This session contains various checks for the formulas and derivations in the paper. It points to other specific Maple session for certain tasks. All are available on the authors' webpages. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwqSSJ4R0YoIiIiKiRGLyEiIkYwKiZGL0YwSSJ5R0YoRjBGMComRi9GMkY0RjJGMDcjRi4=

K:=expand(t*x*y*liste-x*y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCwqKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGMkYxRjZGMUYxKiZGMEYxRjZGMUYxKiZGM0YxRjZGMSEiIkYwRjE3I0Yu

Xsol:=[solve(K,x)];Ysol:=[solve(K,y)];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCosIyIiIiIiI0YyLCZJInlHRihGMiokKSwqKigiIiVGMilJInRHRihGM0YyKUY1IiIkRjIhIiIqKCIiKUYyRjtGMilGNUYzRjJGPyooRjpGMkY7RjJGNUYyRj8qJEZCRjJGMkYxRjJGMkYyRjxGP0Y1Rj8sJkYyRjJGNUYyRj9GMiwkKixGMUYyLCZGNUY/RjZGMkYyRjxGP0Y1Rj9GRUY/Rj83I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ0R0YoISIiKUkieEdGKEYzRjUsKiomRjRGMkY2RjJGNUY0RjVGN0YyKiQpLC4qJilGNEYzRjIpRjciIiVGMkYyKihGM0YyRj5GMkY2RjJGNSooRjNGMkY0RjIpRjciIiRGMkY1KiRGPkYyRjIqKEYzRjJGNEYyRjdGMkY1KiRGNkYyRjJGMUYyRjJGMkYyLCQqKkYxRjIsKkY5RjJGOkYyRjRGMkY3RjVGMkY0RjVGNkY1RjU3I0Yu

Jy:=y/t/(1+y)^2+t/y*(1+y)^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKeUdGKCwmKihJInlHRigiIiJJInRHRighIiIpLCZGMUYxRjBGMSIiI0YzRjEqKEYyRjFGMEYzRjRGMUYxNyNGLg==

factor(subs(y=Ysol[1],Jy)-subs(y=Ysol[2],Jy));

IiIh

Ku:=subs(x=t+t^2*(u+1/u),K);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNLdUdGKCwsKihJInRHRigiIiIpLCZGMEYxKiYpRjAiIiNGMSwmSSJ1R0YoRjEqJEY4ISIiRjFGMUYxRjZGMSlJInlHRihGNkYxRjEqKEYwRjFGMkYxRjxGMUYxKiZGMEYxRjxGMUYxKiZGM0YxRjxGMUY6RjBGMTcjRi4=

with(gfun): map(factor,map(allvalues,algeqtoseries(numer(Ku),t,y,2)));

NyQrKUkidEc2IiokSSJ1R0YlISIiRigqKiwmKiYiIiQiIiIpRiciIiNGLUYtRi9GLUYtLCZGJ0YtRi1GKEYoLCZGJ0YtRi1GLUYoRi5GKCIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YtRi0rKUYkRidGKCwkKiosJiomRi9GLUYuRi1GLUYsRi1GLUYuRi1GMEYoRjFGKEYoRjJGM0Yt

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiZJInRHRighIiIsJiIiIkYzSSJ5R0YoRjNGMUYxNyNGLg==

factor(subs(y=Ysol[1],Gy-x*y)-subs(y=Ysol[2],Gy-x*y));

IiIh

Ly:=S(y)-Gy;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNMeUdGKCwmLUkiU0dGKDYjSSJ5R0YoIiIiKiZJInRHRighIiIsJkYzRjNGMkYzRjZGMzcjRi4=

expr:=(Ly-subs(y=0,Ly))*Jy-a*Ly^3-b*Ly^2-c*Ly-d;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVleHByR0YoLCwqJiwqLUkiU0dGKDYjSSJ5R0YoIiIiKiZJInRHRighIiIsJkY1RjVGNEY1RjhGNS1GMjYjIiIhRjgqJEY3RjhGOEY1LCYqKEY0RjVGN0Y4KUY5IiIjRjhGNSooRjdGNUY0RjhGQEY1RjVGNUY1KiZJImFHRihGNSksJkYxRjVGNkY1IiIkRjVGOComSSJiR0YoRjUpRkZGQUY1RjgqJkkiY0dGKEY1RkZGNUY4SSJkR0YoRjg3I0Yu

