2xlbnNHRig3JCoqKSwmKiQpSSJXR0YoIiIjIiIiISIiRjZGNiNGNkY1RjYpLCZGNEY2RjVGNyIiJEY2LCZGNEY2RjZGNkY2LCYqJkY1RjZGNEY2RjZGNUY3RjcsJEYvRjc3I0Yu

A1squareW:=op(2,isolate(algA1W,AA1^2));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpBMXNxdWFyZVdHRigsJCooKSwmSSJXR0YoIiIiRjNGMyIiJEYzKSwmRjJGMyIiIyEiIiIiJ0YzLCYiIiVGOComRjtGM0YyRjNGM0Y4Rjg3I0Yu

Choosing the right sign A1

LCYiIiUiIiIqKEYjRiRJInRHNiJGJC1JIlVHRic2JCIiIUYrRiRGJA==

A1W:=(1+W)*(2-W)^3/2*sqrt((1+W)/(1-W));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBMVdHRigsJCoqIyIiIiIiI0YxLCZJIldHRihGMUYxRjFGMSksJkYyRjFGNCEiIiIiJEYxKSomRjNGMSwmRjFGMUY0RjdGN0YwRjFGMTcjRi4=

A quick check factor(A1W^2-A1squareW);

IiIh

algA1Z:=collect(op(2,numer(factor(subs(W=WZ,algA1W)))),AA1,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdBMVpHRigsJiooLCZJIlpHRigiIiJGMiEiIkYyKSwmKiQpRjEiIiNGMkYyRjJGMiIiJUYySSRBQTFHRihGMkYyKihGOUYyKSwmRjFGMkYyRjIiIiRGMiksKEY2RjJGMUYzRjJGMkY+RjJGMjcjRi4=

A1Z:=factor(solve(%,AA1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBMVpHRigsJCosIiIlIiIiKSwmSSJaR0YoRjFGMUYxIiIkRjEpLCgqJClGNCIiI0YxRjFGNCEiIkYxRjFGNUYxLCZGNEYxRjFGO0Y7KSwmRjhGMUYxRjFGMEY7Rjs3I0Yu

The expression of the paper A1pap:=(2-W)^3*(1+W)/2*(1+Z)/(1-Z)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZBMXBhcEdGKCwkKiwjIiIiIiIjRjEpLCZGMkYxSSJXR0YoISIiIiIkRjEsJkY1RjFGMUYxRjEsJkkiWkdGKEYxRjFGMUYxLCZGMUYxRjpGNkY2RjE3I0Yu

factor(subs(W=WZ,A1pap)-A1Z)

IiIh

Let us check A1; n:=10: series(subs(U(0,0)=subs(x=0,y=0,User(n)),A1),t,n)

LCYiIiUiIiIqKEYjRiRJInRHNiJGJC1JIlVHRic2JCIiIUYrRiRGJA==

KytJInRHNiIiIiUiIiFGJSIiJCIjVyIiJyIkZyciIio=

series(subs(Z=ZWser(n),W=Wser(n), Vt=Vser(n)^3,A1Z),t,n)

Ky1JInRHNiIiIiUiIiFGJSIiJCIjVyIiJyIkZyciIiotSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzc=

Back to A0 A0sol

LCQqKiMiIiIiIiVGJSwqKigiIztGJSlJJEFBMkc2IiIiJEYlKUkiVkdGLCIiI0YlRiUqJiwmKiYiIilGJSlGLyIiJ0YlISIiRilGN0YlKUYrRjBGJUYlKi5GL0YlLCZGL0YlRjBGJUYlLCYqJClGL0YtRiVGJUYwRiVGJSwmRjxGJSIjNUY3RiUsKCokRi5GJUYlKiZGMEYlRi9GJUY3RiZGJUYlRitGJUYlKihGNkYlRi5GJSwqKiZGLUYlRjVGJUYlKiZGNkYlRj1GJUY3KiZGMEYlKUkkQUExR0YsRjBGJUYlIiNDRjdGJUYlRiVGLkY3LCYqJkYmRiVGK0YlRiUqJkYvRiUsJkY8RiVGJkY3RiVGN0Y3RiU=

A0/V^2 is a series in t^3 A0overV2:=factor(subs(V=Vt^(1/3),Vt=VtW,(factor(subs(AA1=sqrt(A1squareW), AA2=A2sol, A0sol/V^2)))))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlBMG92ZXJWMkdGKCwkKiojIiIiIiIpRjEsLiomIiIlRjEpSSJXR0YoIiImRjFGMSomIiInRjEpRjdGNUYxISIiKiZGOEYxKUY3IiIkRjFGMSomIiM/RjEpRjciIiNGMUY8KiYiIz1GMUY3RjFGMUYyRjFGMUY3RjwsJkYxRjxGN0YxRjxGMTcjRi4=

Degrees of the A_i's: 6, 12, 6 algA0W:=numer(AA0/V^2-A0overV2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdBMFdHRigsMiooIiIlIiIiKUkiVkdGKCIiI0YxKUkiV0dGKCIiJkYxISIiKigiIidGMUYyRjEpRjZGMEYxRjEqKEY3RjFGMkYxKUY2IiIkRjFGOCooIiM/RjFGMkYxKUY2RjRGMUYxKigiIilGMUkkQUEwR0YoRjFGQUYxRjEqKCIjPUYxRjJGMUY2RjFGOCooRkNGMUZERjFGNkYxRjgqJkZDRjFGMkYxRjg3I0Yu

factor(subs(Vt=V^3,resultant(algA0W,algWVt,W)))

LCQqKCIjOyIiIilJIlZHNiIiIiVGJSw2KiQpRiciIzpGJUYlKiYiIihGJSlGJyIjN0YlISIiKiYiJGYiRiUpRiciIipGJUYlKigiI2tGJUkkQUEwR0YoRiUpRidGL0YlRiUqJiIlUzVGJSlGJyIiJ0YlRiUqKCIkRypGJUY5RiVGJkYlRiUqKCIkYyNGJSlGOSIiI0YlKUYnRkRGJUYlKiYiJStTRiUpRiciIiRGJUYlKigiJXdARiVGOUYlRidGJUYlIiUzWUYlRiVGJQ==

factor(resultant(op(3,%),algV,V));degree(%,AA0);

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

IiIn

algA1W:=numer(A1squareW-AA1^2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdBMVdHRigsOCokKUkiV0dGKCIiKiIiIiEiIiomRjJGMylGMSIiKUYzRjMqJiIjRkYzKUYxIiIoRjNGNComIiM6RjMpRjEiIidGM0YzKiYiI3NGMylGMSIiJkYzRjMqJiIkMyJGMylGMSIiJUYzRjQqKEZHRjMpSSRBQTFHRigiIiNGM0YxRjNGNComIiNbRjMpRjEiIiRGM0Y0KiZGR0YzRklGM0YzKiYiJFciRjMpRjFGS0YzRjMiI2tGNDcjRi4=

factor(subs(Vt=V^3,resultant(algA1W,algWVt,W)))

LEQqJClJIlZHNiIiI0YiIiJGKComIiNzRigpRiUiI0NGKEYoKiYiJS9CRigpRiUiI0BGKEYoKiYiJjNJJUYoKUYlIiM9RihGKCooIiVbP0YoKUkkQUExR0YmIiIjRigpRiUiIzpGKEYoKiYiJyc0OyZGKEY6RihGKCooIiUjPilGKEY3RigpRiUiIzdGKEYoKiYiKG8oR1RGKEZARihGKCooIiczaWdGKEY3RigpRiUiIipGKCEiIiomIiknND8/I0YoRkZGKEYoKigiKGcqM3NGKEY3RigpRiUiIidGKEZIKigiKHcmWzVGKClGOCIiJUYoKUYlIiIkRihGKComIilzdVx2RihGTUYoRigqKCIpW3M2Q0YoRjdGKEZTRihGSComIipXXCo0OkYoRlNGKEYoKiYiKDMnKVEpRihGN0YoRkgiKkd4QE0iRig=

factor(resultant(%,algV,V));degree(%,AA1);

LGZwKigiNDNleGFvLnNMQSoiIiIpSSRBQTFHNiIiIzdGJSlJInRHRigiI0ZGJUYlKigiNnc4J3lmOVUlKj4sQEYlKUYnIiM1RiVGKkYlISIiKigiN3MxSS04PSp6SkVZIkYlKUYnIiIpRiVGKkYlRiUqKCI1P0ZhUy1SXWJYPkYlRi9GJSlGKyIjQ0YlRjEqKCI3J1xFVz1DPThIJT4/RiUpRiciIidGJUYqRiVGMSooIjYrSyFHc1JXWV4kcCRGJUY0RiVGOEYlRiUqKCIyb1InKj1xekdnJEYlRi9GJSlGKyIjQEYlRiUqKCI3YzFSL1A6JykqZiZHcUYlKUYnIiIlRiVGKkYlRjEqKCI3c0VcaSdcZ0hLPyk+RiVGPEYlRjhGJUYxKigiNW9SN3YyPiE0MU0iRiVGNEYlRkJGJUYlKigiNy86cU44VyFcLUAmW0YlKUYnIiIjRiVGKkYlRjEqKCI4d2xcdG8sTWJKTDQiRiVGRkYlRjhGJUYlKigiNl9aQlsyeUUqPkE/RiVGPEYlRkJGJUYxKiYiN08mXCNwdTJwLlxCNUYlRipGJUYxKigiMiFbbzc9JT01LCVGJUY0RiUpRisiIz1GJUYxKigiN3dkXmQ8N0soR18yKkYlRk5GJUY4RiVGMSooIjYhMyVwJj4kcFFOeS4jRiVGRkYlRkJGJUYxKigiNCsvXig0NVYlZjMkRiVGPEYlRlhGJUYxKiYiNjdsVGMiZmpYajZNRiVGOEYlRiUqKCIvW0s4ZWx4X0YlRjRGJSlGKyIjOkYlRiUqKCI1Z0heQGwyYSllZCVGJUZORiVGQkYlRiUqKCI1X3Ypbyo0LGRIXmJGJUZGRiVGWEYlRiUqKCIxQz1lXGRZNU5GJUY8RiVGXm9GJUYlKiYiNUMtZUlhRk1GYV1GJUZCRiVGMSooIjUleV0sZUw1Y1IqZUYlRk5GJUZYRiVGJSooIjJXImUyeGQ1LHBGJUZGRiVGXm9GJUYlKigiL3MxSU04KVEiRiVGPEYlKUYrRilGJUYxKiYiNEdgQiZSOzEqeU8lRiVGWEYlRiUqKCI0W0d5ZyhvMj4oPSNGJUZORiVGXm9GJUYxKigiMGNRJipbaVkkUkYlRkZGJUZecEYlRiUqKCIsbyRRKGZWJEYlRjxGJSlGKyIiKkYlRiUqJiIzJ0hOKCp6KmVnRUNGJUZeb0YlRjEqKCIxQ0lmVWxkKjQjRiVGTkYlRl5wRiVGMSooIi9PMllwKmUhUkYlRkZGJUZncEYlRjEqJiIxWz8lPURIdSkqKUYlRl5wRiVGJSooIjBHZEV3OmVKJ0YlRk5GJUZncEYlRiUqKCIsKUcvSVBPRiVGRkYlKUYrRj1GJUYxKiYiMDspeUwkPSI+QUYlRmdwRiVGMSooIi1rW19Kd05GJUZORiVGZXFGJUYlKigiKDMnKVEpRiVGRkYlKUYrIiIkRiVGJSomIi5LTzA1Q18kRiVGZXFGJUYlKigiK0s3KjQjPkYlRk5GJUZcckYlRiUqJiIsL3ohXGhLRiVGXHJGJUYxKiZGW3JGJUZORiVGMSIqR3hATSJGJQ==

IiM3

algA2W:=numer(AA2-A2sol)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdBMldHRigsLiooIiIkIiIiSSJWR0YoRjEpSSJXR0YoRjBGMUYxKigiIiNGMUYyRjEpRjRGNkYxISIiKigiIiVGMUkkQUEyR0YoRjFGNEYxRjEqKEY6RjFGMkYxRjRGMUY4KiZGOkYxRjtGMUY4KiZGOkYxRjJGMUYxNyNGLg==

factor(subs(Vt=V^3,resultant(algA2W,algWVt,W)))

KiZJIlZHNiIiIiIsMiomIiIqRiUpRiMiIzVGJUYlKigiIycqRiVJJEFBMkdGJEYlKUYjIiInRiUhIiIqJiIjKSlGJSlGIyIiKEYlRiUqKCIkYyNGJSlGLSIiI0YlKUYjRjhGJUYlKigiJEciRiVGLUYlKUYjIiIkRiVGMComRjtGJSlGIyIiJUYlRjAqJkY2RiVGLUYlRjAqJkY2RiVGI0YlRiVGJQ==

factor(resultant(op(2,%),algV,V));degree(%,AA2);

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

IiIn

Ixform; Jyform;

LCoqKCwmKiRJIlZHNiIhIiIiIiIqJEkieEdGJ0YoRihGKSksJiomKUYmIiIjRilGK0YpRihGKUYpI0YpRjBGKUkkQUExR0YnRilGKComSSRBQTJHRidGKSwoKiQpRitGMEYoRikqJkkidEdGJ0YoRitGKEYoRitGKEYpRikqJClGNUYwRilGKUkkQUEwR0YnRik=

LCoqJCksKCokKUkieUc2IiIiIyEiIiIiIiomSSJ0R0YpRitGKEYrRitGKEYrRipGLEYsKiZJJEFBMkdGKUYsRiVGLEYsKihJJEFBMUdGKUYsLCYqJEkiVkdGKUYrRiwqJEYoRitGK0YsKSwmKiYpRjVGKkYsRihGLEYrRixGLCNGLEYqRixGLEkkQUEwR0YpRiw=

A0VW:=V^2*A0overV2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVBMFZXR0YoLCQqLCMiIiIiIilGMSlJIlZHRigiIiNGMSwuKiYiIiVGMSlJIldHRigiIiZGMUYxKiYiIidGMSlGOkY4RjEhIiIqJkY7RjEpRjoiIiRGMUYxKiYiIz9GMSlGOkY1RjFGPyomIiM9RjFGOkYxRjFGMkYxRjFGOkY/LCZGMUY/RjpGMUY/RjE3I0Yu

A2sol

LCQqKiMiIiIiIiVGJUkiVkc2IkYlLCoqJiIiJEYlKUkiV0dGKEYrRiVGJSomIiIjRiUpRi1GL0YlISIiKiZGJkYlRi1GJUYxRiZGJUYlLCZGJUYxRi1GJUYxRjE=

A1squareW;A1pap

LCQqKCksJkkiV0c2IiIiIkYoRigiIiRGKCksJkYmRigiIiMhIiIiIidGKCwmIiIlRi0qJkYwRihGJkYoRihGLUYt

LCQqLCMiIiIiIiNGJSksJkYmRiVJIldHNiIhIiIiIiRGJSwmRilGJUYlRiVGJSwmSSJaR0YqRiVGJUYlRiUsJkYlRiVGL0YrRitGJQ==

Expression of DD(0) D0VW:=factor(A2sol/2/t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVEMFZXR0YoLCQqLCMiIiIiIilGMUkiVkdGKEYxLCoqJiIiJEYxKUkiV0dGKEY2RjFGMSomIiIjRjEpRjhGOkYxISIiKiYiIiVGMUY4RjFGPEY+RjFGMSwmRjFGPEY4RjFGPEkidEdGKEY8Rjw3I0Yu

D0W:=factor(subs(V=Vt^(1/3),Vt=VtW,A2sol/2/tV));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSREMFdHRigsJCoqIyIiIiIiJUYxLCoqJiIiJEYxKUkiV0dGKEY1RjFGMSomIiIjRjEpRjdGOUYxISIiKiZGMkYxRjdGMUY7RjJGMUYxLChGOEYxKiZGOUYxRjdGMUY7RjFGO0YxLCZGMUY7RjdGMUY7RjE3I0Yu

We begin with C0=C00, which we find to have degree 6. In fact, this has already been computed by determining the degree of A2=2t D0 factor(resultant(numer(D0VW-DD0),algWVt,W))

LDIqKiIlQzUiIiIpSSRERDBHNiIiIiNGJUkjVnRHRihGJSlJInRHRihGKUYlRiUqLCIkIz5GJUYnRiVJIlZHRihGJSlGKkYpRiVGLEYlISIiKigiIipGJSlGL0YpRiUpRioiIiRGJUYlKiwiJGMjRiVGJ0YlRi9GJUYqRiVGLEYlRjEqKCIjKSlGJUY0RiVGMEYlRiUqKiIkNyZGJUYnRiVGL0YlRixGJUYxKigiJEciRiVGNEYlRipGJUYxKiZGOEYlRjRGJUYl

factor(resultant(subs(Vt=V^3,%),algV,V))

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

algD0:=collect(op(3,%),DD0,factor)

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

Now U(x,0). The degree is at most 24 Ix; Ixform;Deltapx;

KiQpLCwqKiIiIyIiIkkieEc2IkYnSSJ0R0YpRictSSJVR0YpNiRGKCIiIUYnRicqJClGKEYmISIiRicqJkYqRjFGKEYxRjFGKEYnKiZGKkYnLUkjRERHRik2I0YuRidGJ0YmRic=

LCoqKCwmKiRJIlZHNiIhIiIiIiIqJEkieEdGJ0YoRihGKSksJiomKUYmIiIjRilGK0YpRihGKUYpI0YpRjBGKUkkQUExR0YnRilGKComSSRBQTJHRidGKSwoKiQpRitGMEYoRikqJkkidEdGJ0YoRitGKEYoRitGKEYpRikqJClGNUYwRilGKUkkQUEwR0YnRik=

LCYqJilJIlZHNiIiIiMiIiJJInhHRiZGKCEiIkYoRig=

Elimination of the square root that involves x res1:=numer(factor(resultant(II-subs(sqrt(-V^2*x+1)=sDpx,Ixform),sDpx^2-Deltapx,sDpx)));

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

As expected, this involves A1^2 rather than A1 coeff(res1,AA1,1);degree(res1,AA1);

IiIh

IiIj

We replace the series A_1 by their expressions res2:=numer(factor(subs(AA1=sqrt(A1squareW),t=tV,AA2=A2sol,AA0=A0VW,res1)))/V^2: nops(res2)

IiR0Iw==

degree(res2,II);

IiIj

indets(res2)

PCZJI0lJRzYiSSJWR0YkSSJXR0YkSSJ4R0Yk

Elimination of W res3:=factor(resultant(res2,subs(Vt=V^3,algWVt),W)): nops(%);

IiIk

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

NyU8IjwjSSJWRzYiPCVJI0lJR0YmRiVJInhHRiY=

nops(op(3,res3));

