Maple session for the paper "An elementary solution of Gessel's walks in the quadrant"

The number of n step walsk endiing at (i,j) B:=proc(n,i,j) option remember: if n<0 or j<0 or i<0 then 0 elif n=0 and i=0 and j=0 then 1 else B(n-1,i-1,j-1)+B(n-1,i+1,j)+B(n-1,i-1,j)+B(n-1,i+1,j+1)fi: end: The series Q(x,y;t), up to order N (in t) Qser:=proc(N) collect(add(add(add(B(n,i,j)*x^i*y^j*t^n,i=0..n),j=0..n),n=0..N),t): end: Qser(3);

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The series R(x;t) Rser:=proc(n) collect(t*(subs(y=0,Qser(n))-subs(x=0,y=0,Qser(n))),t):end: Rser(3);

LCgqJiwmKiRJInhHNiIiIiQiIiJGJiIiJkYpSSJ0R0YnIiIlRikqJkYmIiIjRitGKEYpKiZGJkYpRitGLkYp

The series S(y;t) Sser:=proc(n) collect(t*(1+y)*subs(x=0,Qser(n)),t,factor): end: Sser(10);

LC4qKCwmIiIiRiVJInlHNiJGJUYlLC4qJiIjVUYlKUYmIiImRiVGJSomIiQ/JUYlKUYmIiIlRiVGJSomIiV3PUYlKUYmIiIkRiVGJSomIiUqKVtGJSlGJiIiI0YlRiUqJiIlXyEpRiVGJkYlRiUiJS8hKUYlRiUpSSJ0R0YnIiM2RiVGJSooRiRGJSwsKiYiIzlGJUYvRiVGJSomIiQ3IkYlRjNGJUYlKiYiJChRRiVGN0YlRiUqJiIkUihGJUYmRiVGJSIkI3lGJUYlKUY9IiIqRiVGJSooRiRGJSwqKiZGLEYlRjNGJUYlKiYiI0lGJUY3RiVGJSomIiN0RiVGJkYlRiUiIyYpRiVGJSlGPSIiKEYlRiUqKEYkRiUsKComRjhGJUY3RiVGJSomIiIpRiVGJkYlRiVGPkYlRiUpRj1GLEYlRiUqKEYkRiUsJkYmRiVGOEYlRiUpRj1GNEYlRiUqJkYkRiVGPUYlRiU=

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

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The main functional equation eqQ:=-Ker*Q+1-t/x*(1+1/y)*Q0y -t/x/y*(Qx0-Q00);

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n:=10:normal(series(subs(Q=Qser(n),Q0y=subs(x=0,Qser(n)),Qx0=subs(y=0,Qser(n)),Q00=subs(x=0,y=0,Qser(n)),eqQ),t,n+1));

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Roots of the kernel factor([solve(Ker,y)]);

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Y0sol:=op(2,%); Y1sol:=op(1,%%);

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LCQqKiMiIiIiIiNGJUkidEc2IiEiIilJInhHRihGJkYpLCoqJkYnRiVGKkYlRilGJ0YpRitGJSokKSomLCpGLUYlKihGJkYlRidGJUYrRiVGJUYnRiVGK0YpRiUsKkYtRiVGMkYpRidGJUYrRilGJUYkRiVGJUYlRiU=

Symmetric functions of the roots expand(factor(Y0sol+Y1sol));expand(factor(Y0sol*Y1sol));

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Y0ser:=proc(N) convert(simplify(normal(series(Y0sol,t,N+1)),symbolic),polynom);end: Y1ser:=proc(N) -1-1/x^2+1/(x*t)-Y0ser(N): end: Y0ser(4); Y1ser(3);

LCgqJkkieEc2IiEiIkkidEdGJSIiIkYoKigpRiQiIiNGJiwmKiRGKkYoRihGKEYoRigpRidGK0YoRigqKClGJCIiJEYmLCgqJClGJCIiJUYoRigqJkYxRihGKkYoRihGKEYoRigpRidGMUYoRig=

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The constant term in Y0^n+Y1^n for n from 0 to 4 do expand(factor(Y0sol^n+Y1sol^n));od;

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LCgiIiIhIiIqJClJInhHNiIiIiNGJEYkKiZJInRHRihGJEYnRiRGIw==

LCwiIiJGIyooIiIjRiNJInRHNiIhIiJJInhHRidGKEYoKiQpRikiIiVGKEYjKihGJUYjRiZGKClGKSIiJEYoRigqJilGJkYlRigpRilGJUYoRiM=

LDIiIiIhIiIqKCIiJEYjSSJ0RzYiRiRJInhHRihGJEYjKihGJkYjRidGJClGKUYmRiRGIyooRiZGIylGJyIiI0YkKUYpRi5GJEYkKiQpRikiIidGJEYkKihGJkYjRidGJClGKSIiJkYkRiMqKEYmRiNGLUYkKUYpIiIlRiRGJComKUYnRiZGJEYrRiRGIw==

LDoiIiJGIyooIiIlRiNJInRHNiIhIiJJInhHRidGKEYoKihGJUYjRiZGKClGKSIiJEYoRigqKCIiJ0YjKUYmIiIjRigpRilGMEYoRiMqKEYlRiNGJkYoKUYpIiImRihGKCooIiIpRiNGL0YoKUYpRiVGKEYjKihGJUYjKUYmRixGKEYrRihGKCokKUYpRjZGKEYjKihGJUYjRiZGKClGKSIiKEYoRigqKEYuRiNGL0YoKUYpRi5GKEYjKihGJUYjRjlGKEYzRihGKComKUYmRiVGKEY3RihGIw==

What remains after canceling the kernel: the "kernel equation" eqker:=S(y)+R(x)-x*y;

LCgtSSJTRzYiNiNJInlHRiUiIiItSSJSR0YlNiNJInhHRiVGKComRixGKEYnRighIiI=

Let's check the four kernel equations (in fact we check only two, thanks to the symmetry in x and 1/x) n:=10: eqkerser:=normal(series(subs(S(y)=Sser(n), R(x)=Rser(n), eqker),t,n+1));

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normal(series(subs({x=x,y=Y0ser(n)},convert(eqkerser,polynom)),t,10));

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normal(series(normal(subs({x=x*convert(Y0ser(n),polynom),y=convert(Y1ser(n),polynom)},convert(eqkerser,polynom))),t,n/2));

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After forming two symmetric expressions in Y0 and Y1, and extracting their non-negative parts, one obtains the following equation in R(x) and R(1/x) eqR:=collect(R(1/x)^2+(1/x+S(0)+2*x-1/t)*R(1/x)+R(x)*R(1/x)+R(x)^2+(x+S(0)+2/x-1/t)*R(x)+S(0)/t-(1/t-1/x)*S(0)-2*D(R)(0)-(1/t-x)*S(0)+1,[R(x),R(1/x), S]);

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We multiply by (R(x)-R(1/x)-x+1/x) to get rid of the product R(x)R(1/x) eqRsep:=collect(eqR*(R(x)-R(1/x)-x+1/x),[R,x]);

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The term with no R's CT:=collect(subs(R(x)=0,R(1/x)=0,eqRsep),x);

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The series P of the paper Pexpr:=op(1,eqRsep)+op(2,eqRsep)+op(3,eqRsep)+op(1,CT)+op(2,CT);

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Expnding P at order 0 normal(subs(R(0)=0,series(Pexpr,x,3)));

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The equation for R(x) eqcatR:=Pexpr-convert(%,polynom)-2*(D(R))(0)*x;