ser:=factor(series(expr,y=-1,3));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzZXJHRigrLywmIiIiRjBJInlHRihGMCwkKiYsJkkidEdGKEYwSSJhR0YoRjBGMClGNSIiJCEiIkY5ISIkLCQqJiwsKihGOEYwRjZGMC1JIlNHRig2I0Y5RjBGMComRj9GMEY1RjBGMComLUZANiMiIiFGMEY1RjBGOUkiYkdGKEYwIiIjRjlGMClGNUZIRjlGOSEiIywkKiYsMioqRjhGMClGP0ZIRjBGNkYwRjVGMEYwKipGSEYwRkdGMEY/RjBGNUYwRjAqKEY4RjAtLUkiREdGJTYjRkBGQUYwRjZGMEYwKiZGUkYwRjVGMEYwRkJGOUZDRjAqJkkiY0dGKEYwRjVGMEYwRjBGMEYwRklGOUY5RjksJCooI0YwRkhGMCw0KipGSEYwKUY/RjhGMEY2RjBGSUYwRjAqKkZIRjBGT0YwRkdGMEZJRjBGMCosIiM3RjBGUkYwRj9GMEY2RjBGNUYwRjAqKkZIRjBGWEYwRj9GMEZJRjBGMCoqIiIlRjBGUkYwRkdGMEY1RjBGMCooRkhGMEkiZEdGKEYwRklGMEYwKihGSEYwRlJGMEY1RjBGOSooRjhGMC0tLUkjQEBHRiU2JEZURkhGVUZBRjBGNkYwRjAqJkZjb0YwRjVGMEYwRjBGSUY5RjlGRiwkKigjRjAiIidGMCw2KiwiIz1GMEZSRjBGT0YwRjZGMEZJRjBGMCosRltvRjBGUkYwRj9GMEZHRjBGSUYwRjAqKkZfcEYwKUZSRkhGMEY2RjBGNUYwRjAqKkZccEYwRlhGMEZSRjBGSUYwRjAqLEZfcEYwRj9GMEZjb0YwRjZGMEY1RjBGMCoqRlxwRjBGY29GMEZHRjBGNUYwRjAqKEY4RjBGY29GMEY1RjBGOSooRjhGMC0tLUZmbzYkRlRGOEZVRkFGMEY2RjBGMComRmhwRjBGNUYwRjAqJkZccEYwRklGMEYwRjBGSUY5RjlGMC1JIk9HRiU2I0YwRkg3I0Yu

eqs:={seq(coeff(ser,1+y,-i),i=0..3)};indets(eqs);

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

PCdJImFHNiJJImJHRiRJImNHRiRJImRHRiRJInRHRiQ=

sol:=op(solve(eqs,[a,b,c,d]));

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

recall that S(-1)=0 sol:=subs(S(-1)=0,sol);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzb2xHRig3Ji9JImFHRigsJEkidEdGKCEiIi9JImJHRigsJiIiIyIiIiomLUkiU0dGKDYjIiIhRjhGMkY4RjgvSSJjR0YoKiYsKEY4RjMqKEY3RjgtLUkiREdGJTYjRjs2I0YzRjhGMkY4RjhGOUYzRjhGMkYzL0kiZEdGKCwkKiZGMkYzLCgqKkY3RjhGQ0Y4RjpGOEYyRjhGOComIiIkRjhGQ0Y4RjgtLS1JI0BAR0YlNiRGRUY3RkZGR0YzRjhGMzcjRi4=

op(2,sol);

L0kiYkc2IiwmIiIjIiIiKiYtSSJTR0YkNiMiIiFGJ0kidEdGJEYnRic=

expand(op(3,sol));

L0kiY0c2IiwoKiRJInRHRiQhIiJGKComIiIjIiIiLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kiU0dGJDYjRihGK0YrLUYzNiMiIiFGKA==

expand(op(4,sol));

L0kiZEc2IiwoKigiIiMiIiItLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSJTR0YkNiMhIiJGKC1GMDYjIiIhRihGMiooIiIkRihGKUYoSSJ0R0YkRjJGMiomLS0tSSNAQEdGLDYkRitGJ0YvRjFGKEY4RjJGKA==

exprexpl:=subs(sol,expr);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlleHByZXhwbEdGKCwsKiYsKi1JIlNHRig2I0kieUdGKCIiIiomSSJ0R0YoISIiLCZGNUY1RjRGNUY4RjUtRjI2IyIiIUY4KiRGN0Y4RjhGNSwmKihGNEY1RjdGOClGOSIiI0Y4RjUqKEY3RjVGNEY4RkBGNUY1RjVGNSomRjdGNSksJkYxRjVGNkY1IiIkRjVGNSomLCZGQUY1KiZGOkY1RjdGNUY1RjUpRkVGQUY1RjgqKCwoRjVGOCooRkFGNS0tSSJER0YlNiNGMjYjRjhGNUY3RjVGNUZJRjhGNUY3RjhGRUY1RjgqJkY3RjgsKCoqRkFGNUZORjVGOkY1RjdGNUY1KiZGRkY1Rk5GNUY1LS0tSSNAQEdGJTYkRlBGQUZRRlJGOEY1RjU3I0Yu