IiR6JA==

degree(res3,II);

IiIl

res4:=op(3,res3):algIV:=res4: res5:=factor(resultant(res4,algV,V)): nops(res5);

IiVlRg==

Hence I(x) has degree 12 over Q(t,x) degree(res5,II);indets(res5);

IiM3

PCVJI0lJRzYiSSJ0R0YkSSJ4R0Yk

algIx:=res5: Ix;

KiQpLCwqKiIiIyIiIkkieEc2IkYnSSJ0R0YpRictSSJVR0YpNiRGKCIiIUYnRicqJClGKEYmISIiRicqJkYqRjFGKEYxRjFGKEYnKiZGKkYnLUkjRERHRik2I0YuRidGJ0YmRic=

Since D0 is in Q(W), we can get back to res2 to determine the degree of U(x,0) res6:=numer(factor(subs(II=(2*x*t*Ux0+1/x^2-1/(t*x)+x+t*D0)^2,D0=A2sol/2/t,t=tV,res2))): nops(%);

IiIk

op(2,res6); op(3,res6);

KiQpSSJWRzYiIiIjIiIi

KiQpSSJ4RzYiIiInIiIi

nops(op(1,res6));

IiRRKQ==

res7:=op(1,res6):algUx0VW:=res7: indets(res7);

PCZJJFV4MEc2IkkiVkdGJEkiV0dGJEkieEdGJA==

Elimination of W res8:=factor(resultant(res7,subs(Vt=V^3,algWVt),W)): nops(%);

IiIj

op(1,res8);

KiQpSSJWRzYiIiM4IiIi

nops(op(2,res8));

IiRuKQ==

res9:=op(2,res8): indets(res9); degree(res9,Ux0);

PCVJJFV4MEc2IkkiVkdGJEkieEdGJA==

IiIp

res10:=factor(resultant(res9,algV,V)): nops(%);

IiUkMyU=

algUx0:=collect(res10,Ux0,factor): degree(%,Ux0);

IiND

Let us check n:=10: normal(series(subs(Ux0=subs(y=0,User(n)),algUx0),t,n+1))

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Now the degree of D(x). We first note that J(x) satisfies the same equation as I(x) over Q(x,t) indets(algIx)

PCVJI0lJRzYiSSJ0R0YkSSJ4R0Yk

Jy

KiYsJiokKSwmKiZJInlHNiIhIiJJInRHRikiIiJGKkYsRiwiIiNGLEYsKigiIiVGLEYoRiwpRitGLUYsRipGLCksJiomRihGLC1JI0RER0YpNiNGKEYsRiwqJkYrRipGKEYqRixGLUYs

We now consider the series D(x) res:=numer(factor(subs(II=JJ,JJ=subs(y=x,Delta)*(DD*x+1/t/x)^2,algIx))): nops(res)

IiIm

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

Nyc8IjwjSSJ0RzYiPCNJInhHRiY8JEYlRig8JUkjRERHRiZGJUYo

nops(op(5,res))

IiVgXA==

algDx:=collect(op(5,res),DD,factor): Let us check n:=10: factor(series(subs(DD=subs(y=x,Dser(n)),algDx),t,n))

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

A1W/A1pap

KigpKiYsJkkiV0c2IiIiIkYoRihGKCwmRihGKEYmISIiRiojRigiIiNGKCwmSSJaR0YnRihGKEYoRiosJkYoRihGLkYqRig=

Degree of C(x,y) eqC

LCwqJiwmIiIiRiUqJkkidEc2IkYlLChJInhHRihGJUkieUdGKEYlKiZGKiEiIkYrRi1GJUYlRi1GJS1JIkNHRig2JEYqRitGJUYtRiVGJSoqRidGJUYqRi1GK0YtLUkjQ21HRig2IyokRipGLUYlRi0qKkYnRiVGKkYtRitGLS1GMzYjKiRGK0YtRiVGLSoqRidGJUYqRi1GK0YtSSRDMDBHRihGJUYt

SUM:=subs(x=2,y=3,DD0=A2sol/2/tV,x*Ux0+y*Uy0+DD0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRTVU1HRigsKCoqIyIiIiIiKUYxLCoqJiIiJEYxKUkiV0dGKEY1RjFGMSomIiIjRjEpRjdGOUYxISIiKiYiIiVGMUY3RjFGO0Y9RjFGMSwmRjFGO0Y3RjFGOywmKiQpSSJWR0YoRjVGMUYxRjlGMUYxRjsqJkY5RjFJJFV4MEdGKEYxRjEqJkY1RjFJJFV5MEdGKEYxRjE3I0Yu

Let us first simplify the equation obtained for U(x,0), by lowering its degree in W degree(algUx0VW,V); degree(algUx0VW,W);nops(algUx0VW)

IiIh

IiIh

IiIi

res0:=factor(rem(algUx0VW,subs(Vt=V^3,algWVt),W)): nops(%)

IiIk

map(indets,[seq(op(i,res0),i=1..nops(res0))])

NyU8IjwjSSJWRzYiPCZJJFV4MEdGJkYlSSJXR0YmSSJ4R0Ym

nops(op(3,res0))

IiRtIw==

algUx0VW:=collect(op(3,res0),Ux0,factor):degree(%,W)

IiIi

Elimination of U(2,0) res1:=factor(resultant(subs(x=2,algUx0VW),SUM-sum,Ux0)): nops(res1)

IiIk

map(indets,[seq(op(i,res1),i=1..nops(res1))])

NyU8IjwmSSRVeTBHNiJJIlZHRiZJIldHRiZJJHN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjwjRig=

res2:=op(2,res1): nops(%)

IiVpOg==

Elimination of U(3,0) res3:=factor(resultant(subs(x=3,Ux0=Uy0,algUx0VW),res2,Uy0)): nops(res3)

IiIl

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

NyY8IjwjSSJWRzYiPCRGJUkiV0dGJjwlRiVGKEkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0Ym

rem(op(3,res3),subs(Vt=V^3,algWVt),W)

LCgqJClJIlZHNiIiIiQiIiJGKComIiIlRihJIldHRiZGKEYoRiohIiI=

nops(op(4,res3))

IiZGMSY=

res4:=op(4,res3): degree(res4,W)

IiNi

res5:=factor(rem(res4,subs(Vt=V^3,algWVt),W)): nops(res5)

IiIj

op(1,res5)

SSJWRzYi

nops(op(2,res5))

IiU3TQ==

res6:=op(2,res5): Elimination of W res7:=factor(resultant(res6, subs(Vt=V^3,algWVt),W)): nops(%)

IiIk

map(indets,[seq(op(i,res7),i=1..nops(res7))])

NyU8IjwjSSJWRzYiPCRGJUkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0Ym

nops(op(3,res7))

IiVKVA==

algsumV:=op(3,res7): degree(%,sum)

IiNL

Elimination of V res8:=factor(resultant(algsumV,algV,V)): nops(%)

IiUhKno=

degree 96 algsum:=res8: degree(%,sum)

IiMnKg==

SUM;

LCgqKiMiIiIiIilGJSwqKiYiIiRGJSlJIldHNiJGKUYlRiUqJiIiI0YlKUYrRi5GJSEiIiomIiIlRiVGK0YlRjBGMkYlRiUsJkYlRjBGK0YlRjAsJiokKUkiVkdGLEYpRiVGJUYuRiVGJUYwKiZGLkYlSSRVeDBHRixGJUYlKiZGKUYlSSRVeTBHRixGJUYl

Let us check the equation n:=20: series(subs(sum=SUM,Ux0=subs(x=2,y=0,User(n)), Uy0=subs(x=3,y=0,User(n)), W=Wser(n), Vt=V^3, V=Vser(n), algsum),t,n+1)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0A=

Jy; Jyform;

KiYsJiokKSwmKiZJInlHNiIhIiJJInRHRikiIiJGKkYsRiwiIiNGLEYsKigiIiVGLEYoRiwpRitGLUYsRipGLCksJiomRihGLC1JI0RER0YpNiNGKEYsRiwqJkYrRipGKEYqRixGLUYs

LCoqJCksKCokKUkieUc2IiIiIyEiIiIiIiomSSJ0R0YpRitGKEYrRitGKEYrRipGLEYsKiZJJEFBMkdGKUYsRiVGLEYsKihJJEFBMUdGKUYsLCYqJEkiVkdGKUYrRiwqJEYoRitGK0YsKSwmKiYpRjVGKkYsRihGLEYrRixGLCNGLEYqRixGLEkkQUEwR0YpRiw=

series(Jy,y,2);

KytJInlHNiIiIiIhIiUsJComIiIjRiVJInRHRiQhIiJGKyEiJCwmKihGKUYlRipGJS1JI0RER0YkNiMiIiFGJUYlKiQpRipGKUYrRiUhIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJUYr

Periodicities: the length is i+j mod 3 map(expand,Cser(4));

Ky1JInRHNiIiIiIiIiEsJkkieEdGJEYlSSJ5R0YkRiVGJSwsKiQpRigiIiNGJUYlKihGLUYlRihGJUYpRiVGJSokKUYpRi1GJUYlKiRGKSEiIkYlKiRGKEYyRiVGLSwwKiQpRigiIiRGJUYlKihGN0YlRixGJUYpRiVGJSooRjdGJUYoRiVGMEYlRiUqJClGKUY3RiVGJSooRi1GJUYoRiVGKUYyRiUiIiVGJSooRi1GJUYoRjJGKUYlRiVGNyw4KiQpRihGPUYlRiUqKEY9RiVGNkYlRilGJUYlKigiIidGJUYsRiVGMEYlRiUqKEY9RiVGKEYlRjtGJUYlKiQpRilGPUYlRiUqKEY3RiVGLEYlRilGMkYlKiYiIipGJUYoRiVGJSomRkpGJUYpRiVGJSooRjdGJUYoRjJGMEYlRiUqJkYtRiVGMEYyRiUqJkYtRiVGLEYyRiVGPQ==

Taking periodicities into account, we introduce a modified version of J(y), namely V^4 J(x/V^2)/Delta(x/V^2). Its square root is xD(x/V^2) +V^4/(tx) Jymod:=factor(expand(subs(y=y/V^2,V^4*Jyform/Delta)));

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

series(%,y,1);

KydJInlHNiIqJilJIlZHRiQiIikiIiIpSSJ0R0YkIiIjISIiISIjLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRilGLQ==

In terms of W JymodW:=map(factor,subs(delta=1-y,collect(subs(sqrt(-y+1)=sqrt(delta),factor(subs(AA0=A0VW,AA2=A2sol,V=Vt^(1/3),Vt=VtW,factor(subs(t=tV,Jymod))))),delta,factor)));

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

This is a series in y with rational coefficients in Z (which is hidden in A1) factor(series(JymodW,y,1));

KydJInlHNiIsJCoqIiNrIiIiKSwmRighIiJJIldHRiRGKCIiI0YoKUYsRi1GKCksKComRi1GKEYuRihGKComRi1GKEYsRihGK0YoRitGLUYoRighIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYr

factor(subs(V=Vt^(1/3),Vt=VtW,V^4/tV))

LCQqKiIiKSIiIkkiV0c2IkYlLCZGJSEiIkYmRiVGJSwoKiYiIiNGJSlGJkYsRiVGJSomRixGJUYmRiVGKUYlRilGJUYl

Mind the sign when taking the square root. serrooty:=subs(csgn((2*W^2-2*W-1)*W*(W-1)/y)=1,factor(simplify(series(sqrt(JymodW),y,3),symbolic)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlzZXJyb290eUdGKCsrSSJ5R0YoLCQqKiIiKSIiIkkiV0dGKEYzLCZGMyEiIkY0RjNGMywoKiYiIiNGMylGNEY5RjNGMyomRjlGM0Y0RjNGNkYzRjZGM0YzRjYsJCoqI0YzIiIlRjMsKiomIiIkRjMpRjRGQkYzRjNGOEY2KiZGP0YzRjRGM0Y2Rj9GM0YzRjdGM0Y1RjZGM0YzLCQqLCNGM0YyRjNGN0YzLDAqJiIiJ0YzKUY0IiImRjNGNiomIiM1RjMpRjRGP0YzRjMqJiIiKEYzRkNGM0YzKihGOUYzSSRBQTFHRihGM0Y0RjNGMyomIiM9RjNGOkYzRjYqJkY5RjNGU0YzRjZGMkYzRjNGNEY2KUY1RjlGNkY2RjktSSJPR0YlNiNGM0ZCNyNGLg==

Introduce D(y) subs(DD(y)=add(DD[i]*y^i,i=0..6), y=y/V^2,t=tV,V^4*Jy/Delta);

KiYpSSJWRzYiIiIlIiIiKSwmKihJInlHRiVGJylGJCIiIyEiIiwwKigmSSNEREdGJTYjIiInRicpRitGNEYnKUYkIiM3Ri5GJyooJkYyNiMiIiZGJylGK0Y7RicpRiQiIzVGLkYnKigmRjI2I0YmRicpRitGJkYnKUYkIiIpRi5GJyooJkYyNiMiIiRGJylGK0ZIRicpRiRGNEYuRicqKCZGMjYjRi1GJylGK0YtRidGI0YuRicqKCZGMjYjRidGJ0YrRidGLEYuRicmRjI2IyIiIUYnRidGJyooRiRGJywmKiQpRiRGSEYnRidGLUYnRidGK0YuRidGLUYn

Again, mind the sign factor(series(simplify(sqrt(%),symbolic),y,4));

KytJInlHNiIqJilJIlZHRiQiIiQiIiIsJiokRiZGKUYpIiIjRilGKSEiIiZJI0RER0YkNiMiIiFGKSomKUYnRixGLSZGLzYjRilGKUYsLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJEY1Rig=

serdiff:=series(%-serrooty,y,3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShzZXJkaWZmR0YoKytJInlHRigsJiomKUkiVkdGKCIiJCIiIiwmKiRGMkY1RjUiIiNGNUY1RjUqKiIiKUY1SSJXR0YoRjUsJkY1ISIiRjtGNUY1LCgqJkY4RjUpRjtGOEY1RjUqJkY4RjVGO0Y1Rj1GNUY9RjVGPUY9LCYmSSNEREdGKDYjIiIhRjUqKiNGNSIiJUY1LCoqJkY0RjUpRjtGNEY1RjVGP0Y9KiZGSUY1RjtGNUY9RklGNUY1Rj5GNUY8Rj1GPUY1LCYqJilGM0Y4Rj0mRkQ2I0Y1RjVGNSosI0Y1RjpGNUY+RjUsMComIiInRjUpRjsiIiZGNUY9KiYiIzVGNSlGO0ZJRjVGNSomIiIoRjVGTEY1RjUqKEY4RjVJJEFBMUdGKEY1RjtGNUY1KiYiIz1GNUZARjVGPSomRjhGNUZqbkY1Rj1GOkY1RjVGO0Y9KUY8RjhGPUY1RjgtSSJPR0YlRlJGNDcjRi4=

Same expression as with Kreweras steps, of course D0W:=factor(solve(coeff(serdiff,y,1),DD[0]));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSREMFdHRigsJCoqIyIiIiIiJUYxLCoqJiIiJEYxKUkiV0dGKEY1RjFGMSomIiIjRjEpRjdGOUYxISIiKiZGMkYxRjdGMUY7RjJGMUYxLChGOEYxKiZGOUYxRjdGMUY7RjFGO0YxLCZGMUY7RjdGMUY7RjE3I0Yu

Let us check n:=10: series(subs(W=Wser(n),Vt=V^3,V=Vser(n),D0W),t,n);

Ky1JInRHNiIiIiIiIiEiIiUiIiQiI1kiIiciJDEoIiIqLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIzc=

subs(y=0,Dser(n));

LCoqJiIkMSgiIiIpSSJ0RzYiIiIqRiVGJSomIiNZRiUpRiciIidGJUYlKiYiIiVGJSlGJyIiJEYlRiVGJUYl

D1WA1:=collect(factor(subs(A1=A1V,solve(coeff(serdiff,y,2),DD[1]))),AA1,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZEMVdBMUdGKCwmKi4jIiIiIiIlRjEpSSJWR0YoIiIjRjEsKComRjVGMSlJIldHRihGNUYxRjEqJkY1RjFGOUYxISIiRjFGO0YxRjlGOywmRjFGO0Y5RjFGO0kkQUExR0YoRjFGOyouI0YxIiIpRjFGNkYxLCwqJiIiJ0YxKUY5IiImRjFGMSomIiM1RjEpRjlGMkYxRjsqJiIiKEYxKUY5IiIkRjFGOyomIiM9RjFGOEYxRjFGQEY7RjFGM0YxRjlGOylGPEY1RjtGMTcjRi4=

A1W

LCQqKiMiIiIiIiNGJSwmSSJXRzYiRiVGJUYlRiUpLCZGJkYlRighIiIiIiRGJSkqJkYnRiUsJkYlRiVGKEYsRixGJEYlRiU=

D1Z:=factor(subs(AA1=A1Z,W=WZ,D1WA1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSREMVpHRigsJCouIyIiIiIiI0YxKUkiVkdGKEYyRjEsLCokKUkiWkdGKCIiJUYxRjEqJkY5RjEpRjgiIiRGMUYxKiYiIidGMSlGOEYyRjEhIiIqJkY5RjFGOEYxRjFGMUYxRjEsNComRjJGMSlGOCIiKkYxRjEqJClGOCIiKUYxRkAqJkY5RjEpRjgiIihGMUZAKiYiIzVGMSlGOEY+RjFGMSomRk1GMUY3RjFGQComRj5GMUY7RjFGMSomRjlGMUY/RjFGMUZBRkBGMUYxRjEpLCZGOEYxRjFGQEY5RkApLCYqJEY/RjFGMUYxRjFGOUZARjE3I0Yu

Dividing by t^2 to get a series in t^3 D1Zovert2:=factor(subs(V=Vt^(1/3),Vt=VtW,W=WZ,factor(D1Z/tV^2)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpEMVpvdmVydDJHRigsJCosIiIjIiIiKSwsKiQpSSJaR0YoIiIlRjFGMSomRjdGMSlGNiIiJEYxRjEqJiIiJ0YxKUY2RjBGMSEiIiomRjdGMUY2RjFGMUYxRjFGOkYxLDQqJkYwRjEpRjYiIipGMUYxKiQpRjYiIilGMUY+KiZGN0YxKUY2IiIoRjFGPiomIiM1RjEpRjZGPEYxRjEqJkZLRjFGNUYxRj4qJkY8RjFGOUYxRjEqJkY3RjFGPUYxRjFGP0Y+RjFGMUYxKSwmRjZGMUYxRj5GN0Y+KSwmKiRGPUYxRjFGMUYxRkZGPkYxNyNGLg==

There is a factor t^3 in this expression algZW

LCgqJkkiV0c2IiIiIilJIlpHRiUiIiNGJkYmRiRGJiomRilGJkYoRiYhIiI=

algZ:=collect(factor(resultant(subs(Vt=V^3,factor(resultant(algZW,algWVt,W))),algV,V)),t,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdaR0YoLCYqKCIiKSIiIiksLCokKUkiWkdGKCIiJUYxRjEqJkY3RjEpRjYiIiRGMUYxKiYiIidGMSlGNiIiI0YxISIiKiZGN0YxRjZGMUYxRjFGMUY6RjEpSSJ0R0YoRjpGMUYxKipGMEYxRjZGMSksJkY2RjFGMUY/Rj5GMSksJiokRj1GMUYxRjFGMUY3RjFGPzcjRi4=

t3V:=solve(%,t^3);

Warning, solving for expressions other than names or functions is not recommended.