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let's check n:=12: normal(series(subs(R(x)=Rser(n+2), S(0)=subs(y=0,Sser(n+1)), D(R)(0)=coeff(Rser(n+2),x,1), ((D@@2)(R))(0)=2*coeff(Rser(n+2),x,2), eqcatR),t,n+1));

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Diff. w. r.t. x0 deqcatR:=diff(subs(R(x)=Rx,eqcatR),Rx);

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Three roots are finite at t=0 factor(subs(t=0,Rx=0,S(0)=0,numer(deqcatR)));

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Dif. w.r.t. x dxeqcatR:=collect(diff(subs(R(x)=Rx,eqcatR),x),Rx,normal);

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We can identify the roots X0 X1 and X2 thanks to the following identity factor(deqcatR+x^2*dxeqcatR);

LCQqLiwmSSJ4RzYiIiIiISIiRidGJywmRiVGJ0YnRidGJywoKiZJInRHRiZGJ0YlIiIjRi1GLEYtRiVGKEYnLCgqJi1JIlNHRiY2IyIiIUYnRiVGJ0YnKiZJI1J4R0YmRidGJUYnRidGJ0YnRidGJSEiI0YsRihGKA==

The discriminant Disc(x) discR:=(factor(discrim(subs(R(x)=Rx,eqcatR),Rx)));

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

Checking symmetry factor(discR/subs(x=1/x,discR));

IiIi

Let us write discR as a polynomial in x+1/x=s reste:=discR: discRs:=0;

IiIh

for i from 6 to 0 by -1 do co:=coeff(reste,x,i); discRs:=discRs+co*s^i: reste:=reste-co*(x+1/x)^i: od: let's check factor(subs(s=x+1/x,discRs)/discR);

IiIi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The discriminant vanishes for s=2, s=-2 and s=1/(2t) algp1:=numer(factor(subs(s=2,S(0)=S0,D(R)(0)=Rp0,((D@@2)(R))(0)=Rp20, discRs)));

LF5xKihJJFJwMEc2IiIiIkkjUzBHRiUiIiVJInRHRiVGKCIiKSooRiQiIiNGJ0YsRilGKCEjSyooSSVScDIwR0YlRiZGJyIiJEYpRihGKComRidGKEYpRihGKiomRiRGMEYpRigiI0sqKkYkRiZGL0YmRidGJkYpRighI3MqKEYkRiZGJ0YwRilGMCIjNyooRiRGJkYnRixGKUYoISNrKiZGJ0YoRilGMCEiKSooRiRGLEYnRiZGKUYwIiNTKiZGJEYsRilGKCIjJyoqKEYkRiZGJ0YsRilGMCIjayomRi9GLEYpRighI0YqKEYvRiZGJ0YsRilGMCIiJyooRi9GJkYnRiZGKUYoRjUqJkYnRihGKUYsRiYqJkYnRjBGKUYwIiM/KiZGJ0YsRilGKEYtKiZGJEYsRilGMCEjJyoqKEYkRiZGL0YmRilGMCIjPSooRiRGJkYnRixGKUYsISIjKihGJEYmRidGJkYpRjBGOSomRiRGJkYpRihGPyooRi9GJkYnRiZGKUYwIiNzKiZGJ0YwRilGLCEjOyomRidGLEYpRjBGQSomRiRGLEYpRiwiIzgqKEYkRiZGJ0YmRilGLEY7KiZGJEYmRilGMCEkIz4qKEYvRiZGJ0YmRilGLCEiJyomRi9GJkYpRjAhIyEqKiZGJ0YwRilGJkYsKiZGJ0YsRilGLCEjQSomRidGJkYpRjAhJC8iKiRGKUYoRjMqKEYkRiZGJ0YmRilGJkYsKiZGJEYmRilGLCIkZSIqJkYvRiZGKUYsIiNPKiZGJ0YsRilGJiEiJSomRidGJkYpRiwiJE8iKiRGKUYwRkwqJkYkRiZGKUYmISNXKiZGL0YmRilGJkZnbyokRidGLEYmKiZGJ0YmRilGJiEjVSokRilGLCIjUEYkRihGJ0YoRilGZ28=

algm1:=numer(factor(subs(s=-2,S(0)=S0,D(R)(0)=Rp0,((D@@2)(R))(0)=Rp20,discRs)));

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

alg3:=numer(factor(subs(s=1/2/t,S(0)=S0,D(R)(0)=Rp0,((D@@2)(R))(0)=Rp20,discRs)));

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

Let us check these three polynomial equations n:=6: normal(series(subs(S0=subs(y=0,Sser(n)), Rp0=coeff(Rser(n),x,1), Rp20=2*coeff(Rser(n),x,2), algp1),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIo

n:=6: normal(series(subs(S0=subs(y=0,Sser(n)), Rp0=coeff(Rser(n),x,1), Rp20=2*coeff(Rser(n),x,2), algm1),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIo

n:=6: normal(series(subs(S0=subs(y=0,Sser(n)), Rp0=coeff(Rser(n),x,1), Rp20=2*coeff(Rser(n),x,2), alg3),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIo

Let us now construct an equation for each of the three series collect(factor(algp1-algm1),Rp20,factor);

LD4qKEkidEc2IiIiIywmKiZJI1MwR0YlIiIiRiRGKkYmRipGKkYqSSVScDIwR0YlRioiI3MqJkYpIiIlRiQiIiQhIzsqKEYpRiZJJFJwMEdGJUYqRiRGLyIkRyIqJkYyRiZGJEYvISQjPiomRilGL0YkRiYhI0sqJkYpRiZGJEYvRjMqKEYpRipGMkYqRiRGJkYwKiZGMkYqRiRGLyEkJVEqJkYpRiZGJEYqISIpKiZGKUYqRiRGJiIkcyMqJEYkRi9GNSomRjJGKkYkRiohIykpRikiIilGJEY9

Rp20sol:=isolate(%,Rp20);

L0klUnAyMEc2IiwkKigsPComSSNTMEdGJCIiJUkidEdGJCIiJCIjOyooSSRScDBHRiQiIiJGKSIiI0YrRiwhJEciKiZGL0YxRitGLCIkIz4qJkYpRixGK0YxIiNLKiZGKUYxRitGLEYyKihGL0YwRilGMEYrRjFGLSomRi9GMEYrRiwiJCVRKiZGKUYxRitGMCIiKSomRilGMEYrRjEhJHMjKiRGK0YsRjQqJkYvRjBGK0YwIiMpKUYpISIpRitGPEYwRishIiMsJiomRilGMEYrRjBGMUYwRjAhIiIjRjAiI3M=

algm1bis:=numer(factor(subs(Rp20sol,algm1)));

LGhwKiZJI1MwRzYiIiIpSSJ0R0YlIiIoIiIlKihJJFJwMEdGJSIiIkYkIiInRidGKCIjayooRisiIiNGJEYpRidGKCIkKUcqJkYkRihGJ0YtIiM7KiZGJEYtRidGKEYuKihGK0YsRiQiIiZGJ0YtIiQjPiooRitGLEYkRilGJ0YoIiR3JiomRitGKUYnRighJUc8KihGK0YwRiQiIiRGJ0YtRjkqJkYkRi1GJ0Y2IiNPKiZGJEY2RidGLUY3KiZGJEYpRidGKEYxKiZGK0Y9RidGKCElN3AqKEYrRixGJEYpRidGNiIkNyQqKEYrRixGJEY9RidGLSIlXzYqKEYrRjBGJEYwRidGNiIkSyUqJkYrRjBGJ0YoISZvLiIqJkYkRjZGJ0YpIiNfKiZGJEYpRidGNiEkcykqJkYkRj1GJ0YtRjkqJkYrRj1GJ0Y2ISRrKSooRitGLEYkRj1GJ0YpIiQvJCooRitGLEYkRjBGJ0Y2IiVPUiomRitGLEYnRihGQyooRitGMEYkRixGJ0YpIiRXIiomRitGMEYnRjYhJVdQKiZGJEYpRidGPUZNKiZGJEY9RidGKSElaz8qJkYkRjBGJ0Y2IiUvTiokRidGKEY7KihGK0YsRiRGMEYnRj0iJG8iKihGK0YsRiRGLEYnRikiJWdMKiZGK0YsRidGNiElJypbKiZGK0YwRidGPSEkVyIqJkYkRj1GJ0YwRj8qJkYkRjBGJ0Y9ISVDNyomRiRGLEYnRikiJTtLKiRGJ0Y2ISU7PyooRitGLEYkRixGJ0YwIiNbKiZGK0YsRidGPUY3KiZGJEYwRidGLEYzKiZGJEYsRidGMCEkZyIqJEYnRj1GWSomRitGLEYnRiwhIilGJEYpRichIiU=

alg3bis:=numer(factor(subs(Rp20sol,alg3)));

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

Equation for S0 factor(resultant(algm1bis,alg3bis,Rp0));

LCQqLEkidEc2IiIjRCwmRiQiIiUhIiIiIiIiIiksJkYkRihGKkYqRissJiomSSNTMEdGJUYqRiRGKiIiI0YqRipGKyxOKiZGL0YrRiQiIigiI0YqJkYvRjNGJCIiJyIkMyIqJkYvRjZGJCIiJiIkKj0qJkYvRihGJEYzISQpRyomRi9GOUYkRihGOiomRi9GKEYkRjkhJF8jKiZGLyIiJEYkRjYhJHcmKiZGL0YwRiRGMyEkNyYqJkYvRihGJEZBIiQ8IiomRi9GQUYkRighJC8mKiZGL0YwRiRGOSIkQycqJkYvRipGJEY2RkQqJEYkRjMhJGMjKiZGL0ZBRiRGMCIjWComRi9GMEYkRkEhJDckKiZGL0YqRiRGKCIkNyoqJEYkRjlGQiomRi9GMEYkRioiIzUqJkYvRipGJEYwISNnKiRGJEZBIiNbRi9GKkYkRilGKiExd0gxUyVlaGwk

algS0:=collect(op(6,%),S0,factor);

LDQqJkkjUzBHNiIiIilJInRHRiUiIigiI0YqJkYkRihGJyIiJyIkMyIqJkYkRitGJyIiJiIkKj0qJkYkRi5GJyIiJUYvKihGJyIiJCwoKiRGJ0YxIiNLKiRGJyIiIyIjRyEjOCIiIkY7RiRGMSEiKiooRidGOCwoRjUiI2tGNyIjYyEiJkY7RjtGJEYzRjwqKEYnRjssKiokRidGKyIkYyNGNSEkNyRGNyIkYyJGQUY7RjtGJEY4ISIjKiosJkY3RjYhIiJGO0Y7LChGN0YxRichIidGO0Y7RjssKEY3RjFGJ0YrRjtGO0Y7RiRGO0ZLKiZGJ0Y7LCpGREZFRjUiJHcmRjchI1tGO0Y7RjtGSw==

The equation for Q(0,0) algQ0:=collect(subs(S0=t*Q0, algS0/t),Q0,factor);

LDoqJkkidEc2IiIjOUkjUTBHRiUiIikiI0YqJkYkIiM3RiciIigiJDMiKiZGJCIjNUYnIiInIiQqPSomRiRGKEYnIiImRjEqKEYkRjAsKCokRiQiIiUiI0sqJEYkIiIjIiNHISM4IiIiRj1GJ0Y3ISIqKihGJEY3LChGNiIja0Y5IiNjISImRj1GPUYnIiIkRj4qKEYkRjosKiokRiRGMCIkYyNGNiEkNyRGOSIkYyJGQ0Y9Rj1GJ0Y6ISIjKiosJkY5RjghIiJGPUY9LChGOUY3RiQhIidGPUY9Rj0sKEY5RjdGJEYwRj1GPUY9RidGPUZORkchJGMjRjYhJHcmRjkiI1tGTkY9

Equation for Rp0: we must choose the correct factor res:=factor(resultant(algm1bis,algS0,S0)):nops(res);

IiIl

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

NiQiKTtzeDsqJEkidEc2IiIjXw==

n:=10:normal(series(subs(Rp0=coeff(Rser(n),x,1),op(3,res)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiND

n:=10:normal(series(subs(Rp0=coeff(Rser(n),x,1),op(4,res)),t,n+1));

KzFJInRHNiIiJSc0JSIiISIoT2ZcJCIiIyIrdyQqUSdlIyIiJSItUzVgJ3ltKCIiJyIwKHArTHZiTD4iIikhMWc8JUgpcD1zaSIjNS1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIzc=

The correct factor (choose it correctly!) is squared op(3,res);

KiQsPComSSRScDBHNiIiIiVJInRHRiYiIiciJEgoKiZGJSIiJEYoRikiJHMqKiZGJSIiI0YoRikhJHkkKiZGJUYsRihGJyIkViMqJkYlIiIiRihGKSEkISkqKiZGJUYvRihGJyEkOCYqJEYoRikhJFYkKiZGJUY0RihGJyIkKlsqJkYlRi9GKEYvIiNGKiRGKEYnIiNQKiZGJUY0RihGLyEjVSokRihGLyEiIkYlRjRGLw==

R'(0) has thus degree 4 only algRp0:=collect(op(1,op(3,res)),Rp0,factor);

LCwqJkkkUnAwRzYiIiIlSSJ0R0YlIiInIiRIKCooRidGJiwmKiRGJyIiI0YmIiIiRi5GLkYkIiIkIiRWIyooRidGLSwoKiRGJ0YmIiM5RiwiIz4hIiJGLkYuRiRGLSEjRioqLCZGLCIjP0Y2Ri5GLiwoRiwiIihGJyEiJ0YuRi5GLiwoRixGPEYnRihGLkYuRi5GJEYuRjYqJkYnRi0sKEYzIiRWJEYsISNQRi5GLkYuRjY=

Finally, an equation for Rp20 res1:=factor(resultant(algp1-algm1,algRp0,Rp0)):nops(res1);

IiIk

op(1,res1),op(2,res1);

NiQiJyNmNSIqJEkidEc2IiIiKg==

res2:=(factor(resultant(op(3,res1),algS0,S0))): nops(res2);

IiIm

op(1,res2),op(2,res2),op(3,res2);

NiUiNSwpR3AwZmFtZEAiKiRJInRHNiIiJDQiKiQsKCokRiUiIiUiJGMjKiRGJSIiIyIkQyMiIiJGMEYu

n:=10:normal(series(subs(Rp20=2*coeff(Rser(n),x,2),op(4,res2)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM2

n:=10:normal(series(subs(Rp20=2*coeff(Rser(n),x,2),op(5,res2)),t,n+1));