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR0M1ZHRigqKkkiWkdGKCIiIiksJkYvRjBGMCEiIiIiI0YwKSwmKiQpRi9GNEYwRjBGMEYwIiIlRjApLCwqJClGL0Y5RjBGMComRjlGMClGLyIiJEYwRjAqJiIiJ0YwRjhGMEYzKiZGOUYwRi9GMEYwRjBGMEZARjM3I0Yu

Zser:=proc(n) op(3,algeqtoseries(algZ,t,Z,n,true)): end: Zser(10);

Ky1JInRHNiIiIiIiIiQiIzkiIiciJFwjIiIqIiVZXCIjNy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiM4

This is lighter D1timest:=factor(subs(V=Vt^(1/3),Vt=VtW,W=WZ,factor(D1Z*tV)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlEMXRpbWVzdEdGKCwkKiwiIiMiIiIsNComRjBGMSlJIlpHRigiIipGMUYxKiQpRjUiIilGMSEiIiomIiIlRjEpRjUiIihGMUY6KiYiIzVGMSlGNSIiJ0YxRjEqJkZARjEpRjVGPEYxRjoqJkZCRjEpRjUiIiRGMUYxKiZGPEYxKUY1RjBGMUYxKiZGPEYxRjVGMUY6RjFGMUYxRjVGMSksJkY1RjFGMUY6RjBGOiksJiokRklGMUYxRjFGMUY8RjpGMTcjRi4=

Let us check n:=16: series(subs(Z=Zser(n), D1timest),t,n);

Ky9JInRHNiIiIiMiIiQiI0MiIiciJHckIiIqIiVDbiIjNyInWy04IiM6LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiM9

expand(coeff(Dser(n),y,1)*t);

LCwqJiInWy04IiIiKUkidEc2IiIjOkYlRiUqJiIlQ25GJSlGJyIjN0YlRiUqJiIkdyRGJSlGJyIiKkYlRiUqJiIjQ0YlKUYnIiInRiVGJSomIiIjRiUpRiciIiRGJUYl

Ix; Ixform;

KiQpLCwqKiIiIyIiIkkieEc2IkYnSSJ0R0YpRictSSJVR0YpNiRGKCIiIUYnRicqJClGKEYmISIiRicqJkYqRjFGKEYxRjFGKEYnKiZGKkYnLUkjRERHRik2I0YuRidGJ0YmRic=

LCoqKCwmKiRJIlZHNiIhIiIiIiIqJEkieEdGJ0YoRihGKSksJiomKUYmIiIjRilGK0YpRihGKUYpI0YpRjBGKUkkQUExR0YnRilGKComSSRBQTJHRidGKSwoKiQpRitGMEYoRikqJkkidEdGJ0YoRitGKEYoRitGKEYpRikqJClGNUYwRilGKUkkQUEwR0YnRik=

Taking periodicities into account, we introduce a variant of I(x), which is a series in V^3. Ixmod:=subs(x=x/V^2,1/V^2*Ixform);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZJeG1vZEdGKComKUkiVkdGKCIiIyEiIiwqKigsJiokRjBGMiIiIiomSSJ4R0YoRjJGL0Y3RjJGNyksJkY5RjJGN0Y3I0Y3RjFGN0kkQUExR0YoRjdGMiomSSRBQTJHRihGNywoKiYpRjlGMUYyKUYwIiIlRjdGNyooSSJ0R0YoRjJGOUYyRi9GN0YyKiZGOUY3Ri9GMkYyRjdGNyokKUZARjFGN0Y3SSRBQTBHRihGN0Y3NyNGLg==

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

Ky9JInhHNiIqJClJIlZHRiQiIiciIiIhIiUsJCooIiIjRikpRiciIiVGKUkidEdGJCEiIkYxISIkKigpRidGLUYpLCYqJkkkQUEyR0YkRikpRjBGLUYpRilGKUYpRilGOEYxISIjKiYsKComSSRBQTFHRiRGKUYwRilGKUY3RjEqJkYtRilGMEYpRjFGKUYwRjFGMSwkKiojRilGLUYpLCoqKEY9RikpRiciIiRGKUYwRilGMSoqRi1GKUkkQUEwR0YkRilGJ0YpRjBGKUYpKihGLUYpRj1GKUYwRilGMSomRi9GKUYnRilGKUYpRkRGMUYwRjFGKSIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YpRik=

IxmodW:=subs(delta=1-x,collect(subs(sqrt(-x+1)=sqrt(delta),factor(subs(AA0=A0VW,AA2=A2sol,V=Vt^(1/3),Vt=VtW,factor(subs(t=tV,Ixmod))))),delta,factor));

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

This is now a series in y with rational coefficients in Z (which is hidden in A1) factor(series(IxmodW,x,1));

KydJInhHNiIsJCooIiM7IiIiKUkiV0dGJCIiI0YoKSwmRighIiJGKkYoRitGKEYoISIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRighIiQ=

Mind the sign when taking the square root. We want the constant term to be 8 not -8. serrootx:=subs(csgn(W*(W-1)/x^2)=-1,factor(simplify(series(sqrt(IxmodW),x,3),symbolic)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlzZXJyb290eEdGKCstSSJ4R0YoLCQqKCIiJSIiIkkiV0dGKEYzLCZGMyEiIkY0RjNGM0Y2ISIjLCgqJkYyRjMpRjQiIiNGM0YzKiZGMkYzRjRGM0Y2RjtGNkY2LCQqKCNGMyIiKUYzLCoqJiIiJEYzKUY0RkNGM0YzKiZGO0YzRjpGM0Y2RjxGNkYyRjNGM0Y1RjZGNiIiISwkKipGP0YzLCZJJEFBMUdGKEYzRjtGNkYzRjRGNkY1RjZGNkYzLUkiT0dGJTYjRjNGOzcjRi4=

subs(U(x,0)=add(UU[i]*x^i,i=0..8),x=x/V^2,t=tV,1/V^2*Ix);

KiYpSSJWRzYiIiIjISIiKSwsKixGJiIiIkkieEdGJUYrRiRGJywmKiQpRiQiIiRGK0YrRiZGK0YnLDQqKCZJI1VVR0YlNiMiIilGKylGLEY2RispRiQiIztGJ0YrKigmRjQ2IyIiKEYrKUYsRj1GKylGJCIjOUYnRisqKCZGNDYjIiInRispRixGREYrKUYkIiM3RidGKyooJkY0NiMiIiZGKylGLEZLRispRiQiIzVGJ0YrKigmRjQ2IyIiJUYrKUYsRlJGKylGJEY2RidGKyooJkY0NiNGMEYrKUYsRjBGKylGJEZERidGKyooJkY0NiNGJkYrKUYsRiZGKylGJEZSRidGKyooJkY0NiNGK0YrRixGK0YjRidGKyZGNDYjIiIhRitGK0YrKiZGZ25GJ0ZobkYrRisqKEYkRitGLUYrRixGJ0YnKiZGLEYrRiNGJ0YrKihGJEYrRi1GJy1JI0RER0YlRl1vRitGK0YmRis=

mind the sign simplify(sqrt(%),symbolic);

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

factor(subs(DD(0)=D0W,series(%,x,5)));

Ky9JInhHNiIqJClJIlZHRiQiIiQiIiIhIiMsJkYlISIiIiIjRixGLCwkKiwjRikiIiVGKSwqKiZGKEYpKUkiV0dGJEYoRilGKSomRi1GKSlGNUYtRilGLComRjFGKUY1RilGLEYxRilGKSwoRjZGKSomRi1GKUY1RilGLEYpRixGKSwmRilGLEY1RilGLCwmRiVGKUYtRilGLEYpIiIhKihGJkYsLChGJUYpKihGLUYpRidGKSZJI1VVR0YkNiNGPUYpRilGLUYpRilGPEYsRiksJCoqRi1GKSlGJ0YxRiwmRkI2I0YpRilGPEYsRilGLS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiRGSEYo

serrootx;

Ky1JInhHNiIsJCooIiIlIiIiSSJXR0YkRigsJkYoISIiRilGKEYoRishIiMsKComRidGKClGKSIiI0YoRigqJkYnRihGKUYoRitGMEYrRissJCooI0YoIiIpRigsKiomIiIkRigpRilGOEYoRigqJkYwRihGL0YoRitGMUYrRidGKEYoRipGK0YrIiIhLCQqKkY0RigsJkkkQUExR0YkRihGMEYrRihGKUYrRipGK0YrRigtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYw

series(%%-%,x,3);

Ky1JInhHNiIsJiokKUkiVkdGJCIiJCIiIkYqKigiIiVGKkkiV0dGJEYqLCZGKiEiIkYtRipGKkYqISIjLChGJkYvKiZGLEYqKUYtIiIjRipGLyomRixGKkYtRipGKkYvLCYqLCNGKkYsRiosKiomRilGKilGLUYpRipGKiomRjRGKkYzRipGL0Y1Ri9GLEYqRiosKEY8RioqJkY0RipGLUYqRi9GKkYvRipGLkYvLCZGJkYqRjRGKkYvRioqKCNGKiIiKUYqRjlGKkYuRi9GKiIiISwmKihGJ0YvLChGJkYqKihGNEYqRihGKiZJI1VVR0YkNiNGQ0YqRipGNEYqRipGP0YvRioqKkZBRiosJkkkQUExR0YkRipGNEYvRipGLUYvRi5GL0YqRiotSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKkY0

diffe:=convert(%,polynom): We check that the first three coefficients are trivial rem(coeff(diffe,x,-2),subs(Vt=V^3,algWVt),W); rem(coeff(diffe,x,-1),subs(Vt=V^3,algWVt),W); rem(numer(coeff(diffe,x,0)),subs(Vt=V^3,algWVt),W);

IiIh

IiIh

IiIh

(solve(coeff(diffe,x,1),UU[0]));;

LCQqLiMiIiIiIztGJSwqKiZJJEFBMUc2IkYlKUkiVkdGKiIiJEYlRiUqJiIiI0YlRitGJSEiIiomIiIpRiUpSSJXR0YqRi9GJUYlKiZGMkYlRjRGJUYwRiVGLEYwRjRGMCwmRiVGMEY0RiVGMCwmKiRGK0YlRiVGL0YlRiVGMA==

U0timestA1:=collect(%*tV,V,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SStVMHRpbWVzdEExR0YoLCYjIiIiIiIjISIiKiwjRjAiIztGMCwmSSRBQTFHRihGMEYxRjJGMEkiV0dGKEYyLCZGMEYyRjhGMEYyKUkiVkdGKCIiJEYwRjI3I0Yu

algWVt;

LCZJI1Z0RzYiIiIiKigiIiVGJUkiV0dGJEYlLCZGJUYlRighIiJGJUYq

U0timestA1:=map(factor,subs(V^3=4*W*(1-W),U0timestA1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SStVMHRpbWVzdEExR0YoLCYiIiIhIiIqJiNGLyIiJUYvSSRBQTFHRihGL0YvNyNGLg==

U0timestW:=subs(AA1=A1W,U0timestA1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpVMHRpbWVzdFdHRigsJiIiIiEiIioqI0YvIiIpRi8sJkkiV0dGKEYvRi9GL0YvKSwmIiIjRi9GNUYwIiIkRi8pKiZGNEYvLCZGL0YvRjVGMEYwI0YvRjhGL0YvNyNGLg==

This is C(-1,0)*t. Let us check n:=16: series(subs(W=Wser(n),V=Vser(n),Vt=Vser(n)^3,U0timestW),t,n);

Ky9JInRHNiIiIiIiIiQiIzYiIiciJGwiIiIqIiV1RyIjNyImI3BhIiM6LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIz0=

expand(coeff(Cmser(n)/t,x,-1));

LCwqJiImI3BhIiIiKUkidEc2IiIjOEYlRiUqJiIldUdGJSlGJyIjNUYlRiUqJiIkbCJGJSlGJyIiKEYlRiUqJiIjNkYlKUYnIiIlRiVGJUYnRiU=

The invariant I(x) is closely related to C(1,1) subs(x=1,y=1,eqC);isolate(%,C00);

LCoqJiwmIiIiRiUqJiIiJEYlSSJ0RzYiRiUhIiJGJS1JIkNHRik2JEYlRiVGJUYqRiVGJSooIiIjRiVGKEYlLUkjQ21HRik2I0YlRiVGKiomRihGJUkkQzAwR0YpRiVGKg==

L0kkQzAwRzYiLCQqJiwoKiYsJiIiIkYqKiYiIiRGKkkidEdGJEYqISIiRiotSSJDR0YkNiRGKkYqRipGKkYqRi4qKCIiI0YqRi1GKi1JI0NtR0YkNiNGKkYqRipGKkYtRi5GLg==

The invariant I(1) can be written in terms of C(1,1) IC11:=subs(C(1,1)=C11,factor(subs(x=1,U(1,0)=Cm(1),DD(0)=C00,%,Ix)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVJQzExR0YoKigpLCYiIiIhIiIqJiIiJEYxSSJ0R0YoRjFGMSIiI0YxKSwmKiZJJEMxMUdGKEYxRjVGMUYxRjFGMUY2RjEpRjVGNkYyNyNGLg==

We have obtained an equation for II=I(x) indets(algIx); degree(algIx,II)

PCVJI0lJRzYiSSJ0R0YkSSJ4R0Yk

IiM3

collect(factor(subs(x=1,algIx)),II,factor): degree(%,II);

IiM3

res:=numer(factor(subs(II=IC11,%%))): nops(res);

IiIl

map(indets,[seq(op(i,res),i=1..4)]);

NyY8IjwjSSJ0RzYiRiQ8JEkkQzExR0YmRiU=

nops(op(4,res));

IiQoXA==

Degree 24 algC11:=collect(op(4,res),C11,factor): degree(%,C11);

IiND

Iof1expr:=subs(x=1,AA0=A0VW,AA2=A2sol,AA1=A1W,Ixform);

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

simplifications? expr:=subsop(1=0,Iof1expr)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVleHByR0YoLCgqLCMiIiIiIiVGMUkiVkdGKEYxLCoqJiIiJEYxKUkiV0dGKEY2RjFGMSomIiIjRjEpRjhGOkYxISIiKiZGMkYxRjhGMUY8RjJGMUYxLCZGMUY8RjhGMUY8SSJ0R0YoRjxGMSokKUY/RjpGPEYxKiwjRjEiIilGMSlGM0Y6RjEsLiomRjJGMSlGOCIiJkYxRjEqJiIiJ0YxKUY4RjJGMUY8KiZGSUYxRjdGMUYxKiYiIz9GMUY7RjFGPComIiM9RjFGOEYxRjFGREYxRjFGOEY8Rj5GPEYxNyNGLg==

expr1:=factor(subs(t=tV,expr))

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

WVt:=op(2,[solve(algWVt,W)])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRXVnRHRigsJiMiIiIiIiNGMComRi9GMCksJkYwRjBJI1Z0R0YoISIiRi9GMEY2NyNGLg==

expr2:=simplify(factor(expand(rationalize(factor(subs(W=WVt,Vt=V^3,expr1)))))) assuming V>0 assuming V<1

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZleHByMkdGKCwkKiwjIiIiIiM7RjEsJkkiVkdGKEYxRjEhIiJGMSksJkY0RjEiIiNGMUY4RjEsJiooKSwmRjFGMUY0RjUjRjFGOEYxKSwoKiQpRjRGOEYxRjFGNEYxRjFGMSMiIiRGOEYxLCoqJClGNEZDRjFGMSomIiIlRjFGQUYxRjUqJkZIRjFGNEYxRjFGSEY1RjFGMSooRkNGMSlGM0Y4RjEsKiokKUY0RkhGMUYxKiYjRjFGQ0YxRkZGMUY1KiYjIiIpRkNGMUY0RjFGMSNGSEZDRjVGMUY1RjFGRkY1RjE3I0Yu

subs(sqrt(1 - V)= sqrt(1-Vt)/(V^2 + V + 1)^(1/2),%)

LCQqLCMiIiIiIztGJSwmSSJWRzYiRiVGJSEiIkYlKSwmRihGJSIiI0YlRi1GJSwmKigpLCZGJUYlSSNWdEdGKUYqI0YlRi1GJSwoKiQpRihGLUYlRiVGKEYlRiVGJUYlLCoqJClGKCIiJEYlRiUqJiIiJUYlRjZGJUYqKiZGPEYlRihGJUYlRjxGKkYlRiUqKEY6RiUpRidGLUYlLCoqJClGKEY8RiVGJSomI0YlRjpGJUY5RiVGKiomIyIiKUY6RiVGKEYlRiUjRjxGOkYqRiVGKkYlRjlGKkYl

We recycle the asymptotic results used in the analysis of Kreweras' case

algV:=-V+t*(V^3+2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdWR0YoLCZJIlZHRighIiIqJkkidEdGKCIiIiwmKiQpRi8iIiRGM0YzIiIjRjNGM0YzNyNGLg==

Here, tt stands for t three (t cubed). This is the equation for V^3=Vt algVt:=Vt-tt*(2+Vt)^3;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdWdEdGKCwmSSNWdEdGKCIiIiomSSN0dEdGKEYwKSwmIiIjRjBGL0YwIiIkRjAhIiI3I0Yu

The only singularity is at tt=1/27 factor(discrim(algVt,Vt));

LCQqKCIiJSIiIkkjdHRHNiJGJSwmKiYiI0ZGJUYmRiVGJUYlISIiRiVGKw==

with(gfun):algeqtoseries(subs(tt=1/27*(1-epsilon),algVt),epsilon,Vt,5);;

NyQrL0koZXBzaWxvbkc2IiEiKSIiISNGJiIiJCIiIiMhJGciIiMiKSIiIyMhJSVlJCIlKD0jRikjISZnIkciJiRvPiIiJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YqIiImLC5GKkYqKiYtSSdSb290T2ZHRjg2IywmKiQpSSNfWkdGOEYuRipGKkYpISIiRiopRiQjRipGLkYqRioqJiNGNUYpRipGJEYqRioqKCMiI04iI2FGKkY/RiopRiQjRilGLkYqRioqJiMiIyEpRi1GKilGJEYuRipGKi1GNzYjKiQpRiQjRjxGLkYqRio=

Vtsing:=subs(RootOf(_V^2-3)=-sqrt(3),op(2,%));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdWdHNpbmdHRigsLiIiIkYvKiYpIiIkI0YvIiIjRi8pSShlcHNpbG9uR0YoRjNGLyEiIiomIyIiJUYyRi9GNkYvRi8qKCMiI04iI2FGL0YxRi8pRjYjRjJGNEYvRjcqJiMiIyEpIiMiKUYvKUY2RjRGL0YvLUkiT0dGJTYjKiQpRjYjIiImRjRGL0YvNyNGLg==

with(plots): implicitplot(algVt,tt=0..1/27,Vt=0..1.5,numpoints=10000);

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series(sqrt(1-Vtsing),epsilon);

LCwqJikiIiQjIiIiIiIlRicpSShlcHNpbG9uRzYiRiZGJ0YnKigjIiIjIiIqRicpRiUjRiVGKEYnKUYqRjFGJyEiIiooRiZGJ0YkRicpRiojIiImRihGJ0YnKigjIiNgIiQnW0YnRjBGJylGKiMiIihGKEYnRjMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YrNiMqJClGKiNGL0YoRidGJw==

Values where Vt=1 subs(Vt=1,algVt)

LCYiIiJGIyomIiNGRiNJI3R0RzYiRiMhIiI=

Now the series W algWVt;

LCZJI1Z0RzYiIiIiKigiIiVGJUkiV0dGJEYlLCZGJUYlRighIiJGJUYq

As a function of Vt, the only singularity is at Vt=1, which happens to be the singular value of Vt in terms of t^3 factor(discrim(%,W));

LCYqJiIjOyIiIkkjVnRHNiJGJSEiIkYkRiU=

op(algeqtoseries(subs(Vt=1-delta,algWVt),delta,W,12));

LCgjIiIiIiIjRiQqJi1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJiIiJUYkKUkjX1pHRilGJUYkRiRGJCEiIkYkKUkmZGVsdGFHRixGI0YkRiQtSSJPR0YpNiMqJClGNSIiJ0YkRiQ=

subs(delta=1-Vtsing,%);

LCgjIiIiIiIjRiQqJi1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJiIiJUYkKUkjX1pHRilGJUYkRiRGJCEiIkYkKSwsKiYpIiIkRiNGJClJKGVwc2lsb25HRixGI0YkRiQqJiNGMEY4RiRGOkYkRjMqKCMiI04iI2FGJEY3RiQpRjojRjhGJUYkRiQqJiMiIyEpIiMiKUYkKUY6RiVGJEYzLUkiT0dGKTYjKiQpRjojIiImRiVGJEYzRiNGJEYkLUZJNiMqJClGNSIiJ0YkRiQ=

series(%,epsilon);

LC4jIiIiIiIjRiQqKC1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJiIiJUYkKUkjX1pHRilGJUYkRiRGJCEiIkYkKSIiJCNGJEYwRiQpSShlcHNpbG9uR0YsRjZGJEYkKiojRiUiIipGJEYnRiQpRjUjRjVGMEYkKUY4Rj1GJEYzKipGNkYkRidGJEY0RiQpRjgjIiImRjBGJEYkKiojIiNgIiQnW0YkRidGJEY8RiQpRjgjIiIoRjBGJEYzLUkiT0dGKTYjKiQpRjgjRjtGMEYkRiQ=

Recall that W is an increasing function of Vt Wsing:=subs(RootOf(4*_V^2-1)=-1/2,%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZXc2luZ0dGKCwuIyIiIiIiI0YwKihGL0YwKSIiJCNGMCIiJUYwKUkoZXBzaWxvbkdGKEY1RjAhIiIqKCNGMCIiKkYwKUY0I0Y0RjZGMClGOEY+RjBGMCooI0YwIiIpRjBGM0YwKUY4IyIiJkY2RjBGOSooIyIjYCIkcypGMEY9RjApRjgjIiIoRjZGMEYwLUkiT0dGJTYjKiQpRjgjRjxGNkYwRjA3I0Yu

series(1-Wsing^2,epsilon);

LDYjIiIkIiIlIiIiKigjRiYiIiNGJilGJCNGJkYlRiYpSShlcHNpbG9uRzYiRitGJkYmKihGK0YmKUYkRihGJilGLUYoRiYhIiIqKCNGJiIiKkYmKUYkRiNGJilGLUYjRiZGMiomI0YmRiRGJkYtRiZGJiooI0YmIiIpRiZGKkYmKUYtIyIiJkYlRiZGJiooIyIjTiIkOyNGJkYwRiYpRi0jRiRGKUYmRjIqKCMiI2AiJHMqRiZGNkYmKUYtIyIiKEYlRiZGMiomIyIjPyIjIilGJilGLUYpRiZGJi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRi42IyokKUYtI0Y1RiVGJkYm

Equation of W over Q(t^3) algW:=factor(resultant(algWVt,algVt,Vt));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdXR0YoLDQqKCIjayIiIilJIldHRigiIidGMUkjdHRHRihGMUYxKigiJCM+RjEpRjMiIiZGMUY1RjEhIiIqKCIjJypGMSlGMyIiJUYxRjVGMUYxKigiJEciRjEpRjMiIiRGMUY1RjFGMSooIiNbRjEpRjMiIiNGMUY1RjFGOiomRj5GMUZFRjFGOiooRkRGMUYzRjFGNUYxRjoqJkY+RjFGM0YxRjEqJiIiKUYxRjVGMUY6NyNGLg==

implicitplot(algW,tt=0..1/27,W=0..1,numpoints=10000);

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Finally, the series Z factor(discrim(algZW,Z));

LCQqKCIiJSIiIiwmRiUhIiJJIldHNiJGJUYlLCZGKEYlRiVGJUYlRic=

implicitplot(algZW,W=0..1/2,Z=-2..2);

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noB!#;7$$"2$=&Q$pm8$\%!#<$"27kq%=X3tB!#<7$$"#X!"#$"2K%pyAK9xB!#<7$$"1c!eE3u">X!#;$"1f'pAL%\)Q#!#;7$$"2$*oH1jn*RX!#<$"2&fG=kZ#3S#!#<7$$"2jYUnK%=XX!#<$"2Ou2!=;#RS#!#<7$$"$b%!"$$"0<Dmf"y1C!#:7$$"1-g(fDn6d%!#;$"2#)=F$pkO>C!#<7$$"201Suh>Sf%!#<$"1%fusmyHV#!#;7$$"1*QCahArf%!#;$"1J1<]!H[V#!#;7$$"#Y!"#$"2u?r8KXlV#!#<7$$"1P?+S-0BY!#;$"2[*4qm%4.X#!#<7$$"2/BJSgAyk%!#<$"1HjOqD8lC!#;7$$"21nDG*)\*[Y!#<$"2dr_7(y!eY#!#<7$$"$l%!"$$"2<%)H2*pVmC!#<7$$"2Cd1jQ@[n%!#<$"2G)*fWQC8[#!#<7$$"#Z!"#$"2kVbtDfk\#!#<7$$"14<7/Xm+Z!#;$"1%RNa9fo\#!#;7$$"2c67ibu8q%!#<$"2N1eMZ'G(\#!#<7$$"29uHU.zks%!#<$"283Q0H7C^#!#<7$$"$v%!"$$"1J%\\$[hED!#;7$$"/7B"ykAv%!#9$"2&e^UXR)z_#!#<7$$"1"o'="[tYv%!#;$"2')z\lPS%HD!#<7$$"2%\ir5:-yZ!#<$"2xWS3EuNa#!#<7$$"#[!"#$"2%4*[/]1pb#!#<7$$"0CwY/\P![!#:$"2ZSd7N$=fD!#<7$$"2WqIwU<x![!#<$"1L:kzUfhD!#;7$$"1L,:IrWH[!#;$"1E(\$z8"[d#!#;7$$"$&[!"$$"2uC6j4Pte#!#<7$$"1N&QEg:^&[!#;$"2CO.4Xe/f#!#<7$$"2m!ev&[/0'[!#<$"1oKt#=[Pf#!#;7$$"1"Q*=%>a2)[!#;$"2wSK#QZ71E!#<7$$"#\!"#$"2'*)***G`4zh#!#<7$$"2jg#[]FO1\!#<$"22rw0M5=i#!#<7$$"2P$3O(yKI"\!#<$"2E+De3-fi#!#<7$$"1E3g"*4%>$\!#;$"2%**\%zV:vj#!#<7$$"$&\!"$$"21kx4zE'[E!#<7$$"21$**eq()[d\!#<$"0CS[7SKl#!#:7$$"1K2(e]+`'\!#;$"1Pn"*))f0eE!#;7$$"1;[4)z0I)\!#;$"2&\.M(e%)*oE!#<7$$""&!""$"1D7JC>\zE!#;-%&COLORG6&%$RGBG$")C)eq%!")$""!!""$"('>!\&!")-&%&_AXISG6#"""6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-&%&_AXISG6#""#6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-%+AXESLABELSG6$-I#miG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"65Q"W6"/%'familyGQ!6"/%%sizeGQ#106"/%%boldGQ&false6"/%'italicGQ%true6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"-I#miG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"65Q"Z6"/%'familyGQ!6"/%%sizeGQ#106"/%%boldGQ&false6"/%'italicGQ%true6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"-%)_VISIBLEG6#"""-%%ROOTG6'-%)BOUNDS_XG6#$"$g(!""-%)BOUNDS_YG6#$"#]!""-%-BOUNDS_WIDTHG6#$"%qI!""-%.BOUNDS_HEIGHTG6#$"%+N!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+S!""-%.BOUNDS_HEIGHTG6#$"%+S!""-%)CHILDRENG6"G6"

What about the series Z? (will not be needed, in fact) algeqtoseries(subs(W=1/2-kappa,algZW),kappa,Z,6);;

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

solve(_Z^2-4*_Z+1); evalf([%]);

NiQsJiIiIyIiIiokKSIiJCNGJUYkRiVGJSwmRiRGJUYmISIi

NyQkIiszMzBLUCEiKiQiKiM+XHpFRiU=

Zkappasing:=subs(RootOf(_Z^2-4*_Z+1)=2-sqrt(3),op(%%%));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SStaa2FwcGFzaW5nR0YoKzFJJmthcHBhR0YoLCYiIiMiIiIqJCkiIiQjRjJGMUYyISIiIiIhLCYiIiVGMiomIyIiKUY1RjJGNEYyRjdGMiwmRj1GMiomIyIjUyIiKkYyRjRGMkY3RjEsJiIjO0YyKiYjIiRjIyIjRkYyRjRGMkY3RjUsJiIjS0YyKiYjIiVzOSIjIilGMkY0RjJGN0Y6LCYiI2tGMiomIyIlKTMqIiRWI0YyRjRGMkY3IiImLUkiT0dGJTYjRjIiIic3I0Yu

Zsing:=series(subs(kappa=1/2-Wsing,Zkappasing),epsilon);

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

evalf(%);

LDAkIiojPlx6RSEiKiIiIiomJCIreHIlPjIlISM1RiYpSShlcHNpbG9uRzYiI0YmIiIlRiYhIiIqJiQiKzpHbzI4RipGJilGLCNGJiIiI0YmRiYqJiQiKlo7cGokRipGJilGLCMiIiRGL0YmRiYqJiQiKEIjb0NGJUYmRixGJkYwKiYkIitsJ1I/KGUhIzZGJilGLCMiIiZGL0YmRjAtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YtNiMqJClGLCNGPEY2RiZGJg==

We need to study all dominant sigularities of V algV;

LCZJIlZHNiIhIiIqJkkidEdGJCIiIiwmKiQpRiMiIiRGKEYoIiIjRihGKEYo

factor(discrim(algV,V));solve(%);

LCQqKiIiJSIiIkkidEc2IkYlLCZGJSEiIiomIiIkRiVGJkYlRiVGJSwoKiYiIipGJSlGJiIiI0YlRiVGKkYlRiVGJUYlRik=

NiYiIiEjIiIiIiIkLCYjRiUiIichIiIqJiomRihGJV4jRiVGJUYlKUYmI0YlIiIjRiVGJSwmRihGKiomLCRGLEYlRiVGLkYlRio=

Asymptotics of Vt=V^3 around 1/27 (already seen) Vtsing;

LC4iIiJGIyomKSIiJCNGIyIiI0YjKUkoZXBzaWxvbkc2IkYnRiMhIiIqJiMiIiVGJkYjRipGI0YjKigjIiNOIiNhRiNGJUYjKUYqI0YmRihGI0YsKiYjIiMhKSIjIilGIylGKkYoRiNGIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRis2IyokKUYqIyIiJkYoRiNGIw==

Using the defining equation of V Vsing:=t*(2+Vtsing);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZWc2luZ0dGKComSSJ0R0YoIiIiLC4iIiRGMComKUYyI0YwIiIjRjApSShlcHNpbG9uR0YoRjVGMCEiIiomIyIiJUYyRjBGOEYwRjAqKCMiI04iI2FGMEY0RjApRjgjRjJGNkYwRjkqJiMiIyEpIiMiKUYwKUY4RjZGMEYwLUkiT0dGJTYjKiQpRjgjIiImRjZGMEYwRjA3I0Yu

At 1/3 (note that the meaning of epsilon is changing here) Vsing0:=series(subs(epsilon=1-27*t^3,t=1/3*(1-epsilon),Vsing),epsilon);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdWc2luZzBHRigsLiIiIkYvKiQpSShlcHNpbG9uR0YoI0YvIiIjRi8hIiIqJiNGLyIiJEYvRjJGL0YvKiYjIiIlIiIqRi8pRjIjRjhGNEYvRjUqJiMiIikiI0ZGLylGMkY0Ri9GLy1JIk9HRiU2IyokKUYyIyIiJkY0Ri9GLzcjRi4=

At (-1+ I sqrt(3))/6 Vsing1:=map(simplify,series(subs(epsilon=1-27*t^3,t=1/3*(-1+I*sqrt(3))/2*(1-epsilon),convert(Vsing,polynom)),epsilon,2));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdWc2luZzFHRigsLiMiIiIiIiMhIiIqJiomRi9GMF4jRjBGMEYwKSIiJEYvRjBGMCooRi9GMCwmRjBGMiomRjVGMEY2RjBGMEYwKUkoZXBzaWxvbkdGKEYvRjBGMiooI0YwIiInRjBGOUYwRjxGMEYwKigjRjEiIipGMEY5RjApRjwjRjdGMUYwRjItSSJPR0YlNiMqJClGPEYxRjBGMDcjRi4=

map(factor,series(Vsing1/(-1+I*sqrt(3))*2,epsilon))

LCwiIiJGIyokKUkoZXBzaWxvbkc2IiNGIyIiI0YjISIiKiYjRiMiIiRGI0YmRiNGIyomIyIiJSIiKkYjKUYmI0YtRilGI0YqLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJzYjKiQpRiZGKUYjRiM=

Vsing2:=subs(I=-I,Vsing1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdWc2luZzJHRigsLiMiIiIiIiMhIiIqJiwkKiZGL0YwXiNGMEYwRjBGMCkiIiRGL0YwRjIqKEYvRjAsJiomLCRGNkYwRjBGN0YwRjJGMEYyRjApSShlcHNpbG9uR0YoRi9GMEYyKigjRjAiIidGMEY6RjBGPkYwRjAqKCNGMSIiKkYwRjpGMClGPiNGOEYxRjBGMi1JIk9HRiU2IyokKUY+RjFGMEYwNyNGLg==

IC11; Iof1expr

KigpLCYiIiIhIiIqJiIiJEYlSSJ0RzYiRiVGJSIiI0YlKSwmKiZJJEMxMUdGKkYlRilGJUYlRiVGJUYrRiUpRilGK0Ym

LCoqLiMiIiIiIiNGJSwmKiRJIlZHNiIhIiJGJUYlRitGJSksJiokKUYpRiZGJUYrRiVGJUYkRiUsJkkiV0dGKkYlRiVGJUYlKSwmRiZGJUYxRisiIiRGJSkqJkYwRiUsJkYlRiVGMUYrRitGJEYlRisqLCNGJSIiJUYlRilGJSwqKiZGNEYlKUYxRjRGJUYlKiZGJkYlKUYxRiZGJUYrKiZGOkYlRjFGJUYrRjpGJUYlLCZGJUYrRjFGJUYrSSJ0R0YqRitGJSokKUZCRiZGK0YlKiwjRiUiIilGJUYvRiUsLiomRjpGJSlGMSIiJkYlRiUqJiIiJ0YlKUYxRjpGJUYrKiZGS0YlRj1GJUYlKiYiIz9GJUY/RiVGKyomIiM9RiVGMUYlRiVGR0YlRiVGMUYrRkFGK0Yl

At t=1/3 (the meaning of epsilon changes here) map(simplify,series(subs(W=Wsing, epsilon=1-27*t^3,t=1/3*(1-epsilon),V=Vsing0,sqrt(Iof1expr)), epsilon,1));

LCgqLCMiIioiIikiIiIpIiIjI0YnRilGJykiIiQjRiciIiVGJyksJkYpRicqJEYoRichIiJGKkYnKUkoZXBzaWxvbkc2IiNGLEYmRidGJyoqI0YlIiNLRidGK0YnRi9GMilGNCMiIihGJkYnRjItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0Y1NiMqJClGNEYkRidGJw==

The constant const:=subs(epsilon=1,op(1,%))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjb25zdEdGKCwkKiojIiIqIiIpIiIiKSIiIyNGM0Y1RjMpIiIkI0YzIiIlRjMpLCZGNUYzKiRGNEYzISIiRjZGM0YzNyNGLg==

constpap:=9/4*(9/2-3*sqrt(2))^(1/4)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSljb25zdHBhcEdGKCwkKiYjIiIqIiIlIiIiKSwmI0YxIiIjRjMqJiIiJEYzKUY3I0YzRjdGMyEiIiNGM0YyRjNGMzcjRi4=

simplify(const/constpap)

IiIi

At the other dominant singularities : no contribution at first order eps1:=expand(subs(t=1/3*(-1+I*sqrt(3))/2*(1-epsilon),1-27*t^3))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcHMxR0YoLCgqJClJKGVwc2lsb25HRigiIiQiIiJGMyomRjJGMylGMSIiI0YzISIiKiZGMkYzRjFGM0YzNyNGLg==

map(simplify,series(subs(W=Wsing, epsilon=eps1, t=1/3*(-1+I*sqrt(3))/2*(1-epsilon),V=Vsing1, sqrt(Iof1expr)), epsilon,1));