KzFJInRHNiIiPSsrKysrK0gvSDUvYGM3IiIhIkArKytjWS9aY11TeVgmKilRNSIiIyJCK2doXiZwSCozcWpLNmtRKltPIiIlIkRXRCEqPUV6IVs/LUEkb3goeWxldCIiJyJHbjVpdzJ2Plp3U0FxXWNZTXk9NiIiKSJJM0Uob2EnW3YmUSE+KnorXSRveiI0JHk5IiM1LUkiT0clKnByb3RlY3RlZEc2IyIiIiIjNw==

algRp20:=collect(op(4,res2),Rp20,factor);

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

JSFH

JSFH

t2Tm:=(T-1)*(T+3)^3/256/T^3;

LCQqKCwmSSJURzYiIiIiISIiRidGJywmRiVGJyIiJEYnRipGJSEiJCNGJyIkYyM=

with(gfun):algeqtoseries(numer(t^2-t2Tm),t,T,10);

NyQrL0kidEc2IiIiIiIiISIiJSIiIyIjT0YoIiQnUiIiJyIlKXklIiIpLUkiT0clKnByb3RlY3RlZEc2I0YmIiM1LDAhIiRGJiomLUknUm9vdE9mRzYkRjFJKF9zeXNsaWJHRiU2IywmKiRJI19aR0Y5IiIkRiYhJUc8RiZGJkYkI0YpRj9GJiomRjdGKUYkI0YoRj8jISIiRigqJEYkRikiIyUpKiZGN0YmRiQjRi5GPyMhI1BGPyomRjdGKUYkI0YzRj8jIiIoRiwtRjA2IyokRiQjIiM2Rj9GJg==

Tmser:=op(1,%);

Ky9JInRHNiIiIiIiIiEiIiUiIiMiI09GJyIkJ1IiIiciJSl5JSIiKS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNQ==

res:=numer(factor(expand(subs(t=sqrt(t2Tm),algRp0))));

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

Choosing the correct factor normal(series(subs(T=Tmser,Rp0=subs(x=0,diff(Rser(n),x)),op(1,res)),t)); normal(series(subs(T=Tmser,Rp0=subs(x=0,diff(Rser(n),x)),op(2,res)),t));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

KytJInRHNiIiJ1dARSIiISEoJ0hmQiIiIyEpb1YoKT0iIiUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

Rp0Tm:=factor(solve(op(1,res),Rp0));

KigsJkkiVEc2IiIiIiEiIkYmRiYsKCokRiQiIiNGJkYkISInIiNARiZGJiwmRiRGJiIiJEYmISIk

Then S(0)= t Q(0,0) algQ0Tm:=collect(op(1,numer(factor(subs(t=sqrt(t2Tm),algQ0)))),Q0,factor);

LCgqKCwmSSJURzYiIiIiIiIkRiciIicsJkYlRichIiJGJ0YnSSNRMEdGJiIiI0YnKihGJUYoRiRGKEYsRiciJGMjKiZGJUYoLCoqJEYlRihGJyokRiVGLSEiJEYlIiNGISIqRidGJyElQzU=

factor(discrim(algQ0Tm,Q0));

LCQqKCwoKiRJIlRHNiIiIiMiIiJGJiIiJyEiJEYpRigsJkYmRikiIiRGKUYqRiZGLSIlJzQl

factor([solve(factor(subs(T=Z^2,algQ0Tm)),Q0)]);

NyQsJCoqSSJaRzYiIiIkLCoqJEYlRiciIiIqJEYlIiIjRidGJUYnISIkRipGKiwmRiVGKiEiIkYqRi8sJkYrRipGJ0YqRi0hI0ssJCoqRiVGJywqRilGKkYrRi1GJUYnRidGKkYqLCZGJUYqRipGKkYvRjBGLSIjSw==

We take Z=1 + O(t), then Q00 is given by: Q0Zm:=op(2,%);

LCQqKkkiWkc2IiIiJCwqKiRGJEYmIiIiKiRGJCIiIyEiJEYkRiZGJkYpRiksJkYkRilGKUYpISIiLCZGKkYpRiZGKUYsIiNL

t2Zm:=factor(subs(T=Z^2,t2Tm));

LCQqKiwmSSJaRzYiIiIiISIiRidGJywmRiVGJ0YnRidGJywmKiRGJSIiI0YnIiIkRidGLUYlISInI0YnIiRjIw==

algeqtoseries(numer(t^2-t2Zm),t,Z,20,true): Zmser:=subs(RootOf(_Z^2-1)=1,op(1,%));

KzlJInRHNiIiIiIiIiEiIiNGJyIjOyIiJSIkbSIiIiciJU0+IiIpIiZ3UyMiIzUiJytISiIjNyIoRzo+JSIjOSIpW1tVZEYoIipHPl4rKSIjPS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjPw==

Expression Rp0 in terms of Z= sqrt T Rp0Zm:=subs(T=Z^2,Rp0Tm);

KigsJiokSSJaRzYiIiIjIiIiISIiRihGKCwoKiRGJSIiJUYoRiQhIiciI0BGKEYoLCZGJEYoIiIkRighIiQ=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Finally, Rp20 = 2 t coeff(x^2, Q(x,0)) r1:=numer(factor(expand(subs(Rp20=2*t*Q2,t=sqrt(t2Zm),algRp20)))): nops(%);

IiIl

op(1,r1);

KiQqKiwmSSJaRzYiIiIiISIiRidGJywmRiVGJ0YnRidGJywmKiRGJSIiI0YnIiIkRidGLUYlISInI0YnRiw=

normal(series(subs(Z=Zmser,Q2=coeff(subs(y=0,Qser(n)),x,2),op(2,r1)),t)); normal(series(subs(Z=Zmser,Q2=coeff(subs(y=0,Qser(n)),x,2),op(3,r1)),t)); normal(series(subs(Z=Zmser,Q2=coeff(subs(y=0,Qser(n)),x,2),op(4,r1)),t));

KytJInRHNiIiJSM+KSIiISInJ1wxIiIiIyIoTyIqXCIiIiUtSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

KytJInRHNiIiPmtPaiYqZjxPcVtseXorJ28iIiEiQHcmR1pNSVVcUkNdcklPIylmIiIjIkIvZiYqR251TSxpOSlmInpXSUEkIiIlLUkiT0clKnByb3RlY3RlZEc2IyIiIiIiJw==

Q2Zm:=factor(solve(op(3,r1),Q2));

LCQqLEkiWkc2IiIiJCwmRiQiIiIhIiJGKEYoLC4qJEYkIiInRigqJEYkIiImISIjKiRGJCIiJUYpKiRGJCIiIyIiKEYkRjNGKEYoRigsJkYkRihGKEYoRiksJkYyRihGJkYoISInIiQ3Jg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

We start with R'(0), which has degree 4 algRp0s:=expand(subs(t=sqrt(s),algRp0));

LDwqKCIkSCgiIiIpSSRScDBHNiIiIiVGJSlJInNHRigiIiRGJUYlKigiJHMqRiUpRidGLEYlRipGJUYlKigiJFYjRiVGL0YlKUYrIiIjRiVGJSooIiR5JEYlKUYnRjNGJUYqRiUhIiIqKCIkOCZGJUY2RiVGMkYlRjcqKCIkISkqRiVGJ0YlRipGJUY3KigiI0ZGJUY2RiVGK0YlRiUqKCIkKltGJUYnRiVGMkYlRiUqJiIkViRGJUYqRiVGNyooIiNVRiVGJ0YlRitGJUY3KiYiI1BGJUYyRiVGJUYnRiVGK0Y3

with(algcurves): p1:=factor(parametrization( algRp0s,s,Rp0,T));