LCgqJiMiIiQiIiUiIiIpKiYsKComLCQqJiIjRkYnXiNGJ0YnRidGJyksJiIiJ0YnKiYqJiIiI0YnRi9GJ0YnKUYlI0YnRjVGJ0YnRjdGJyEiIiomLCQqJiIjN0YnRi9GJ0YnRidGNkYnRjgiIycqRidGJyksJkYnRjgqJkYvRidGNkYnRidGNUY4RjdGJ0YnKjAjRiUiIztGJ0YoRicsKComIiIoRidGNkYnRjgqJkYyRidGMEYnRicqJiIiKkYnRi9GJ0YnRidGP0YnKUkoZXBzaWxvbkc2IkYkRicsKEZARicqJiomI0ZJRiZGJ0YvRidGJ0YwRidGJyIiKUY4RjgsJkYlRidGQEYnRjhGOC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRkw2I0ZLRic=

map(evalf,%)

LCgsJiQiK1tSITNUJCEiKiIiIiomJCIrU0wva0hGJkYnXiNGJ0YnRidGJyomLCYkIispKlxQLEkhIzUhIiIqJiQiKykqKjRMZydGMEYnRitGJ0YxRicpSShlcHNpbG9uRzYiIyIiJCIiJUYnRictSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0Y3NiNGNkYn

Let us check. evalf(const/GAMMA(5/8))

JCIrTyRccjYiISIq

n:=100: ser:=subs(x=1,y=1,Cser(n)): plot([seq([1/i,coeff(ser,t,i)/3^(i)*i^(3/8)],i=1..n)],a=0..0.1,b=1.01..1.13,style=point);

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Ig==

algV:=-V+t*(V^3+2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdWR0YoLCZJIlZHRighIiIqJkkidEdGKCIiIiwmKiQpRi8iIiRGM0YzIiIjRjNGM0YzNyNGLg==

Here, tt stands for t three (t cubed). This is the equation for V^3=Vt algVt:=Vt-tt*(2+Vt)^3;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdWdEdGKCwmSSNWdEdGKCIiIiomSSN0dEdGKEYwKSwmIiIjRjBGL0YwIiIkRjAhIiI3I0Yu

The only singularity is at tt=1/27 factor(discrim(algVt,Vt));

LCQqKCIiJSIiIkkjdHRHNiJGJSwmKiYiI0ZGJUYmRiVGJUYlISIiRiVGKw==

with(gfun):algeqtoseries(subs(tt=1/27*(1-epsilon),algVt),epsilon,Vt,5);;

NyQrL0koZXBzaWxvbkc2IiEiKSIiISNGJiIiJCIiIiMhJGciIiMiKSIiIyMhJSVlJCIlKD0jRikjISZnIkciJiRvPiIiJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YqIiImLC5GKkYqKiYtSSdSb290T2ZHRjg2IywmKiQpSSNfWkdGOEYuRipGKkYpISIiRiopRiQjRipGLkYqRioqJiNGNUYpRipGJEYqRioqKCMiI04iI2FGKkY/RiopRiQjRilGLkYqRioqJiMiIyEpRi1GKilGJEYuRipGKi1GNzYjKiQpRiQjRjxGLkYqRio=

Vtsing:=subs(RootOf(_V^2-3)=-sqrt(3),op(2,%));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdWdHNpbmdHRigsLiIiIkYvKiYpIiIkI0YvIiIjRi8pSShlcHNpbG9uR0YoRjNGLyEiIiomIyIiJUYyRi9GNkYvRi8qKCMiI04iI2FGL0YxRi8pRjYjRjJGNEYvRjcqJiMiIyEpIiMiKUYvKUY2RjRGL0YvLUkiT0dGJTYjKiQpRjYjIiImRjRGL0YvNyNGLg==

with(plots): implicitplot(algVt,tt=0..1/27,Vt=0..1.5,numpoints=10000);

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series(sqrt(1-Vtsing),epsilon);

LCwqJikiIiQjIiIiIiIlRicpSShlcHNpbG9uRzYiRiZGJ0YnKigjIiIjIiIqRicpRiUjRiVGKEYnKUYqRjFGJyEiIiooRiZGJ0YkRicpRiojIiImRihGJ0YnKigjIiNgIiQnW0YnRjBGJylGKiMiIihGKEYnRjMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YrNiMqJClGKiNGL0YoRidGJw==

Values where Vt=1 subs(Vt=1,algVt)

LCYiIiJGIyomIiNGRiNJI3R0RzYiRiMhIiI=

Now the series W algWVt;

LCZJI1Z0RzYiIiIiKigiIiVGJUkiV0dGJEYlLCZGJUYlRighIiJGJUYq

As a function of Vt, the only singularity is at Vt=1, which happens to be the singular value of Vt in terms of t^3 factor(discrim(%,W));

LCYiIzsiIiIqJkYjRiRJI1Z0RzYiRiQhIiI=

op(algeqtoseries(subs(Vt=1-delta,algWVt),delta,W,12));

LCgjIiIiIiIjRiQqJi1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJiIiJUYkKUkjX1pHRilGJUYkRiRGJCEiIkYkKUkmZGVsdGFHRixGI0YkRiQtSSJPR0YpNiMqJClGNSIiJ0YkRiQ=

subs(delta=1-Vtsing,%);

LCgjIiIiIiIjRiQqJi1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJiIiJUYkKUkjX1pHRilGJUYkRiRGJCEiIkYkKSwsKiYpIiIkRiNGJClJKGVwc2lsb25HRixGI0YkRiQqJiNGMEY4RiRGOkYkRjMqKCMiI04iI2FGJEY3RiQpRjojRjhGJUYkRiQqJiMiIyEpIiMiKUYkKUY6RiVGJEYzLUkiT0dGKTYjKiQpRjojIiImRiVGJEYzRiNGJEYkLUZJNiMqJClGNSIiJ0YkRiQ=

series(%,epsilon);

LC4jIiIiIiIjRiQqKC1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYqJiIiJUYkKUkjX1pHRilGJUYkRiRGJCEiIkYkKSIiJCNGJEYwRiQpSShlcHNpbG9uR0YsRjZGJEYkKiojRiUiIipGJEYnRiQpRjUjRjVGMEYkKUY4Rj1GJEYzKipGNkYkRidGJEY0RiQpRjgjIiImRjBGJEYkKiojIiNgIiQnW0YkRidGJEY8RiQpRjgjIiIoRjBGJEYzLUkiT0dGKTYjKiQpRjgjRjtGMEYkRiQ=

Recall that W is an increasing function of Vt Wsing:=subs(RootOf(4*_V^2-1)=-1/2,%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZXc2luZ0dGKCwuIyIiIiIiI0YwKihGL0YwKSIiJCNGMCIiJUYwKUkoZXBzaWxvbkdGKEY1RjAhIiIqKCNGMCIiKkYwKUY0I0Y0RjZGMClGOEY+RjBGMCooI0YwIiIpRjBGM0YwKUY4IyIiJkY2RjBGOSooIyIjYCIkcypGMEY9RjApRjgjIiIoRjZGMEYwLUkiT0dGJTYjKiQpRjgjRjxGNkYwRjA3I0Yu

series(1-Wsing^2,epsilon);

LDYjIiIkIiIlIiIiKigjRiYiIiNGJilGJCNGJkYlRiYpSShlcHNpbG9uRzYiRitGJkYmKihGK0YmKUYkRihGJilGLUYoRiYhIiIqKCNGJiIiKkYmKUYkRiNGJilGLUYjRiZGMiomI0YmRiRGJkYtRiZGJiooI0YmIiIpRiZGKkYmKUYtIyIiJkYlRiZGJiooIyIjTiIkOyNGJkYwRiYpRi0jRiRGKUYmRjIqKCMiI2AiJHMqRiZGNkYmKUYtIyIiKEYlRiZGMiomIyIjPyIjIilGJilGLUYpRiZGJi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRi42IyokKUYtI0Y1RiVGJkYm

Equation of W over Q(t^3) algW:=factor(resultant(algWVt,algVt,Vt));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdXR0YoLDQqKCIjayIiIilJIldHRigiIidGMUkjdHRHRihGMUYxKigiJCM+RjEpRjMiIiZGMUY1RjEhIiIqKCIjJypGMSlGMyIiJUYxRjVGMUYxKigiJEciRjEpRjMiIiRGMUY1RjFGMSooIiNbRjEpRjMiIiNGMUY1RjFGOiomRj5GMUZFRjFGOiooRkRGMUYzRjFGNUYxRjoqJkY+RjFGM0YxRjEqJiIiKUYxRjVGMUY6NyNGLg==

implicitplot(algW,tt=0..1/27,W=0..1,numpoints=10000);

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Finally, the series Z factor(discrim(algZW,Z));

LCQqKCIiJSIiIiwmRiUhIiJJIldHNiJGJUYlLCZGKEYlRiVGJUYlRic=

implicitplot(algZW,W=0..1/2,Z=-2..2);

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

What about the series Z? (will not be needed, in fact) algeqtoseries(subs(W=1/2-kappa,algZW),kappa,Z,6);;

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

solve(_Z^2-4*_Z+1); evalf([%]);

NiQsJiIiIyIiIiokKSIiJCNGJUYkRiVGJSwmRiRGJUYmISIi

NyQkIiszMzBLUCEiKiQiKiM+XHpFRiU=

Zkappasing:=subs(RootOf(_Z^2-4*_Z+1)=2-sqrt(3),op(%%%));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SStaa2FwcGFzaW5nR0YoKzFJJmthcHBhR0YoLCYiIiMiIiIqJCkiIiQjRjJGMUYyISIiIiIhLCYiIiVGMiomIyIiKUY1RjJGNEYyRjdGMiwmRj1GMiomIyIjUyIiKkYyRjRGMkY3RjEsJiIjO0YyKiYjIiRjIyIjRkYyRjRGMkY3RjUsJiIjS0YyKiYjIiVzOSIjIilGMkY0RjJGN0Y6LCYiI2tGMiomIyIlKTMqIiRWI0YyRjRGMkY3IiImLUkiT0dGJTYjRjIiIic3I0Yu

Zsing:=series(subs(kappa=1/2-Wsing,Zkappasing),epsilon);

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

evalf(%);

LDAkIiojPlx6RSEiKiIiIiomJCIreHIlPjIlISM1RiYpSShlcHNpbG9uRzYiI0YmIiIlRiYhIiIqJiQiKzpHbzI4RipGJilGLCNGJiIiI0YmRiYqJiQiKlo7cGokRipGJilGLCMiIiRGL0YmRiYqJiQiKEIjb0NGJUYmRixGJkYwKiYkIitsJ1I/KGUhIzZGJilGLCMiIiZGL0YmRjAtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YtNiMqJClGLCNGPEY2RiZGJg==

We need to study all dominant singularities of V algV;

LCZJIlZHNiIhIiIqJkkidEdGJCIiIiwmKiQpRiMiIiRGKEYoIiIjRihGKEYo

factor(discrim(algV,V));solve(%);

LCQqKiIiJSIiIkkidEc2IkYlLCYqJiIiJEYlRiZGJUYlRiUhIiJGJSwoKiYiIipGJSlGJiIiI0YlRiVGKUYlRiVGJUYlRis=

NiYiIiEjIiIiIiIkLCYjRiUiIichIiIqJiomRihGJV4jRiVGJUYlKUYmI0YlIiIjRiVGJSwmRihGKiomLCRGLEYlRiVGLkYlRio=

Asymptotics of Vt=V^3 around 1/27 (already seen) Vtsing;

LC4iIiJGIyomKSIiJCNGIyIiI0YjKUkoZXBzaWxvbkc2IkYnRiMhIiIqJiMiIiVGJkYjRipGI0YjKigjIiNOIiNhRiNGJUYjKUYqI0YmRihGI0YsKiYjIiMhKSIjIilGIylGKkYoRiNGIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRis2IyokKUYqIyIiJkYoRiNGIw==

Using the defining equation of V Vsing:=t*(2+Vtsing);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZWc2luZ0dGKComSSJ0R0YoIiIiLC4iIiRGMComKUYyI0YwIiIjRjApSShlcHNpbG9uR0YoRjVGMCEiIiomIyIiJUYyRjBGOEYwRjAqKCMiI04iI2FGMEY0RjApRjgjRjJGNkYwRjkqJiMiIyEpIiMiKUYwKUY4RjZGMEYwLUkiT0dGJTYjKiQpRjgjIiImRjZGMEYwRjA3I0Yu

At 1/3 (note that the meaning of epsilon is changing here) Vsing0:=series(subs(epsilon=1-27*t^3,t=1/3*(1-epsilon),Vsing),epsilon);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdWc2luZzBHRigsLiIiIkYvKiQpSShlcHNpbG9uR0YoI0YvIiIjRi8hIiIqJiNGLyIiJEYvRjJGL0YvKiYjIiIlIiIqRi8pRjIjRjhGNEYvRjUqJiMiIikiI0ZGLylGMkY0Ri9GLy1JIk9HRiU2IyokKUYyIyIiJkY0Ri9GLzcjRi4=

At (-1+ I sqrt(3))/6 Vsing1:=map(simplify,series(subs(epsilon=1-27*t^3,t=1/3*(-1+I*sqrt(3))/2*(1-epsilon),convert(Vsing,polynom)),epsilon,2));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdWc2luZzFHRigsLiMiIiIiIiMhIiIqJiomRi9GMF4jRjBGMEYwKSIiJEYvRjBGMCooRi9GMCwmRjBGMiomRjVGMEY2RjBGMEYwKUkoZXBzaWxvbkdGKEYvRjBGMiooI0YwIiInRjBGOUYwRjxGMEYwKigjRjEiIipGMEY5RjApRjwjRjdGMUYwRjItSSJPR0YlNiMqJClGPEYxRjBGMDcjRi4=

map(factor,series(Vsing1/(-1+I*sqrt(3))*2,epsilon))

LCwiIiJGIyokKUkoZXBzaWxvbkc2IiNGIyIiI0YjISIiKiYjRiMiIiRGI0YmRiNGIyomIyIiJSIiKkYjKUYmI0YtRilGI0YqLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJzYjKiQpRiZGKUYjRiM=

Vsing2:=subs(I=-I,Vsing1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdWc2luZzJHRigsLiMiIiIiIiMhIiIqJiwkKiZGL0YwXiNGMEYwRjBGMCkiIiRGL0YwRjIqKEYvRjAsJiomLCRGNkYwRjBGN0YwRjJGMEYyRjApSShlcHNpbG9uR0YoRi9GMEYyKigjRjAiIidGMEY6RjBGPkYwRjAqKCNGMSIiKkYwRjpGMClGPiNGOEYxRjBGMi1JIk9HRiU2IyokKUY+RjFGMEYwNyNGLg==

Walks ending on the diagonal Here we introduce a variant of J(y), namely t^4 J(y/t^2) /Delta(y/t^2), which is a function of t^3 factor(subs(y=y/t^2,t^4*Jy/Delta));

KiYpLCYqJilJInlHNiIiIiMiIiItSSNEREdGKDYjKiZGJ0YqKUkidEdGKEYpISIiRipGKiokKUYwIiIkRipGKkYpRipGJkYx

Jyform

LCoqJCksKCokKUkieUc2IiIiIyEiIiIiIiomSSJ0R0YpRitGKEYrRitGKEYrRipGLEYsKiZJJEFBMkdGKUYsRiVGLEYsKihJJEFBMUdGKUYsLCYqJEkiVkdGKUYrRiwqJEYoRitGK0YsKSwmKiYpRjVGKkYsRihGLEYrRixGLCNGLEYqRixGLEkkQUEwR0YpRiw=

The same series, in terms of Vt=V^3 and W Jynew:=subs(V=Vt^(1/3),map(factor,subs(AA0=A0VW,AA2=A2sol,AA1=A1W,y=y/t^2,t=tV,t^4*Jyform/Delta))): indets(%)

PCdJI1Z0RzYiSSJXR0YkSSJ5R0YkKiQpLCQqJiwmRiUiIiJGLEYsRiwsJkYsISIiRiVGLEYuRi4jRiwiIiNGLCokKSwqKiYpRiNGMEYsRiZGLEYuKigiIiVGLEYjRixGJkYsRi4qJkY3RixGJkYsRi5GLEYsRi9GLA==

res:=map(factor,series(subs(Vt=Vtsing,W=Wsing,Jynew),epsilon,1));

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

Let us examine the first two coefficients d0:=subs(delta=1-9*y,collect(subs(sqrt(1-9*y)=sqrt(delta),op(1,%)),delta,factor));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkMEdGKCwmKi4jIiMiKSIjSyIiIikiIiQjRjMiIiNGM0kieUdGKEYzLCYqJiIjT0YzRjhGM0YzRjMhIiJGPCwmKiYiIipGM0Y4RjNGM0YzRjxGPCksJkYzRjNGPkY8RjZGM0Y8KiwjRjMiJms7IkYzRj1GMywqKiZGREYzKUY4RjVGM0YzKiYiJXRFRjMpRjhGN0YzRjMqJiIkSyVGM0Y4RjNGMyIjO0Y8RjNGSkY8RjlGPEY8NyNGLg==

d1:=subs(epsilon=1,collect(subs(sqrt(1-9*y)=sqrt(delta),op(2,res)),delta,factor));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkMUdGKCwmKi4jIiIqIiIpIiIiKSIiJCNGMyIiJUYzSSJ5R0YoRjMsJiomIiNPRjNGOEYzRjNGMyEiIkY8LCYqJkYxRjNGOEYzRjNGM0Y8RjwpSSZkZWx0YUdGKCNGMyIiI0YzRjMqLCNGMyIkVyJGMylGNSNGNUY3RjMsKiomIiVLZUYzKUY4RjVGM0YzKiYiJFYjRjMpRjhGQkYzRjMqJiIkOyNGM0Y4RjNGM0YyRjxGMylGPUZCRjxGOUY8Rjw3I0Yu

We need to take the square root, and divide by y. First, consider the reciprocal of d0 collect(subs(sqrt(1-9*y)=sqrt(delta),factor(rationalize(1/d0))),delta,factor);

LCYqLiIrc11ceDgiIiIpIiIkI0YlIiIjRiUpSSJ5RzYiIiImRiUpLCYqJiIiKkYlRitGJUYlRilGJUYpISIiKSwqKiYiJms7IkYlKUYrRidGJUYlKiYiJSw8RiUpRitGKUYlRjIqJiIkSyVGJUYrRiVGJSIjO0YyRilGMilJJmRlbHRhR0YsRihGJUYlKi4iJmNtJUYlKSwmRjBGJUYlRjJGKUYlLCpGNUYlKiYiJXRFRiVGOkYlRiVGO0YlRj1GMkYlRjpGJUYuRjJGM0YyRjI=

collect(simplify(sqrt(%),symbolic),delta)

LCQqLCIkMyIiIiIpIiIjI0YlRidGJUkieUc2IkYlKSwwKioiJlwhZkYlKUkmZGVsdGFHRipGKEYlKSIiJEYoRiUpRilGMkYlRiUqJiIobyYqKT1GJSlGKSIiJkYlISIiKiYiJkFKIkYlKUYpIiIlRiVGOComIiU7SEYlRjNGJUYlKiYiJil6N0YlKUYpRidGJUYlKiYiJVM5RiVGKUYlRjgiI0tGJUYoRiUsLComIid3XDVGJUY7RiVGJSomIiU+ISlGJUYzRiVGJSomIiQnW0YlRkFGJUYlKiYiJD8oRiVGKUYlRiVGREY4RjhGJQ==

We want the positive root c1:=-d1*%/2/y;

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

series(subs(delta=1-9*y,c1),y,3)

KytJInlHNiIsJComIyIiJCIiJSIiIilGKEYnRiohIiIiIiEsJiomIyIkViMiIztGKilGKCNGKkYpRipGKiomIyIjIilGKUYqRitGKkYsRiosJiomIyImQk0nIiRHIkYqRitGKkYsKiYjIiYkbz4iI0tGKkYzRipGKiIiIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YqRig=

Hdexpl:=-3^(7/4)*subs(y=y/9,delta=1-y,c1);

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

simplify(series(Hdexpl,y,5))

Ky9JInlHNiIsJComIyIjRiIiJSIiIikiIiQjRioiIiNGKkYqIiIhLCYqJiMiIyIpRilGKkYrRipGKiMiJFYjIiM7ISIiRiosJiomIyIlWnEiJEciRipGK0YqRiojIiUoPSMiI0tGN0YuLCYjIic8aDciJDcmRjcqJiMiJlQuIiIja0YqRitGKkYqRiwsJiMiJ3YzKSkiJUM1RjcqJiMiKFhoRCUiJSM+KUYqRitGKkYqRiktSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKiIiJg==

Hdexpl

LCQqLiIkaSIiIiIpIiIkI0YnIiIlRiUsJiouI0YlIiIpRiUpRicjRiVGKUYlSSJ5RzYiRiUsJiomRilGJUYwRiVGJUYlISIiRjQsJkYwRiVGJUY0RjQpLCZGJUYlRjBGNCNGJSIiI0YlRiUqLCNGJSIkVyJGJUYmRiUsKiomRi1GJSlGMEYnRiVGJSomRidGJSlGMEY5RiVGJSomIiNDRiVGMEYlRiVGLUY0RiUpRjVGOUY0RjJGNEY0RiUpRjlGOEYlKSwwKioiIyIpRiVGNkYlKUYnRjhGJUY/RiVGJSomIiNLRiUpRjAiIiZGJUY0KiZGOUYlKUYwRilGJUY0KiZGKUYlRj9GJUYlKiYiJGUiRiVGQUYlRiUqJiIkZyJGJUYwRiVGNEZMRiVGOEYlLCwqJiIjO0YlRlBGJUYlKiYiIzZGJUY/RiVGJSomIiInRiVGQUYlRiUqJiIjISlGJUYwRiVGJUZMRjRGNEYl

subs(sqrt(1-y)=s,H^2-Hdexpl^2)

LCYqJClJIkhHNiIiIiMiIiJGKCosIidrdTpGKCkiIiQjRihGJ0YoKSwmKi4jRigiIilGKClGLCNGKCIiJUYoSSJ5R0YmRigsJiomRjVGKEY2RihGKEYoISIiRjksJkY2RihGKEY5RjlJInNHRiZGKEYoKiwjRigiJFciRigpRiwjRixGNUYoLCoqJkYyRigpRjZGLEYoRigqJkYsRigpRjZGJ0YoRigqJiIjQ0YoRjZGKEYoRjJGOUYoKUY6RidGOUY3RjlGOUYnRigsMCoqIiMiKUYoRjtGKEYrRihGQ0YoRigqJiIjS0YoKUY2IiImRihGOSomRidGKClGNkY1RihGOSomRjVGKEZDRihGKComIiRlIkYoRkVGKEYoKiYiJGciRihGNkYoRjlGTUYoRigpLCwqJiIjO0YoRlFGKEYoKiYiIzZGKEZDRihGKComIiInRihGRUYoRigqJiIjISlGKEY2RihGKEZNRjlGJ0Y5Rjk=

res0:=factor(resultant(numer(%),s^2-(1-y),s)); nops(res0)

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

IiIm

algHd:=collect(op(5,res0),H,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdIZEdGKCwoKioiJUM1IiIiLCYqJiIiJUYxSSJ5R0YoRjFGMUYxISIiRjEpLCZGNUYxRjFGNiIiKEYxKUkiSEdGKEY0RjFGMSooIidPKnojRjEpRjgiIiZGMSlGOyIiI0YxRjEqJiIocEh5JUYxKSwmRjVGMUZBRjFGQUYxRjE3I0Yu

Hssolens:=[solve(subs(H=sqrt(Hs),algHd),Hs)]

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlIc3NvbGVuc0dGKDckLCQqKiMiJSg9IyIjSyIiIiwqKiYiIiNGNClJInlHRihGN0Y0ISIiKiYiIiRGNCksJiomRjxGNClGOUY8RjRGOiomRjxGNEY4RjRGNCNGNEY3RjRGNComIiIlRjRGOUY0RjRGN0Y6RjQsKComRkRGNEY4RjRGNComIiImRjRGOUY0RjpGNEY0RjopLCZGOUY0RjRGOkY8RjpGNCwkKipGMUY0LCpGNkY0RjtGNEZDRjpGN0Y0RjRGRUY6RklGOkY6NyNGLg==

subs(csgn(y)=1,map(series,%,y,3))

NyQrK0kieUc2IiMiJSg9IyIjOyIiISwmIyIlaGwiIikiIiIqJiNGLCIjS0YuKSIiJCNGLiIiI0YuISIiRi4sJiMiJlwhZkYoRi4qJiMiJjolKSoiI2tGLkYyRi5GNkY1LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRi5GMysrRiRGJkYpLCZGK0YuRi9GLkYuLCZGOEYuRjpGLkY1Rj5GMw==

evalf(%)

NyQrK0kieUc2IiQiKytdKG9PIiEiKCIiISQiK3lYK11ZRigiIiIkIitMJUdyLSIhIiciIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGLCIiJCsrRiRGJkYpJCIrVSZcXzwiRi9GLCQiK25sKlJOJ0YvRjBGMUY3

map(Z->series(sqrt(Z),y),%)

NyQrK0kieUc2IiQiKyZITSJwNiEiKSIiISQiKyYpR2wpKT5GKCIiIiQiKyFmdzhxI0YoIiIjLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiwiIiQrK0YkRiZGKSQiKyUpRzpFXUYoRiwkIitnRSxQOyEiKEYvRjBGNg==

Hssol:=factor(op(1,Hssolens))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIc3NvbEdGKCwkKiojIiUoPSMiI0siIiIsKiomIiIjRjMpSSJ5R0YoRjZGMyEiIiomIiIkRjMpLCQqKEY7RjNGN0YzLCZGOEYzRjNGOUYzRjkjRjNGNkYzRjMqJiIiJUYzRjhGM0YzRjZGOUYzLCZGQUYzRjNGOUY5KUY/RkJGOUYzNyNGLg==

Hdfinal:=sqrt(%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShIZGZpbmFsR0YoLCQqKCMiI0YiIikiIiIpIiInI0YzIiIjRjMpKigsKiomRjdGMylJInlHRihGN0YzISIiKiYiIiRGMyksJCooRkBGM0Y8RjMsJkY9RjNGM0Y+RjNGPkY2RjNGMyomIiIlRjNGPUYzRjNGN0Y+RjMsJkZFRjNGM0Y+Rj4pRkRGRkY+RjZGM0YzNyNGLg==

Hdexpl

LCQqLiIkaSIiIiIpIiIkI0YnIiIlRiUsJiouI0YlIiIpRiUpRicjRiVGKUYlSSJ5RzYiRiUsJiomRilGJUYwRiVGJUYlISIiRjQsJkYwRiVGJUY0RjQpLCZGJUYlRjBGNCNGJSIiI0YlRiUqLCNGJSIkVyJGJUYmRiUsKiomRi1GJSlGMEYnRiVGJSomRidGJSlGMEY5RiVGJSomIiNDRiVGMEYlRiVGLUY0RiUpRjVGOUY0RjJGNEY0RiUpRjlGOEYlKSwwKioiIyIpRiVGNkYlKUYnRjhGJUY/RiVGJSomIiNLRiUpRjAiIiZGJUY0KiZGOUYlKUYwRilGJUY0KiZGKUYlRj9GJUYlKiYiJGUiRiVGQUYlRiUqJiIkZyJGJUYwRiVGNEZMRiVGOEYlLCwqJiIjO0YlRlBGJUYlKiYiIzZGJUY/RiVGJSomIiInRiVGQUYlRiUqJiIjISlGJUYwRiVGJUZMRjRGNEYl

simplify(series(Hdexpl-Hdfinal,y,10)) assuming y>0

KyVJInlHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

ifactor(2187)

KiQpLUkhRzYiNiMiIiQiIigiIiI=

Hdspap:=3^7/2^5/(1-4*y)/(1-y)^2*(2-3*y*sqrt(3)/(1-y)^(3/2))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdIZHNwYXBHRigsJCoqIyIlKD0jIiNLIiIiLCZGM0YzKiYiIiVGM0kieUdGKEYzISIiRjgpLCZGM0YzRjdGOCIiI0Y4LCZGO0YzKioiIiRGM0Y3RjMpRj4jRjNGO0YzKUY6I0Y+RjtGOEY4RjNGMzcjRi4=

simplify(series(Hssol-Hdspap,y,10)) assuming y>0

KyVJInlHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

Hdpap:=27*sqrt(3)/4/(1-y)/sqrt(1-4*y)*sqrt(1-3*y*sqrt(3)/2/(1-y)^(3/2))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIZHBhcEdGKCwkKiwjIiNGIiIpIiIiKSIiJCNGMyIiI0YzLCZGM0YzSSJ5R0YoISIiRjopLCZGM0YzKiYiIiVGM0Y5RjNGOkY2RjopLCZGPkYzKioiIidGM0Y5RjNGNEYzKUY4I0Y1RjdGOkY6RjZGM0YzNyNGLg==

simplify(series(Hdexpl/Hdpap,y,10)) assuming y>0

KydJInlHNiIiIiIiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNQ==

Hdpap

LCQqLCMiI0YiIikiIiIpIiIkI0YnIiIjRicsJkYnRidJInlHNiIhIiJGLyksJkYnRicqJiIiJUYnRi1GJ0YvRipGLyksJkYzRicqKiIiJ0YnRi1GJ0YoRicpRiwjRilGK0YvRi9GKkYnRic=

Let us check this numerically First, at (0,0) N:=100: i:=0: ser:=coeff(Dser(N), y,i): plot([seq([1/n,-coeff(ser,t,n)/3^n*n^(7/4)*GAMMA(-3/4)],n=1..N)],a=0..0.1,b=9..12,style=point);

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NiI=

series(Hdpap,y,1);evalf(%);

KydJInlHNiIsJComIyIjRiIiJSIiIikiIiQjRioiIiNGKkYqIiIhLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRipGKg==

KydJInlHNiIkIismSE0icDYhIikiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGLw==

At (1,1) N:=150: i:=1: ser:=coeff(Dser(N), y,i): plot([seq([1/n,-coeff(ser,t,n)/3^n*n^(7/4)*GAMMA(-3/4)],n=1..N)],a=0..0.1,b=18..20,style=point);

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NiI=

series(subs(delta=1-9*y,Hdpap),y,2);evalf(%);

KylJInlHNiIsJComIyIjRiIiJSIiIikiIiQjRioiIiNGKkYqIiIhLCYqJiMiIyIpRilGKkYrRipGKiMiJFYjIiM7ISIiRiotSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKkYu

KylJInlHNiIkIismSE0icDYhIikiIiEkIisnKUdsKSk+RiciIiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKyIiIw==

At (2,2) N:=150: i:=2: ser:=coeff(Dser(N), y,i): plot([seq([1/n,-coeff(ser,t,n)/3^n*n^(7/4)*GAMMA(-3/4)],n=1..N)],a=0..0.1,b=18..27,style=point);

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

simplify(series(subs(delta=1-9*y,Hdpap),y,3));evalf(%);

KytJInlHNiIsJComIyIjRiIiJSIiIikiIiQjRioiIiNGKkYqIiIhLCYqJiMiIyIpRilGKkYrRipGKiMiJFYjIiM7ISIiRiosJiomIyIlWnEiJEciRipGK0YqRiojIiUoPSMiI0tGN0YuLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRipGLA==

KytJInlHNiIkIismSE0icDYhIikiIiEkIisnKUdsKSk+RiciIiIkIisoZnc4cSNGJyIiIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YrIiIk

Walks ending on the boundary (negative x-axis) Ixform ;

LCoqKCwmKiRJIlZHNiIhIiIiIiIqJEkieEdGJ0YoRihGKSksJiomKUYmIiIjRilGK0YpRihGKUYpI0YpRjBGKUkkQUExR0YnRilGKComSSRBQTJHRidGKSwoKiQpRitGMEYoRikqJkkidEdGJ0YoRitGKEYoRitGKEYpRikqJClGNUYwRilGKUkkQUEwR0YnRik=

We consider this variant of I(x), which is a series in t^3. It is 1/t^2 I(x/t^2) factor(subs(U(x,0)=1/x*Cm(x),x=x/t^2,1/t^2*Ix));

KigpLCwqKClJInRHNiIiIiQiIiItSSNEREdGKDYjIiIhRiopSSJ4R0YoIiIjRipGKioqRjFGKkYmRiotSSNDbUdGKDYjKiZGMEYqKUYnRjEhIiJGKkYvRipGKiokKUYnIiInRipGKiomRiZGKkYwRipGOCokKUYwRilGKkYqRjFGKkY6RjgpRjAiIiVGOA==

The same series, in terms of Vt=V^3 and W Ixnew:=subs(V=Vt^(1/3),map(factor,subs(AA0=A0VW,AA2=A2sol,AA1=A1W,x=x/t^2,t=tV,1/t^2*Ixform))): indets(%)

PCdJI1Z0RzYiSSJXR0YkSSJ4R0YkKiQpLCQqJiwmRiUiIiJGLEYsRiwsJkYsISIiRiVGLEYuRi4jRiwiIiNGLCokKSwqKiYpRiNGMEYsRiZGLEYuKigiIiVGLEYjRixGJkYsRi4qJkY3RixGJkYsRi5GLEYsRi9GLA==

res:=map(factor,series(subs(Vt=Vtsing,W=Wsing,Ixnew),epsilon,1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyZXNHRigsKCoqIyIiIiImR0wjRjEsJkYxISIiKiYiIipGMUkieEdGKEYxRjFGMSwwKioiJlwhZkYxKSwmRjVGNEYxRjEjRjEiIiNGMSkiIiRGPUYxKUY3RkBGMUYxKiYiKG8mKik9RjEpRjciIiZGMUY0KiYiJkFKIkYxKUY3IiIlRjFGNComIiU7SEYxRkFGMUYxKiYiJil6N0YxKUY3Rj5GMUYxKiYiJVM5RjFGN0YxRjQiI0tGMUYxRkhGNEY0KiwjRjEiJFciRjEpRkAjRjFGSUYxRk5GNCwuKigiJUtlRjFGP0YxRkFGMUYxKigiJFYjRjFGP0YxRk5GMUYxKigiJWU5RjFGO0YxRk5GMUYxKigiJDsjRjFGP0YxRjdGMUYxKigiJGkiRjFGO0YxRjdGMUY0KiYiIilGMUY/RjFGNEYxKUkoZXBzaWxvbkdGKCNGQEZJRjFGMS1JIk9HRiU2I0Zfb0YxNyNGLg==

Let us examine the first two coefficients d0:=subs(delta=1-9*x,collect(subs(sqrt(1-9*x)=sqrt(delta),op(1,res)),delta,factor));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkMEdGKCwmKiwjIiMiKSIjSyIiIiwmRjMhIiIqJiIiKkYzSSJ4R0YoRjNGM0YzKSIiJCNGMyIiI0YzRjhGNSksJkY2RjVGM0YzRjtGM0Y1KiojRjMiJms7IkYzKUY0RjpGMywqKiZGQUYzKUY4RjpGM0YzKiYiJXRFRjMpRjhGPEYzRjMqJiIkSyVGM0Y4RjNGMyIjO0Y1RjMpRjgiIiVGNUYzNyNGLg==

d1:=subs(epsilon=1,collect(subs(sqrt(1-9*x)=sqrt(delta),op(2,res)),delta,factor));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkMUdGKCwmKiwjIiIqIiIpIiIiKSIiJCNGMyIiJUYzSSJ4R0YoISIiLCZGM0Y5KiZGMUYzRjhGM0YzRjMpSSZkZWx0YUdGKCNGMyIiI0YzRjMqKiNGMyIkVyJGMylGNSNGNUY3RjMsKiomIiVLZUYzKUY4RjVGM0YzKiYiJFYjRjMpRjhGP0YzRjMqJiIkOyNGM0Y4RjNGM0YyRjlGM0ZLRjlGMzcjRi4=

We need to take the square root, and divide by 2. First, consider the reciprocal of d0 collect(subs(sqrt(1-9*x)=sqrt(delta),factor(rationalize(1/d0))),delta,factor);