NyQsJCoqIiUvQiIiIiksJiomRiVGJkkiVEc2IkYmRiYiJiQ+KiohIiIiIiRGJiwmRilGJiInKFwsIkYtRi0pLCZGKUYmIiYiRyMqRi1GLkYtRiYsJCooRiVGJiwoKiYiKDslM2BGJilGKiIiI0YmRiYqJiIqI2YxSVpGJkYqRiZGLSIsOCYpelMwIkYmRiZGJ0YtRiY=

It takes some fiddling p2:=factor(subs(T=T/2304,p1));

NyQsJCoqIiUvQiIiIiksJkkiVEc2IkYmIiYkPioqISIiIiIkRiYsJkYpRiYiJyhcLCJGLEYsKSwmRilGJiImIkcjKkYsRi1GLEYmLCQqKEYlRiYsKCokKUYpIiIjRiZGJiomIicpSDAjRiZGKUYmRiwiLDgmKXpTMCJGJkYmRidGLEYm

p3:=factor(subs(T=T+99193,p2));

NyQsJCoqIiUvQiIiIilJIlRHNiIiIiRGJiwmRihGJkYlISIiRiwpLCZGKEYmIiU3cEYmRipGLEYmLCQqKEYlRiYsKCokKUYoIiIjRiZGJiomRi9GJkYoRiZGLCIpW18jZiJGJkYmRidGLEYm

6912/2304;

IiIk

p4:=factor(subs(T=T*2304,p3));

NyQqKClJIlRHNiIiIiQiIiIsJkYlRihGKCEiIkYqKSwmRiVGKEYnRihGJ0YqKiYsKCokKUYlIiIjRihGKComRidGKEYlRihGKkYnRihGKEYkRio=

with(gfun):algeqtoseries(numer(t^2-op(1,p4)),t,T,6);

NyQrK0kidEc2IiIiIiEiIyEiKSIiISEjPSIiIy1JIk9HJSpwcm90ZWN0ZWRHNiNGJiIiJSwoKiYtSSdSb290T2ZHNiRGLkkoX3N5c2xpYkdGJTYjLCYqJClJI19aR0Y1IiIkRiZGJiIjRkYmRiYpRiQjRitGPEYmRiYqJiIiJ0YmKUYkRitGJkYmLUYtNiMqJClGJCMiIilGPEYmRiY=

p5:=factor(subs(T=1/T,p4));

NyQsJCooSSJURzYiIiIiLCZGJUYnRichIiJGKSksJkYnRicqJiIiJEYnRiVGJ0YnRi1GKUYpKiYsKComRi1GJylGJSIiI0YnRidGLEYpRidGJ0YnRiVGJw==

algeqtoseries(numer(t^2-op(1,p5)),t,T,10);

NyQrL0kidEc2IiIiIiIiIyIiKSIiJSIjIykiIiciJFcqRigiJkg7IiIjNS1JIk9HJSpwcm90ZWN0ZWRHNiNGJiIjNywsKiYtSSdSb290T2ZHNiRGMUkoX3N5c2xpYkdGJTYjLCYqJiIjRkYmKUkjX1pHRjgiIiRGJkYmRiZGJkYmKUYkI0YnRkAhIiJGJiooRitGJilGNkYnRiZGQUYmRkMqKCNGKEZARiZGNkYmKUYkI0YpRkBGJkZDKiYjRiZGQEYmKUYkRidGJkZDLUYwNiMqJClGJEZHRiZGJg==

Tser:=op(1,%);

Ky9JInRHNiIiIiIiIiMiIikiIiUiIyMpIiInIiRXKkYnIiZIOyIiIzUtSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzc=

Let us check normal(series(subs(T=Tser,op(2,p5)),t,16));

Ky9JInRHNiIiIiIiIiMiIiYiIiUiI1AiIiciJEskIiIpIiVWTCIjNS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIjNw==

coeff(Rser(10),x,1);

LCwqJiIlVkwiIiIpSSJ0RzYiIiM1RiVGJSomIiRLJEYlKUYnIiIpRiVGJSomIiNQRiUpRiciIidGJUYlKiYiIiZGJSlGJyIiJUYlRiUqJClGJyIiI0YlRiU=

t2T:=op(1,p5);

LCQqKEkiVEc2IiIiIiwmRiRGJkYmISIiRigpLCZGJkYmKiYiIiRGJkYkRiZGJkYsRihGKA==

Rp0T:=op(2,p5);

KiYsKComIiIkIiIiKUkiVEc2IiIiI0YmRiYqJkYlRiZGKEYmISIiRiZGJkYmRihGJg==

Then Q(0,0) res:=(numer(factor(subs(t=sqrt(t2T),algQ0)))):nops(res);

IiIk

normal(series(subs(T=Tser,Q0=subs(x=0,y=0,Qser(n)),op(1,res)),t)); normal(series(subs(T=Tser,Q0=subs(x=0,y=0,Qser(n)),op(2,res)),t)); normal(series(subs(T=Tser,Q0=subs(x=0,y=0,Qser(n)),op(3,res)),t));

KyVJInRHNiIhIiIiIiE=

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

KytJInRHNiIiIiIiIiEiIiciIiMhI0AiIiUtSSJPRyUqcHJvdGVjdGVkRzYjRiVGJw==

op(2,res);

LDwqJiIkViMiIiIpSSJURzYiIiIoRiVGJSomRiRGJSlGJyIiJ0YlISIiKigiI0ZGJUkjUTBHRihGJSlGJyIiJUYlRi0qJiIjIilGJSlGJyIiJkYlRi0qJiIkPCJGJUYxRiVGJSomKUYwIiIjRiVGJ0YlRiUqKCIjPUYlRjBGJSlGJ0Y7RiVGJSomIiIqRiUpRiciIiRGJUYlKigiIilGJUYwRiVGJ0YlRiUqJiIjTEYlRj5GJUYtRjBGJSomIiM2RiVGJ0YlRi1GJUYt

algQ0T:=collect(op(2,res),Q0,factor);

LCgqJilJI1EwRzYiIiIjIiIiSSJUR0YmRihGKCooRiVGKCwmRilGKEYoISIiRigpLCZGKEYoKiYiIiRGKEYpRihGKEYwRihGLCooRitGKCwqKiYiIipGKClGKUYwRihGKComRjRGKClGKUYnRihGLEYvRihGKEYoRihGLUYoRig=

factor(discrim(algQ0T,Q0));

LCQqKCwmSSJURzYiIiIiRichIiJGJyksKComIiIkRicpRiUiIiNGJ0YnKiYiIidGJ0YlRidGKEYnRihGLkYnKSwmRidGJyomRixGJ0YlRidGJ0YsRidGKA==

factor([solve(factor(subs(T=Z/(1-Z+Z^2),algQ0T)),Q0)]);

NyQsJCoqLCZJIlpHNiIiIiJGKCEiIkYoLCoqJClGJiIiJEYoRigqJkYtRigpRiYiIiNGKEYoKiZGLUYoRiZGKEYpRihGKEYoKSwmRiZGKEYoRihGLUYoKSwoKiRGL0YoRihGJkYpRihGKEYtRilGKSosRiVGKCwqRitGKEYuRilGMUYoRihGKEYoRjJGKEY0RilGJkYp

t2Z:=factor(subs(T=Z/(1-Z+Z^2),t2T));

KiopLCgqJClJIlpHNiIiIiMiIiJGKkYnISIiRipGKiIiJEYqRidGKiksJkYnRipGKkYqIiInRispLCZGJ0YqRipGK0YpRis=

Zser:=op(3,algeqtoseries(numer(t^2-t2Z),t,Z,10));

Ky9JInRHNiIiIiIiIiMiIigiIiUiI28iIiciJGcoIiIpIiUnPSoiIzUtSSJPRyUqcHJvdGVjdGVkRzYjRiUiIzc=

Let us write R'(0) in terms of Z Rp0Z:=factor(solve(op(1,numer(factor(expand(subs(t=sqrt(t2Z),algRp0))))),Rp0));

KihJIlpHNiIiIiIsLCokKUYjIiIlRiVGJSomIiImRiUpRiMiIiRGJSEiIiomIiIqRiUpRiMiIiNGJUYlKiZGK0YlRiNGJUYuRiVGJUYlKSwoKiRGMUYlRiVGI0YuRiVGJUYtRi4=

Finally, R''(0), or rather Q2 = coeff(x^2,Q(x,0)) r1:=numer(factor(expand(subs(Rp20=2*t*Q2,t=sqrt(t2Z),algRp20)))): nops(%);

IiIm

op(1,r1),op(2,r1);

NiQiJGMjKiQpKiopLCgqJClJIlpHNiIiIiMiIiJGLkYrISIiRi5GLiIiJEYuRitGLiksJkYrRi5GLkYuIiInRi8pLCZGK0YuRi5GL0YtRi8jRi5GLUYu

normal(series(subs(Z=Zser,Q2=coeff(subs(y=0,Qser(n)),x,2),op(3,r1)),t)); normal(series(subs(Z=Zser,Q2=coeff(subs(y=0,Qser(n)),x,2),op(4,r1)),t)); normal(series(subs(Z=Zser,Q2=coeff(subs(y=0,Qser(n)),x,2),op(5,r1)),t));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIn

KylJInRHNiIiIiIiIiFGJSIiJS1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJw==

KytJInRHNiIiJyIpcEEiIiEiKGglKjMjIiIjIilAIXksKSIiJS1JIk9HJSpwcm90ZWN0ZWRHNiMiIiIiIic=

Q2Z:=factor(solve(op(3,r1),Q2));

LCQqLEkiWkc2IiIiIiwuKiQpRiQiIidGJkYmKiYiIiRGJilGJCIiJUYmRiYqJkYuRiYpRiRGLEYmRiYqJkYsRiYpRiQiIiNGJkYmKiZGLkYmRiRGJiEiIkYmRiZGJiksJkYkRiZGJkY1RixGJiksJkYkRiZGJkYmRixGJiksKCokRjJGJkYmRiRGNUYmRiZGKkY1RjU=

JSFH

JSFH

For parity reasons, we consider Q(xt,0) (which is even) Rser(3);subs(x=x*t,%);

LCgqJiwmKiQpSSJ4RzYiIiIkIiIiRioqJiIiJkYqRidGKkYqRiopSSJ0R0YoIiIlRipGKiomKUYnIiIjRiopRi5GKUYqRioqJkYnRiopRi5GMkYqRio=

LCgqJiwmKiYpSSJ0RzYiIiIkIiIiKUkieEdGKEYpRipGKiooIiImRipGJ0YqRixGKkYqRiopRiciIiVGKkYqKiYpRidGLkYqKUYsIiIjRipGKiomRiZGKkYsRipGKg==

the equation for R(x) eqcatR;

LDIqJC1JIlJHNiI2I0kieEdGJiIiJCIiIiomLCgtSSJTR0YmNiMiIiFGKiokRighIiJGKSokSSJ0R0YmRjJGMkYqRiQiIiNGKiomLC4tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjRiVGLyEiIyokRihGNUYyKiZGNEYyRihGKkYqKiZGLUYqRjRGMkYyKiYsJkYtRjVGM0YyRipGKEYyRioqJEYoRj9GNUYqRiRGKkYqKiZGLUYqRihGNUYyKiYsKEY4RjVGQkYqRjJGKkYqRihGKkYqKiZGOEYqRihGMkY/KiYsKCooRi1GKkY4RipGNEYqRjUqJi0tLUkjQEBHRjs2JEY6RjVGPkYvRipGNEYqRipGOEYyRipGNEYyRjIqJkY4RipGKEYqRj8=

We use here van Hoeij's parametrization algQxtZm:=collect(factor(expand(numer(factor(subs(S(0)=t*Q0Zm,D(R)(0)=Rp0Zm, ((D@@2)(R))(0)=2*t*Q2Zm,R(x)=t*(Qxt-Q0Zm),x=x*t,t=sqrt(t2Zm),eqcatR))))),Qxt,factor);

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

collect(algQxtZm,[Qxt,x],factor);

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

Let's check. Careful since with this parametrization, Zm=O(1) n:=7: normal(series(subs(Qxt=subs(x=x*t,y=0,Qser(2*n+2)),t=sqrt(t2Zm),algQxtZm),Z=1,n));

KyUsJkkiWkc2IiIiIkYmISIiLUkiT0clKnByb3RlY3RlZEc2I0YmIiIo

Ker;

LCYiIiJGIyomSSJ0RzYiRiMsJiomSSJ4R0YmRiMsJkYjRiNJInlHRiZGI0YjRiMqJkYpISIiLCZGI0YjKiRGK0YtRiNGI0YjRiNGLQ==

We eliminate x to obtain an equation for S(y) or Q(0,y) r2:=numer(factor(resultant(numer(subs(Qxt=Q0Zm+Rxt/t,Rxt=t*x*y-S(y),algQxtZm)), subs(x=x*t,numer(Ker)),x))): nops(r2);

IiIo

op(1,r2),op(2,r2),op(3,r2),op(4,r2),op(5,r2),op(6,r2);

NighIiIqJCwmSSJaRzYiIiIiRihGKCIiKSokLCYqJEYmIiIjRigiIiRGKCIjPSokSSJ0R0YnRilJInlHRicsJkYoRihGMkYo

Let's check n:=8: normal(series(subs(Z=Zmser,S(y)=Sser(n),op(7,r2)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiIq

We move back to Q(0,y) and to Z instead of t r3:=factor(subs(S(y)=t*(1+y)*Qy,op(7,r2))):nops(r3);

IiIk

op(1,r3),op(2,r3);

NiQqJEkidEc2IiIiJCwmIiIiRihJInlHRiVGKA==

n:=8: normal(series(subs(Z=Zmser,Qy=subs(x=0,Qser(n)),op(3,r3)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM1

r4:=op(3,r3): degree(r4,t);

IiM1

r5:=numer(factor(subs(t=sqrt(t2Zm),r4))): nops(r5);

IiIm

op(1,r5),op(2,r5),op(3,r5),op(4,r5);

NiYqJCwmSSJaRzYiIiIiISIiRiciIiQqJCwmRiVGJ0YnRidGKSokLCYqJEYlIiIjRidGKUYnIiIqLCZGJ0YnSSJ5R0YmRic=

n:=12: normal(series(subs(Z=Zmser,Qy=subs(x=0,Qser(n)),op(5,r5)),t,n+1));

KyVJInRHNiItSSJPRyUqcHJvdGVjdGVkRzYjIiIiIiM5

The result is a square nops(op(5,r5));

IiIj

algQyZm:=collect(op(1,op(5,r5)),Qy,factor);

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x in terms of T and an additional series U xTUm:=factor(solve(16*T^2*(U^2-T)-x*(-2*T+U+U*T)*(-9*T+U^2+8*T*U+T^2-T*U^2),x));