LCYqMiIrc11ceDgiIiIpIiIkI0YlIiIjRiUpSSJ4RzYiIiIoRiUsJkYlISIiKiYiI09GJUYrRiVGJUYvKSwmKiYiIipGJUYrRiVGJUYpRiVGKUYvKSwmRiVGL0Y0RiVGKUYvKSwqKiYiJms7IkYlKUYrRidGJUYlKiYiJSw8RiUpRitGKUYlRi8qJiIkSyVGJUYrRiVGJSIjO0YvRilGLylJJmRlbHRhR0YsRihGJUYlKi4iJmNtJUYlKUYrIiIlRiUsKkY6RiUqJiIldEVGJUY/RiVGJUZARiVGQkYvRiVGOEYvRjJGL0YuRi9GJQ==

simplify(sqrt(%)) assuming x<1/1000 assuming x>-1/1000

LCQqMiIkMyIiIiIpIiIjI0YlRidGJSlJInhHNiJGJ0YlKSwmRiVGJSomIiNPRiVGKkYlISIiRihGMCksMCoqIiZcIWZGJSlJJmRlbHRhR0YrRihGJSkiIiRGKEYlKUYqRjhGJUYwKiYiKG8mKik9RiUpRioiIiZGJUYwKiYiJkFKIkYlKUYqIiIlRiVGMComIiU7SEYlRjlGJUYlKiYiJil6N0YlRilGJUYlKiYiJVM5RiVGKkYlRjAiI0tGJUYoRiUsJkYlRjAqJiIiKkYlRipGJUYlRjAsKiomIiZrOyJGJUY5RiVGJSomIiUsPEYlRilGJUYwKiYiJEslRiVGKkYlRiUiIztGMEYwLCZGSkYlRidGJUYwRiU=

We want the positive root c1:=d1*%/4;

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

simplify(series(subs(delta=1-9*x,c1),x,3))

KytJInhHNiIsJComIyIiJCIiKSIiIilGKCNGKCIiJUYqISIiIiIhLCQqJiMiJFYjIiNLRiopRigjRipGLUYqRi5GKiwkKigjIiVobCIkYyNGKkY1RiosJiokKUYoI0YqIiIjRipGKkYtRi5GKkYqRkAtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKkYo

We have determined the singular part of the following series collect(simplify(sqrt(factor(subs(U(x,0)=1/x*Cm(x),x=x/t^2,1/t^2*Ix/4))),symbolic),[Cm,DD],factor)

LCgtSSNDbUc2IjYjKiZJInhHRiUiIiIpSSJ0R0YlIiIjISIiRikqJiNGKUYsRiktSSNEREdGJTYjIiIhRilGKSoqRi9GKSwoKiQpRisiIidGKUYpKiYpRisiIiRGKUYoRilGLSokKUYoRjtGKUYpRilGOkYtKUYoRixGLUYp

Gmexpl:=-3^(7/4)*subs(x=x/9,delta=1-x,c1);

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

simplify(series(Gmexpl,x,5))

Ky9JInhHNiIsJComIyIjRiIiKSIiIikiIiQjRioiIiNGKkYqIiIhIyIkViMiI0tGKiwmKiYjIiRIKCIkYyNGKkYrRiohIiIjRjYiI2tGKkYuLCYjIiZUNiMiJUM1RioqJiMiJSg9I0Y3RipGK0YqRjhGLCwmIyImcEkqIiVbP0YqKiYjIideJHkkIiYlUTtGKkYrRipGOCIiJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YqIiIm

One need to subtract the constant term (ocrresponding to D0/2) to have only the part that is relevant for Cm Hmexpl:=Gmexpl-27*sqrt(3)*(1/8)

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

Let us check at (-1,0) N:=70: i:=1: ser:=coeff(subs(x=1/x,Cmser(N)), x,i): plot([seq([1/n,-coeff(ser,t,n)/3^n*n^(7/4)*GAMMA(-3/4)],n=1..N)],a=0..0.1,b=5..7.6,style=point);

NistJSdDVVJWRVNHNiQ3Ym83JCQiIzUhIiIkIiIhISIiNyQkIiImISIiJCIrLFNuMT0hIio3JCQiK0xMTExMISM1JCIiISEiIjckJCIjRCEiIyQiIiEhIiI3JCQiIiMhIiIkIitmY11lTyEiKjckJCIrbm1tbTshIzUkIiIhISIiNyQkIitIOWRHOSEjNSQiIiEhIiI3JCQiJEQiISIkJCIqKCl5amklISIpNyQkIis2NjY2NiEjNSQiIiEhIiI3JCQiIiIhIiIkIiIhISIiNyQkIisiNDQ0NCohIzYkIjIwKyshNG8nM0AmISM7NyQkIitMTExMJCkhIzYkIiIhISIiNyQkIisjcDJCcCghIzYkIiIhISIiNyQkIitWciZHOSghIzYkIipjZTVnJiEiKTckJCIrbm1tbW0hIzYkIiIhISIiNyQkIiREJyEiJSQiIiEhIiI3JCQiMjErKzUlSE4jKWUhIz0kIjImKioqKipSIzMiKnplISM7NyQkIitjYmJiYiEjNiQiIiEhIiI3JCQiKyYqeTpqXyEjNiQiIiEhIiI3JCQiIiYhIiMkIisqXGEiKjMnISIqNyQkIitpWiE+dyUhIzYkIiIhISIiNyQkIitYWFhYWCEjNiQiIiEhIiI3JCQiKygzRXlNJSEjNiQiKylRMD9EJyEiKjckJCIrbm1tbVQhIzYkIiIhISIiNyQkIiIlISIjJCIiISEiIjckJCIrWVE6WVEhIzYkIitVO1IjUSchIio3JCQiKy9QcS5QISM2JCIiISEiIjckJCIrciZHOWQkISM2JCIiISEiIjckJCIraWVGW00hIzYkIitsPjwqWychIio3JCQiK0xMTExMISM2JCIiISEiIjckJCIrX2shZUEkISM2JCIiISEiIjckJCIlREohIiYkIitYJ1sjeWwhIio3JCQiKi4uLi4kISM1JCIiISEiIjckJCIrcms8VEghIzYkIiIhISIiNyQkIitkRzlkRyEjNiQiK1VkcWBtISIqNyQkIit5eHh4RiEjNiQiIiEhIiI3JCQiKy5GcS1GISM2JCIiISEiIjckJCIrWip5OmojISM2JCIrdCNmJT1uISIqNyQkIitrRDVrRCEjNiQiIiEhIiI3JCQiI0QhIiQkIiIhISIiNyQkIipSQyFSQyEjNSQiMSoqKioqUixaWXgnISM6NyQkIisiUV80USMhIzYkIiIhISIiNyQkIismUiJlREIhIzYkIiIhISIiNyQkIit0c3NzQSEjNiQiK3I/KFEjbyEiKjckJCIrQUFBQUEhIzYkIiIhISIiNyQkIitWSSJSPCMhIzYkIiIhISIiNyQkIit1JmZ3NyMhIzYkIjEqKioqKlJEZ3QnbyEjOjckJCIrTExMJDMjISM2JCIiISEiIjckJCIrRmoiMy8jISM2JCIiISEiIjckJCIiIyEiIyQiKzZdMTFwISIqNyQkIis5VnlnPiEjNiQiIiEhIiI3JCQiK0JwMkI+ISM2JCIiISEiIjckJCIrYEN6Jyk9ISM2JCIrXCJSMiVwISIqNyQkIitfPSY9Jj0hIzYkIiIhISIiNyQkIis9PT09PSEjNiQiIiEhIiI3JCQiKydHOWR5IiEjNiQiK0ZdKT4ocCEiKjckJCIrbGZRYTwhIzYkIiIhISIiNyQkIitKejhDPCEjNiQiIiEhIiI3JCQiK2FfIlxwIiEjNiQiKyJSIUgrcSEiKjckJCIrbm1tbTshIzYkIiIhISIiNyQkIitpVU1SOyEjNiQiIiEhIiI3JCQiK0VLIUhoIiEjNiQiKztXMEVxISIqNyQkIisoZSx0ZSIhIzYkIiIhISIiNyQkIiZEYyIhIickIiIhISIiNyQkIitROllROiEjNiQiK3dwZ1xxISIqNyQkIis6Ojo6OiEjNiQiIiEhIiI3JCQiKzh0YCNcIiEjNiQiIiEhIiI3JCQiK04jKWVxOSEjNiQiK29JQXJxISIqNyQkIitpYEZcOSEjNiQiIiEhIiI3JCQiK0g5ZEc5ISM2JCIiISEiIi0lJkNPTE9SRzYnJSRSR0JHJCIpQyllcSUhIikkIiIhISIiJCIoJz4hXCYhIiktJStfQVRUUklCVVRFRzYjLyUnc291cmNlRyUsbWF0aGRlZmF1bHRHLSUlVklFV0c2JDskIiIhISIiJCIiIiEiIjskIiNdISIiJCIjdyEiIi0mJSZfQVhJU0c2IyIiIjYmLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSpMSU5FU1RZTEVHNiMiIiEtJSpUSElDS05FU1NHNiMiIiEtJS1UUkFOU1BBUkVOQ1lHNiMkIiIhISIiLSYlJl9BWElTRzYjIiIjNiYtJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIiISEiIi0lKkxJTkVTVFlMRUc2IyIiIS0lKlRISUNLTkVTU0c2IyIiIS0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJStBWEVTTEFCRUxTRzYkLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVEiYTYiLyUnZmFtaWx5R1EhNiIvJSVzaXplR1EjMTA2Ii8lJWJvbGRHUSZmYWxzZTYiLyUnaXRhbGljR1EldHJ1ZTYiLyUqdW5kZXJsaW5lR1EmZmFsc2U2Ii8lKnN1YnNjcmlwdEdRJmZhbHNlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjdXRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJmZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ2l0YWxpYzYiLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVEiYjYiLyUnZmFtaWx5R1EhNiIvJSVzaXplR1EjMTA2Ii8lJWJvbGRHUSZmYWxzZTYiLyUnaXRhbGljR1EldHJ1ZTYiLyUqdW5kZXJsaW5lR1EmZmFsc2U2Ii8lKnN1YnNjcmlwdEdRJmZhbHNlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjdXRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJmZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ2l0YWxpYzYiLSUmU1RZTEVHNiMlJlBPSU5URy0lKV9WSVNJQkxFRzYjIiIiLSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIkKychIiItJSlCT1VORFNfWUc2IyQiI10hIiItJS1CT1VORFNfV0lEVEhHNiMkIiUhPSQhIiItJS5CT1VORFNfSEVJR0hURzYjJCIlK04hIiItJSlDSElMRFJFTkc2Ii0lK0FOTk9UQVRJT05HNictJSlCT1VORFNfWEc2IyQiIiEhIiItJSlCT1VORFNfWUc2IyQiIiEhIiItJS1CT1VORFNfV0lEVEhHNiMkIiUrUyEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiUrUyEiIi0lKUNISUxEUkVORzYiRzYi

simplify(series(Hmexpl,x,3));evalf(%);

KylJInhHNiIjIiRWIyIjSyIiIiwmKiYjIiRIKCIkYyNGKCkiIiQjRigiIiNGKCEiIiNGLCIja0YoRjEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYv

KylJInhHNiIkIisrK3YkZighIioiIiIkIisicFIkZWtGJyIiIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIk

at (-2,0) N:=90: i:=2: ser:=coeff(subs(x=1/x,Cmser(N)), x,i): plot([seq([1/n,-coeff(ser,t,n)/3^n*n^(7/4)*GAMMA(-3/4)],n=1..N)],a=0..0.1,b=4..7,style=point);

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Ig==

simplify(series(Hmexpl,x,4));evalf(%);

KytJInhHNiIjIiRWIyIjSyIiIiwmKiYjIiRIKCIkYyNGKCkiIiQjRigiIiNGKCEiIiNGLCIja0YoRjEsJiMiJlQ2IyIlQzVGKComIyIlKD0jRi1GKEYuRihGMkYvLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIiU=

KytJInhHNiIkIisrK3YkZighIioiIiIkIisicFIkZWtGJyIiIyQiKik9bFtlISIpIiIkLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIiU=

Gmexpl

LCQqNCkiIiQjRiUiIiUiIiIsJiosIyIjIikiIilGKClGJSNGKEYnRihJInhHNiIhIiIsJkYoRjJGMEYoRigpLCZGMEYyRihGKCNGKCIiI0YoRigqKiMiIioiIztGKEYkRigsKiomRi1GKClGMEYlRihGKComRiVGKClGMEY3RihGKComIiNDRihGMEYoRihGLUYyRihGQEYyRihGKClGN0Y2RihGQEYoKSwmRihGKComRidGKEYwRihGMkY2RjIpLDAqKkYsRihGNEYoKUYlRjZGKEY+RihGMiomIiNLRigpRjAiIiZGKEYyKiZGN0YoKUYwRidGKEYyKiZGJ0YoRj5GKEYoKiYiJGUiRihGQEYoRigqJiIkZyJGKEYwRihGMkZMRihGNkYoRjNGMiwqKiZGO0YoRj5GKEYoKiYiI0BGKEZARihGMiomIiNbRihGMEYoRihGO0YyRjIsJkYwRihGN0YoRjJGMg==

subs(sqrt(1-x)=s,G^2-Gmexpl^2)

LCYqJClJIkdHNiIiIiMiIiJGKCo0IiInRigpIiIkI0YoRidGKCksJiosIyIjIikiIilGKClGLCNGKCIiJUYoSSJ4R0YmISIiLCZGKEY4RjdGKEYoSSJzR0YmRihGKCoqIyIiKiIjO0YoKUYsI0YsRjZGKCwqKiZGM0YoKUY3RixGKEYoKiZGLEYoKUY3RidGKEYoKiYiI0NGKEY3RihGKEYzRjhGKEZFRjhGKEYnRigpRjdGNkYoLCZGKEYoKiZGNkYoRjdGKEY4RjgsMCoqRjJGKEY6RihGK0YoRkNGKEY4KiYiI0tGKClGNyIiJkYoRjgqJkYnRihGSEYoRjgqJkY2RihGQ0YoRigqJiIkZSJGKEZFRihGKComIiRnIkYoRjdGKEY4Rk5GKEYoKUY5RidGOCksKiomRj5GKEZDRihGKComIiNARihGRUYoRjgqJiIjW0YoRjdGKEYoRj5GOEYnRjgpLCZGN0YoRidGKEYnRjhGOA==

res0:=factor(resultant(numer(%),s^2-(1-x),s)); nops(res0)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVyZXMwR0YoKi4pSSJ4R0YoIiIpIiIiKSwmRjIhIiIqJiIiJUYyRjBGMkYyIiIjRjIpLCZGMEYyRjhGMkY3RjIpLCoqJiIjO0YyKUYwIiIkRjJGMiomIiNARjIpRjBGOEYyRjUqJiIjW0YyRjBGMkYyRj5GNUY3RjIsJkYyRjVGMEYyRjIsOiooIiYlUTtGMilJIkdHRihGN0YyRj9GMkYyKigiJl8iXEYyRkpGMkZDRjJGNSooRk1GMkZKRjJGMEYyRjIqKCIoVyg+NkYyKUZLRjhGMkY/RjJGNSomRklGMkZKRjJGNSooIihLI2ZMRjJGUUYyRkNGMkYyKihGVEYyRlFGMkYwRjJGNSomIil3PTg+RjJGP0YyRjIqJkZQRjJGUUYyRjIqJiIpTlh1ckYyRkNGMkYyKiYiKUdjUmRGMkYwRjJGMkZXRjVGMjcjRi4=

IiIn

algGm:=collect(op(6,res0),G,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdHbUdGKCwoKigiJiVROyIiIiksJkYxISIiSSJ4R0YoRjEiIiRGMSlJIkdHRigiIiVGMUYxKigiKFcoPjZGMUYyRjEpRjgiIiNGMUY0KigiKHBIeSVGMSwmRjFGNComRjlGMUY1RjFGMUYxKSwmRjVGMUY9RjFGPUYxRjE3I0Yu

Gssolens:=[solve(subs(G=sqrt(Gs),algGm),Gs)]

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlHc3NvbGVuc0dGKDckLCQqKCMiJSg9IyIkRyIiIiIsKiomIiIjRjQpSSJ4R0YoRjdGNEY0KiYiIiRGNCksJiomRjtGNClGOUY7RjQhIiIqJkY7RjRGOEY0RjQjRjRGN0Y0RjQqJiIiJUY0RjlGNEZARjdGNEY0KSwmRjRGQEY5RjRGN0ZARjQsJCooRjFGNCwqRjZGQEY6RjRGQ0Y0RjdGQEY0RkVGQEZANyNGLg==

subs(csgn(x)=1,map(series,%,x,3))

NyQrK0kieEc2IiMiJSg9IyIjayIiISwkKiYjIiVobCIkRyIiIiIpIiIkI0YvIiIjRi9GL0YvLCQqJiMiJiRvPiIkYyNGL0YwRi9GL0YzLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRi9GMSsrRiRGJkYpLCRGKyEiIkYvLCRGNUZBRjNGOUYx

evalf(%)

NyQrK0kieEc2IiQiKyt2PTxNISIpIiIhJCIrY042eSkpRigiIiIkIitMcXJKOCEiKCIiIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YsIiIkKytGJEYmRikkIStjTjZ5KSlGKEYsJCErTHFySjhGL0YwRjFGNw==

map(Z->series(sqrt(Z),x),%)

NyQrK0kieEc2IiQiK3c5blhlISIqIiIhJCIrLSt2JGYoRigiIiIkIishcFIkZWtGKCIiIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YsIiIkKytGJEYmRikkISstK3YkZihGKEYsJCErSjVISzshIilGL0YwRjY=

Gssol:=simplify(factor(op(1,Gssolens)),symbolic)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZHc3NvbEdGKCwkKigjIiIiIiRHIkYxLCYqKCIlaGxGMUkieEdGKEYxKSwmIiIkRjEqJkY5RjFGNkYxISIiI0YxIiIjRjFGMSomIiV1VkYxKSwmRjFGO0Y2RjFGPUYxRjFGMUZARjtGMTcjRi4=

Gmfinal:=sqrt(%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShHbWZpbmFsR0YoLCQqKCMiIiIiIztGMSkiIiMjRjFGNEYxKSomLCYqKCIlaGxGMUkieEdGKEYxKSwmIiIkRjEqJkY+RjFGO0YxISIiRjVGMUYxKiYiJXVWRjEpLCZGMUZARjtGMUY0RjFGMUYxRkNGQEY1RjFGMTcjRi4=

Gmexpl

LCQqNCkiIiQjRiUiIiUiIiIsJiosIyIjIikiIilGKClGJSNGKEYnRihJInhHNiIhIiIsJkYoRjJGMEYoRigpLCZGMEYyRihGKCNGKCIiI0YoRigqKiMiIioiIztGKEYkRigsKiomRi1GKClGMEYlRihGKComRiVGKClGMEY3RihGKComIiNDRihGMEYoRihGLUYyRihGQEYyRihGKClGN0Y2RihGQEYoKSwmRihGKComRidGKEYwRihGMkY2RjIpLDAqKkYsRihGNEYoKUYlRjZGKEY+RihGMiomIiNLRigpRjAiIiZGKEYyKiZGN0YoKUYwRidGKEYyKiZGJ0YoRj5GKEYoKiYiJGUiRihGQEYoRigqJiIkZyJGKEYwRihGMkZMRihGNkYoRjNGMiwqKiZGO0YoRj5GKEYoKiYiI0BGKEZARihGMiomIiNbRihGMEYoRihGO0YyRjIsJkYwRihGN0YoRjJGMg==

Hmfinal:=Gmfinal-27*sqrt(3)*(1/8)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShIbWZpbmFsR0YoLCYqKCMiIiIiIztGMSkiIiMjRjFGNEYxKSomLCYqKCIlaGxGMUkieEdGKEYxKSwmIiIkRjEqJkY+RjFGO0YxISIiRjVGMUYxKiYiJXVWRjEpLCZGMUZARjtGMUY0RjFGMUYxRkNGQEY1RjFGMSomIyIjRiIiKUYxKUY+RjVGMUZANyNGLg==

simplify(series(Hmexpl-Hmfinal,x,10)) assuming x>0

KyVJInhHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

ifactor(4374/128)

KiYpLUkhRzYiNiMiIiQiIigiIiIpLUYlNiMiIiMiIichIiI=

ifactor(6561/128)

KiYpLUkhRzYiNiMiIiQiIikiIiIpLUYlNiMiIiMiIighIiI=

Gmspap:=3^7/2^6*(1+3*sqrt(3)*x/2/(1-x)^(3/2))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdHbXNwYXBHRigsJiMiJSg9IyIjayIiIioqIyIlaGwiJEciRjIpIiIkI0YyIiIjRjJJInhHRihGMiksJkY7ISIiRjJGMiNGOEY6Rj5GMjcjRi4=

simplify(series(Gssol-Gmspap,x,10)) assuming x>0

KyVJInhHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

Hmpap:=81/8*sqrt(1/3+x*sqrt(3)/2/(1-x)^(3/2))-27*sqrt(3)*(1/8)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIbXBhcEdGKCwmKiYjIiNGIiM7IiIiKSwmIiM3RjMqKiIjPUYzKSIiJCNGMyIiI0YzSSJ4R0YoRjMpLCZGPSEiIkYzRjMjRjpGPEZARjNGO0YzRjMqJiNGMSIiKUYzRjlGM0ZANyNGLg==

simplify(series(Hmexpl/Hmpap,x,10)) assuming x>0

KydJInhHNiIiIiIiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIiKg==

Hmpap:=27*sqrt(3)/8*(sqrt(1+3*x*sqrt(3)/2/(1-x)^(3/2))-1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIbXBhcEdGKCwkKigjIiNGIiIpIiIiKSIiJCNGMyIiI0YzLCYqJkY2RjMpLCYiIiVGMyoqIiInRjNGNEYzSSJ4R0YoRjMpLCZGPyEiIkYzRjMjRjVGN0ZCRjNGNkYzRjNGM0ZCRjNGMzcjRi4=

simplify(series(Hmexpl/Hmpap,x,10)) assuming x>0

KydJInhHNiIiIiIiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIiKg==

A series in t^3 map(factor,subs(x=x/t^2,t=tV,Qx0))

LCQqKiMiIiIiIiNGJSw0KiYpSSJWRzYiIiIqRiVJInhHRitGJSEiIioqRiZGJSksKiomKUYqIiInRiVGLUYlRi4qKCIiJUYlKUYqIiIkRiVGLUYlRi4qJkY2RiVGLUYlRi5GJUYlRiRGJUYzRiVGLUYlRiUqKEY0RiVGM0YlRi1GJUYuKioiIilGJUYwRiVGN0YlRi1GJUYlKihGJkYlRjBGJUY3RiVGLiooIiM3RiVGN0YlRi1GJUYuKiZGJkYlRjdGJUYlKihGPEYlRjBGJUYtRiVGJSomRjxGJUYtRiVGLkYlKSwmKiRGN0YlRiVGJkYlRjhGLilGLUYmRi5GLg==

subs(V=Vt^(1/3),%)

LCQqKiMiIiIiIiNGJSw0KiYpSSNWdEc2IiIiJEYlSSJ4R0YrRiUhIiIqKkYmRiUpLCoqJilGKkYmRiVGLUYlRi4qKCIiJUYlRipGJUYtRiVGLiomRjVGJUYtRiVGLkYlRiVGJEYlRjNGJUYtRiVGJSooIiInRiVGM0YlRi1GJUYuKioiIilGJUYwRiVGKkYlRi1GJUYlKihGJkYlRjBGJUYqRiVGLiooIiM3RiVGKkYlRi1GJUYuKiZGJkYlRipGJUYlKihGOkYlRjBGJUYtRiVGJSomRjpGJUYtRiVGLkYlKSwmRiZGJUYqRiVGLEYuKUYtRiZGLkYu

map(factor,series(subs(Vt=Vtsing,%),epsilon,2))

LCoqKCMiIiIiI2FGJSwqKiYiI0ZGJUkieEc2IkYlISIiKigiIz1GJSksJiomIiIqRiVGKkYlRixGJUYlI0YlIiIjRiVGKkYlRiUqJkY0RiVGL0YlRixGNEYlRiUpRipGNEYsRiwqLEYkRiUsKEYxRiVGNUYlRjRGLEYlSShlcHNpbG9uR0YrRiVGL0YsRjZGLEYlKiwjRiUiIiRGJSlGPEYzRiUpRjkjRjxGNEYlLCZGJUYsRjFGJUYsRi9GLEYsLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKzYjKiQpRjlGNEYlRiU=

c:=subs(epsilon=1,op(3,%))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJjR0YoLCQqKiMiIiIiIiRGMSlGMiNGMSIiI0YxLCZGMSEiIiomIiIqRjFJInhHRihGMUYxRjcpLCZGOEY3RjFGMUY0RjdGNzcjRi4=

Hx:=3^(5/2)*subs(x=x/9,c);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNIeEdGKCwkKigiIioiIiIsJkYxISIiSSJ4R0YoRjFGMyksJkY0RjNGMUYxI0YxIiIjRjNGMzcjRi4=

Let us check Qser(5)

Ky9JInRHNiIiIiIiIiEqJkkieEdGJEYlSSJ5R0YkRiVGJSwoKiYpRigiIiNGJSlGKUYtRiVGJUYoRiVGKUYlRi0sKiomKUYoIiIkRiUpRilGMkYlRiUqKEYtRiVGLEYlRilGJUYlKihGLUYlRihGJUYuRiVGJUYtRiVGMiwuKiYpRigiIiVGJSlGKUY5RiVGJSooRjJGJUYxRiVGLkYlRiUqKEYyRiVGLEYlRjNGJUYlKiZGLUYlRixGJUYlKigiIidGJUYoRiVGKUYlRiUqJkYtRiVGLkYlRiVGOSwyKiYpRigiIiZGJSlGKUZERiVGJSooRjlGJUY4RiVGM0YlRiUqKEY5RiVGMUYlRjpGJUYlKihGREYlRjFGJUYpRiVGJSooIiM3RiVGLEYlRi5GJUYlKihGREYlRihGJUYzRiVGJSomIiIpRiVGKEYlRiUqJkZNRiVGKUYlRiVGRA==

N:=100: i:=0: ser:=coeff(subs(y=0,convert(Qser(N),polynom)), x,i): plot([seq([1/n,coeff(ser,t,n)/3^n*n^(5/2)*GAMMA(-3/2)],n=1..N)],a=0..0.1,b=7..9,style=point);

NistJSdDVVJWRVNHNiQ3YHE3JCQiIzUhIiIkIiIhISIiNyQkIiImISIiJCIiISEiIjckJCIrTExMTEwhIzUkIitBNygpR0YhIio3JCQiI0QhIiMkIiIhISIiNyQkIiIjISIiJCIiISEiIjckJCIrbm1tbTshIzUkIisnZnVRZCUhIio3JCQiK0g5ZEc5ISM1JCIiISEiIjckJCIkRCIhIiQkIiIhISIiNyQkIis2NjY2NiEjNSQiMSwrK2AlSD1nJiEjOjckJCIiIiEiIiQiIiEhIiI3JCQiKyI0NDQ0KiEjNiQiIiEhIiI3JCQiK0xMTEwkKSEjNiQiK2ApNG1DJyEiKjckJCIrI3AyQnAoISM2JCIiISEiIjckJCIrVnImRzkoISM2JCIiISEiIjckJCIrbm1tbW0hIzYkIjImKioqKipIbFlxbychIzs3JCQiJEQnISIlJCIiISEiIjckJCIyMSsrNSVITiMpZSEjPSQiIiEhIiI3JCQiK2NiYmJiISM2JCIyMCsrXWhAbCsoISM7NyQkIismKnk6al8hIzYkIiIhISIiNyQkIiImISIjJCIiISEiIjckJCIraVohPnclISM2JCIxKioqKioqR2VvW3MhIzo3JCQiK1hYWFhYISM2JCIiISEiIjckJCIrKDNFeU0lISM2JCIiISEiIjckJCIrbm1tbVQhIzYkIismKWZdUXUhIio3JCQiIiUhIiMkIiIhISIiNyQkIitZUTpZUSEjNiQiIiEhIiI3JCQiKy9QcS5QISM2JCIxKioqKipwJ3BFImYoISM6NyQkIityJkc5ZCQhIzYkIiIhISIiNyQkIitpZUZbTSEjNiQiIiEhIiI3JCQiK0xMTExMISM2JCIrXjAlb3IoISIqNyQkIitfayFlQSQhIzYkIiIhISIiNyQkIiVESiEiJiQiIiEhIiI3JCQiKi4uLi4kISM1JCIrN0kpPSN5ISIqNyQkIityazxUSCEjNiQiIiEhIiI3JCQiK2RHOWRHISM2JCIiISEiIjckJCIreXh4eEYhIzYkIisiNFc1InohIio3JCQiKy5GcS1GISM2JCIiISEiIjckJCIrWip5OmojISM2JCIiISEiIjckJCIra0Q1a0QhIzYkIitCJHB3KXohIio3JCQiI0QhIiQkIiIhISIiNyQkIipSQyFSQyEjNSQiIiEhIiI3JCQiKyJRXzRRIyEjNiQiK2ZwQWEhKSEiKjckJCIrJlIiZURCISM2JCIiISEiIjckJCIrdHNzc0EhIzYkIiIhISIiNyQkIitBQUFBQSEjNiQiKylleEQ2KSEiKjckJCIrVkkiUjwjISM2JCIiISEiIjckJCIrdSZmdzcjISM2JCIiISEiIjckJCIrTExMJDMjISM2JCIrQS06ayIpISIqNyQkIitGaiIzLyMhIzYkIiIhISIiNyQkIiIjISIjJCIiISEiIjckJCIrOVZ5Zz4hIzYkIitiLzE1IykhIio3JCQiK0JwMkI+ISM2JCIiISEiIjckJCIrYEN6Jyk9ISM2JCIiISEiIjckJCIrXz0mPSY9ISM2JCIrPT4+XiMpISIqNyQkIis9PT09PSEjNiQiIiEhIiI3JCQiKydHOWR5IiEjNiQiIiEhIiI3JCQiK2xmUWE8ISM2JCIrYE5EKUcpISIqNyQkIitKejhDPCEjNiQiIiEhIiI3JCQiK2FfIlxwIiEjNiQiIiEhIiI3JCQiK25tbW07ISM2JCIxLCsrOjIjPUspISM6NyQkIitpVU1SOyEjNiQiIiEhIiI3JCQiK0VLIUhoIiEjNiQiIiEhIiI3JCQiKyhlLHRlIiEjNiQiK1FfT18kKSEiKjckJCImRGMiISInJCIiISEiIjckJCIrUTpZUTohIzYkIiIhISIiNyQkIis6Ojo6OiEjNiQiKypveC1RKSEiKjckJCIrOHRgI1wiISM2JCIiISEiIjckJCIrTiMpZXE5ISM2JCIiISEiIjckJCIraWBGXDkhIzYkIitLVCllUykhIio3JCQiK0g5ZEc5ISM2JCIiISEiIjckJCIrLzJYMzkhIzYkIiIhISIiNyQkIisqKSkpKSkpUSIhIzYkIitcKmUlSCUpISIqNyQkIis5SScpcDghIzYkIiIhISIiNyQkIiteOE5eOCEjNiQiIiEhIiI3JCQiK0xMTEw4ISM2JCIrT1lCXiUpISIqNyQkIit1JSp5OjghIzYkIiIhISIiNyQkIisqSCwoKUgiISM2JCIiISEiIjckJCIrI0deP0ciISM2JCIrUCY0OVopISIqNyQkIismeUFlRSIhIzYkIiIhISIiNyQkIiREIiEiJSQiIiEhIiI3JCQiKyx6Y003ISM2JCIxLCsrISlROiFcKSEjOjckJCIrJj43Jj43ISM2JCIiISEiIjckJCIreCM+Wz8iISM2JCIiISEiIjckJCIqPncvPiIhIzUkIismZTl3XSkhIio3JCQiKyllcWs8IiEjNiQiIiEhIiI3JCQiKylwIXppNiEjNiQiIiEhIiI3JCQiKyhHRCVcNiEjNiQiKjc+Ul8pISIpNyQkIitPT09PNiEjNiQiIiEhIiI3JCQiKzFiZkI2ISM2JCIiISEiIjckJCIrNjY2NjYhIzYkIitHJ3kiUiYpISIqNyQkIisqNCwqKTQiISM2JCIiISEiIjckJCIrQWwmcDMiISM2JCIiISEiIjckJCIrPClvXzIiISM2JCIxLCsrVi9cYCYpISM6NyQkIisoeUhRMSIhIzYkIiIhISIiNyQkIit6OmpfNSEjNiQiIiEhIiI3JCQiK25tbVQ1ISM2JCIreCwlcGMpISIqNyQkIitOeSM0LiIhIzYkIiIhISIiNyQkIitqIjMvLSIhIzYkIiIhISIiNyQkIiosLCwsIiEjNSQiK1JNZ3omKSEiKjckJCIiIiEiIyQiIiEhIiItJSZDT0xPUkc2JyUkUkdCRyQiKUMpZXElISIpJCIiISEiIiQiKCc+IVwmISIpLSUrX0FUVFJJQlVURUc2Iy8lJ3NvdXJjZUclLG1hdGhkZWZhdWx0Ry0lJVZJRVdHNiQ7JCIiISEiIiQiIiIhIiI7JCIjcSEiIiQiIyEqISIiLSYlJl9BWElTRzYjIiIiNiYtJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIiISEiIi0lKkxJTkVTVFlMRUc2IyIiIS0lKlRISUNLTkVTU0c2IyIiIS0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJiUmX0FYSVNHNiMiIiM2Ji0lJkNPTE9SRzYmJSRSR0JHJCIiISEiIiQiIiEhIiIkIiIhISIiLSUqTElORVNUWUxFRzYjIiIhLSUqVEhJQ0tORVNTRzYjIiIhLSUtVFJBTlNQQVJFTkNZRzYjJCIiISEiIi0lK0FYRVNMQUJFTFNHNiQtSSNtaUc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjY1USJhNiIvJSdmYW1pbHlHUSE2Ii8lJXNpemVHUSMxMDYiLyUlYm9sZEdRJmZhbHNlNiIvJSdpdGFsaWNHUSV0cnVlNiIvJSp1bmRlcmxpbmVHUSZmYWxzZTYiLyUqc3Vic2NyaXB0R1EmZmFsc2U2Ii8lLHN1cGVyc2NyaXB0R1EmZmFsc2U2Ii8lK2ZvcmVncm91bmRHUShbMCwwLDBdNiIvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XTYiLyUnb3BhcXVlR1EmZmFsc2U2Ii8lK2V4ZWN1dGFibGVHUSZmYWxzZTYiLyUpcmVhZG9ubHlHUSZmYWxzZTYiLyUpY29tcG9zZWRHUSZmYWxzZTYiLyUqY29udmVydGVkR1EmZmFsc2U2Ii8lK2ltc2VsZWN0ZWRHUSZmYWxzZTYiLyUscGxhY2Vob2xkZXJHUSZmYWxzZTYiLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lLG1hdGh2YXJpYW50R1EnaXRhbGljNiItSSNtaUc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjY1USJiNiIvJSdmYW1pbHlHUSE2Ii8lJXNpemVHUSMxMDYiLyUlYm9sZEdRJmZhbHNlNiIvJSdpdGFsaWNHUSV0cnVlNiIvJSp1bmRlcmxpbmVHUSZmYWxzZTYiLyUqc3Vic2NyaXB0R1EmZmFsc2U2Ii8lLHN1cGVyc2NyaXB0R1EmZmFsc2U2Ii8lK2ZvcmVncm91bmRHUShbMCwwLDBdNiIvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XTYiLyUnb3BhcXVlR1EmZmFsc2U2Ii8lK2V4ZWN1dGFibGVHUSZmYWxzZTYiLyUpcmVhZG9ubHlHUSZmYWxzZTYiLyUpY29tcG9zZWRHUSZmYWxzZTYiLyUqY29udmVydGVkR1EmZmFsc2U2Ii8lK2ltc2VsZWN0ZWRHUSZmYWxzZTYiLyUscGxhY2Vob2xkZXJHUSZmYWxzZTYiLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lLG1hdGh2YXJpYW50R1EnaXRhbGljNiItJSZTVFlMRUc2IyUmUE9JTlRHLSUpX1ZJU0lCTEVHNiMiIiItJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiQrJyEiIi0lKUJPVU5EU19ZRzYjJCIkUyIhIiItJS1CT1VORFNfV0lEVEhHNiMkIiUhPSQhIiItJS5CT1VORFNfSEVJR0hURzYjJCIlP00hIiItJSlDSElMRFJFTkc2Ii0lK0FOTk9UQVRJT05HNictJSlCT1VORFNfWEc2IyQiIiEhIiItJSlCT1VORFNfWUc2IyQiIiEhIiItJS1CT1VORFNfV0lEVEhHNiMkIiUrUyEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiUrUyEiIi0lKUNISUxEUkVORzYiRzYi

series(Hx,x,1);evalf(%);

KydJInhHNiIiIioiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGLQ==

KydJInhHNiIkIiIqIiIhRictSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGLg==

N:=100: i:=1: ser:=coeff(subs(y=0,convert(Qser(N),polynom)), x,i): plot([seq([1/n,coeff(ser,t,n)/3^n*n^(5/2)*GAMMA(-3/2)],n=1..N)],a=0..0.1,b=11..13,style=point);

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series(Hx,x,2);evalf(%);

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N:=130: i:=2: ser:=coeff(subs(y=0,convert(Qser(N),polynom)), x,i): plot([seq([1/n,coeff(ser,t,n)/3^n*n^(5/2)*GAMMA(-3/2)],n=1..N)],a=0..0.1,b=14..17,style=point);

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NiI=

series(Hx,x,3);evalf(%);

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JSFH