LCQqLCIjOyIiIilJIlRHNiIiIiNGJSwmKiQpSSJVR0YoRilGJSEiIkYnRiVGJSwoKiZGJ0YlRi1GJUYlKiZGKUYlRidGJUYuRi1GJUYuLCwqJkYnRiVGLEYlRi4qJEYmRiVGJSooIiIpRiVGJ0YlRi1GJUYlRitGJSomIiIqRiVGJ0YlRi5GLkYu

xZUm:=factor(subs(T=Z^2,xTUm));

LCQqMCIjOyIiIilJIlpHNiIiIiVGJSwmSSJVR0YoRiVGJyEiIkYlLCZGK0YlRidGJUYlLCgqJkYrRiUpRiciIiNGJUYlKiZGMUYlRjBGJUYsRitGJUYsLCoqJkYrRiVGJ0YlRiUqJEYwRiVGJUYrRiUqJiIiJEYlRidGJUYsRiwsKkY0RiVGNUYsRitGLEY2RixGLEYs

We plug the rational expression of x in the equation found for Q(xt,0) res:=numer(factor(subs(x=xZUm,t=sqrt(t2Zm),algQxtZm))): nops(res);

IiIl

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

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degree(op(3,res),Qxt),degree(op(4,res),Qxt);;

NiQiIiIiIiM=

QxtZUm:=factor(solve(op(3,res),Qxt));

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

The expression of Q(xt,0) in terms of x and Zm1 n:=4: normal(series(subs(y=0,x=x*t,t=sqrt(t2Zm),Z=1+Zm1,Qser(2*n)),Zm1,n+1));

Ky9JJFptMUc2IiIiIiIiISwmRiVGJSomI0YlIiIjRiVJInhHRiRGJUYlRiUsKCMiIiYiIiUhIiIqJiMiIiRGL0YlRitGJUYwKiYjRiVGL0YlKUYrRipGJUYlRiosKiNGJSIiKUYwKiZGNUYlRitGJUYlKiYjIiIoRjlGJUY2RiVGMComRjhGJSlGK0YzRiVGJUYzLCwjIiM2RjlGJSomI0Y9IiM7RiVGK0YlRiUqJiMiI0RGRUYlRjZGJUYlKiYjRi5GOUYlRj9GJUYwKiYjRiVGRUYlKUYrRi9GJUYlRi8tSSJPRyUqcHJvdGVjdGVkRzYjRiVGLg==

How should we choose U ? there are three values, only one is correct algeqtoseries(subs(Z=1+Zm1,numer(xZUm-x)),Zm1,U,3);

NyQrK0kkWm0xRzYiLCQqKCIiJSIiIiwmSSJ4R0YlRilGKSEiIkYpRitGLEYpRiwqKEYrRiwsKComIiImRikpRisiIiNGKUYpKiYiIzxGKUYrRilGLCIjNUYpRilGKkYsIiIhKihGK0YsLCwqJClGK0YoRilGKSomIiM3RikpRisiIiRGKUYsKiYiI0FGKUYxRilGKUYzRixGMEYpRikpRipGPkYsRiktSSJPRyUqcHJvdGVjdGVkRzYjRilGMisrRiQtSSdSb290T2ZHNiRGREkoX3N5c2xpYkdGJTYjLCoqJkYqRikpSSNfWkdGSUYyRilGKSooRjJGKUYrRilGT0YpRixGKUYpRitGKUY2LCQqKCNGKUYyRiksMComRkdGKUY9RilGKSooIiIoRilGR0YpRjFGKUYsKiRGPUYpRiwqKEZXRilGK0YpRkdGKUYpRi9GKSomRjJGKUZHRilGLComRj5GKUYrRilGLEYpKUYqRjJGLEYsRiksJCosI0YpRihGKSwyKihGMkYpRkdGKUY6RilGKSooIiIqRilGR0YpRj1GKUYsKiZGMkYpRjpGKUYsKiZGXW9GKUY9RilGKSooIiIpRilGK0YpRkdGKUYpKiZGPkYpRkdGKUYsKiZGYW9GKUYrRilGLEY+RilGKUYqRiwsKkZYRikqJkY+RilGMUYpRixGZW5GKUYpRixGLEYrRilGLEYyRkJGPg==

Not the first one map(factor,op(1,algeqtoseries(subs(Z=1+Zm1,numer(xZUm-x)),Zm1,U,5)));

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

normal(series(subs(U=%,Z=1+Zm1,QxtZUm),Zm1));

KytJJFptMUc2IiwkKigsJiomIiIjIiIiSSJ4R0YkRipGKkYpISIiRipGK0YsLCZGK0YqRipGLEYsRixGLCooLCgqJClGKyIiJEYqRioqJiIiKUYqRitGKkYsRjRGKkYqRitGLEYtRiwiIiEsJCoqI0YqRilGKiwsKiZGKUYqKUYrIiIlRipGKiomIiM/RipGMUYqRiwqJiIjZEYqKUYrRilGKkYqKiZGQEYqRitGKkYsIiM+RipGKkYrRiwpRi1GMkYsRipGKi1JIk9HJSpwcm90ZWN0ZWRHNiNGKkYp

The other two values are formal power series in Zm1 Usolens:=factor([allvalues(op(2,algeqtoseries(subs(Z=1+Zm1,numer(xZUm-x)),Zm1,U,6)))]);

NyQrL0kkWm0xRzYiIiIiIiIhRiZGJiwkKiYjRiYiIiVGJkkieEdGJUYmRiYiIiQsJCooI0YmIiIpRiYsJkYsRiZGKyEiIkYmRixGJkYmRissJCoqI0YmIiM7RiYsJkYsRiZGLUYzRiZGMkYmRixGJkYmIiImLUkiT0clKnByb3RlY3RlZEc2I0YmIiInKzFGJComLCZGJkYmRixGJkYmLCZGLEYmRiZGM0YzRicqJiksJiomIiIjRiZGLEYmRiZGJkYzRkdGJilGQkYtRjNGJiwkKi4jRiZGR0YmRkVGJkZBRiYsKCokKUYsRkdGJkYmKiZGOUYmRixGJkYzRi1GJkYmRixGJilGQkY5RjNGM0ZHLCQqKkYqRiYsMiokKUYsIiIoRiZGJiomRjFGJilGLEY+RiZGMyomIiNURiYpRixGOUYmRiYqJiIkPSJGJilGLEYrRiZGMyomIiMnKkYmKUYsRi1GJkYmKiZGK0YmRk5GJkYmKiYiI0dGJkYsRiZGM0ZWRiZGJkYsRiYpRkJGVkYzRjNGLSwkKipGMEYmLDgqJClGLCIjNUYmRiYqJiIjOEYmKUYsIiIqRiZGMyomIiNyRiYpRixGMUYmRiYqJiIkSSNGJkZVRiZGMyomIiQ1J0YmRlhGJkYmKiYiJWk4RiZGZW5GJkYzKiYiJV46RiZGaG5GJkYmKiYiJDwoRiZGW29GJkYzKiZGMUYmRk5GJkYmKiYiIyQpRiZGLEYmRiZGN0YzRiZGLEYmKUZCRmlvRjNGM0YrLCQqKkY2RiYsPiokKUYsRmdvRiZGJiomIiM9RiYpRiwiIzdGJkYzKiYiJFciRiYpRiwiIzZGJkYmKiYiJHonRiZGZG9GJkYzKiYiJSU0I0YmRmhvRiZGJiomIiUrWEYmRlxwRiZGMyomIiUqKnpGJkZVRiZGJiomIiY6VCJGJkZYRiZGMyomIiYmKXoiRiZGZW5GJkYmKiYiJnlHIkYmRmhuRiZGMyomIiUqSCVGJkZbb0YmRiYqJiIkJz1GJkZORiZGMyomIiRDI0YmRixGJkYzIiNPRiZGJkYsRiYpRkJGZ3FGM0YzRjlGOkY+

normal(series(subs(U=op(1,%),Z=1+Zm1,QxtZUm),Zm1));

Ky1JJFptMUc2IiIiIiIiISwmRiVGJSomI0YlIiIjRiVJInhHRiRGJUYlRiUsKCMiIiYiIiUhIiIqJiMiIiRGL0YlRitGJUYwKiYjRiVGL0YlKUYrRipGJUYlRiosKiNGJSIiKUYwKiZGNUYlRitGJUYlKiYjIiIoRjlGJUY2RiVGMComRjhGJSlGK0YzRiVGJUYzLUkiT0clKnByb3RlY3RlZEc2I0YlRi8=

normal(series(subs(U=op(2,%%),Z=1+Zm1,QxtZUm),Zm1));

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

Hence the correct value of U is the one with CT equal to 1 Umser:=op(1,Usolens);

Ky9JJFptMUc2IiIiIiIiIUYlRiUsJComI0YlIiIlRiVJInhHRiRGJUYlIiIkLCQqKCNGJSIiKUYlLCZGK0YlRiohIiJGJUYrRiVGJUYqLCQqKiNGJSIjO0YlLCZGK0YlRixGMkYlRjFGJUYrRiVGJSIiJi1JIk9HJSpwcm90ZWN0ZWRHNiNGJSIiJw==

We now look at the equation for Q(y) yTVm:=(1-T+3*V+V*T)/V^2/(3+V+T-V*T);

KigsKiomSSJURzYiIiIiSSJWR0YmRidGJ0YlISIiKiYiIiRGJ0YoRidGJ0YnRidGJylGKCIiI0YpLCpGJEYpRiVGJ0YoRidGK0YnRik=

yZVm:=factor(subs(T=Z^2,yTVm));

LCQqKCwqKiZJIlZHNiIiIiIpSSJaR0YnIiIjRihGKCokRilGKCEiIiomIiIkRihGJkYoRihGKEYoRigpRiZGK0YtLCpGJUYoRixGLUYmRi1GL0YtRi1GLQ==

res:=numer(factor(subs(y=yZVm,algQyZm))): nops(res);

IiIk

op(1,res);

ISIi

QyZVm:=factor(solve(op(2,res),Qy));

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

degree(op(3,res),Qy);

IiIj

n:=4: normal(series(subs(x=0,t=sqrt(t2Zm),Z=1+Zm1,Qser(2*n)),Zm1,n+1));

Ky9JJFptMUc2IiIiIiIiISwmKiYjRiUiIiNGJUkieUdGJEYlRiVGJUYlRiUsJiomRilGJSlGK0YqRiVGJSMiIiYiIiUhIiJGKiwqKiYjRjAiIilGJSlGKyIiJEYlRiUqJiNGJUYxRiVGLkYlRjIqJkYvRiVGK0YlRjIjRiVGNkYyRjgsLComIyIiKEY2RiUpRitGMUYlRiUqJkYpRiVGN0YlRjIqJiMiI0QiIztGJUYuRiVGMiomI0Y4RjFGJUYrRiVGJSMiIzZGNkYlRjEtSSJPRyUqcHJvdGVjdGVkRzYjRiVGMA==

solensVm:=algeqtoseries(subs(Z=1+Zm1,numer(y-yZVm)),Zm1,V,4);

NyUrLUkkWm0xRzYiIiIjISIiLCQqJEkieUdGJUYnRiciIiEsJCooIyIiIkYmRi8sJiokKUYqRiZGL0YvRi9GJ0YvRjJGJ0YvRi8sJCooI0YvIiIlRi8sKCokKUYqIiIkRi9GL0YxRidGJkYvRi9GOUYnRidGJi1JIk9HJSpwcm90ZWN0ZWRHNiNGL0Y6Ky1GJEYpRissJEYtRidGLywkKihGNUYvLCgqJClGKkY2Ri9GL0YxRi9GJkYnRi9GOUYnRidGJiwkKigjRi8iIilGLywqKiZGJkYvKUYqIiInRi9GL0ZERicqJkY2Ri9GMkYvRi8iIiZGJ0YvRkVGJ0YnRjpGO0Y2Ky1GJEYuRi8sJComRjVGL0YqRi9GL0YmLCZGSEYnKiZGNUYvRjJGL0YvRjosKComI0ZPIiM7Ri9GOUYvRi8qJiNGOkZYRi9GKkYvRicjRi9GWEYvRjZGO0ZP

normal(series(subs(V=solensVm[1],Z=1+Zm1,QyZVm),Zm1,10)); normal(series(subs(V=solensVm[2],Z=1+Zm1,QyZVm),Zm1,10)); normal(series(subs(V=solensVm[3],Z=1+Zm1,QyZVm),Zm1,10));

KydJJFptMUc2IiwkKigiIiMiIiIsJkkieUdGJEYoRighIiJGKCksJkYoRihGKkYoRidGK0YoRistSSJPRyUqcHJvdGVjdGVkRzYjRigiIiE=

Ky1JJFptMUc2IiwkKigiIiMiIiJJInlHRiRGKCksJkYoRihGKUYoRichIiJGKEYsLCQqKEYpRiwsKiokKUYpIiIkRihGKComIiInRigpRilGJ0YoRiwqJkYnRihGKUYoRihGKEYoRihGKkYsRiwiIiEsJCoqI0YoRidGKEY1RiwsLiokKUYpIiImRihGKComIiIlRigpRilGQEYoRigqJiIjOUYoRjFGKEYsKiZGPkYoRjVGKEYoKiZGQEYoRilGKEYoRihGKEYoRipGLEYsRigsJCoqI0YoRkBGKEYxRiwsMiomRidGKClGKSIiKEYoRigqJkZARigpRilGNEYoRigqJkYyRihGPUYoRiwqJiIiKkYoRkFGKEYsKiYiIzVGKEYxRihGLComRjJGKEY1RihGLEZFRihGJ0YoRihGKkYsRixGJy1JIk9HJSpwcm90ZWN0ZWRHNiNGKEYy

Ky1JJFptMUc2IiIiIiIiISwmKiYjRiUiIiNGJUkieUdGJEYlRiVGJUYlRiUsJiomRilGJSlGK0YqRiVGJSMiIiYiIiUhIiJGKiwqKiYjRjAiIilGJSlGKyIiJEYlRiUqJiNGJUYxRiVGLkYlRjIqJkYvRiVGK0YlRjIjRiVGNkYyRjgtSSJPRyUqcHJvdGVjdGVkRzYjRiVGMQ==

hence the correct V is V = O(t) Vmser:=op(3,solensVm);

Ky1JJFptMUc2IiMiIiIiIiNGJiwkKiYjRiYiIiVGJkkieUdGJEYmRiZGJywmI0YmIiIpISIiKiZGKkYmKUYsRidGJkYmIiIkLCgqJiMiIiYiIztGJilGLEYzRiZGJiomI0YzRjhGJkYsRiZGMCNGJkY4RiZGKy1JIk9HJSpwcm90ZWN0ZWRHNiNGJkY3

JSFH

JSFH

JSFH

JSFH