Syexpr:=t*(1+y)*Q(y); diff(Syexpr,y); subs(y=-1,%);diff(Syexpr,y$2); subs(y=-1,%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdTeWV4cHJHRigqKEkidEdGKCIiIiwmRjBGMEkieUdGKEYwRjAtSSJRR0YoNiNGMkYwNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsJiomSSJ0R0YoIiIiLUkiUUdGKDYjSSJ5R0YoRi9GLyooRi5GLywmRi9GL0YzRi9GLy1JJWRpZmZHRiY2JEYwRjNGL0YvRis=

KiZJInRHNiIiIiItSSJRR0YkNiMhIiJGJQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsJiooIiIjIiIiSSJ0R0YoRi8tSSVkaWZmR0YmNiQtSSJRR0YoNiNJInlHRihGN0YvRi8qKEYwRi8sJkYvRi9GN0YvRi8tRjI2JEY0LUkiJEdGJjYkRjdGLkYvRi83IywmRi1GLyooRjBGL0Y5Ri8tRjI2JEYxRjdGL0Yv

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsJCooIiIjIiIiSSJ0R0YoRi8tSSVkaWZmR0YmNiQtSSJRR0YoNiMhIiJGN0YvRi9GKw==

eqQ:=collect(numer(factor(subs(S(y)=t*(1+y)*Q(y), S(0)=t*A1, (D(S))(-1)=t*A2, ((D@@2)(S))(-1)=2*t*A3,exprexpl))),Q,factor);

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

The value of A1 is forced:(in terms of Q) factor(subs(y=0,eqQ));

LCQqJilJInRHNiIiIiMiIiIsJkkjQTFHRiZGKC1JIlFHRiY2IyIiISEiIkYoRi8=

The value of A2 is forced:(in terms of Q) factor(subs(y=-1,eqQ));

LCYqJiIiIyIiIi1JIlFHNiI2IyEiIkYlRioqJkYkRiVJI0EyR0YoRiVGJQ==

The value of A3 is forced:(in terms of Q) factor(subs(A2=Q(-1),series(eqQ,y=-1,2)));

KycsJiIiIkYkSSJ5RzYiRiQsJiomIiIjRiRJI0EzR0YmRiRGJComRilGJC0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNJIlFHRiY2IyEiIkYkRjVGJC1JIk9HRi82I0YkRik=

eqQ;

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

restart: Now we want to sow that there exists a unique solution where Q(y) has polynomial coefficients eqQ:=t^4*y*(1+y)^4*Q(y)^3-t^2*y*(1+y)^2*(A1*t^2*y+A1*t^2+2*y-1)*Q(y)^2+(A1*t^2*y^3-2*A2*t^2*y^3+t^2*y^4-4*A2*t^2*y^2+4*t^2*y^3-A1*t^2*y-2*A2*t^2*y+6*t^2*y^2+4*t^2*y+y^3+t^2-y^2)*Q(y)+2*A1*A2*t^2*y^2-t^2*A1*y^3+2*A1*A2*t^2*y-3*A1*t^2*y^2-3*t^2*A1*y-t^2*A1+3*A2*y^2-2*A3*y^2-y^3+A2*y-2*A3*y-2*y^2-y;

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

Qser:=proc(N) add(q[n](y)*t^n,n=0..N): end: Qser(2);

LCgtJkkicUc2IjYjIiIhNiNJInlHRiYiIiIqJi0mRiU2I0YrRilGK0kidEdGJkYrRisqJi0mRiU2IyIiI0YpRispRjBGNUYrRis=

A1ser:=proc(n) subs(y=0,Qser(n)): end: A2ser:=proc(n) subs(y=-1,Qser(n)): end: A3ser:=proc(N) add(D(q[n])(-1)*t^n,n=0..N): end: A1ser(2); A2ser(2);A3ser(2);

LCgtJkkicUc2IjYjIiIhRiciIiIqJi0mRiU2I0YpRidGKUkidEdGJkYpRikqJi0mRiU2IyIiI0YnRikpRi5GM0YpRik=

LCgtJkkicUc2IjYjIiIhNiMhIiIiIiIqJi0mRiU2I0YrRilGK0kidEdGJkYrRisqJi0mRiU2IyIiI0YpRispRjBGNUYrRis=

LCgtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjJkkicUdGKTYjIiIhNiMhIiIiIiIqJi0tRiU2IyZGLDYjRjFGL0YxSSJ0R0YpRjFGMSomLS1GJTYjJkYsNiMiIiNGL0YxKUY4Rj9GMUYx

n:=0: from now on we can iterate the following lines n:=n+1: factor(series(subs(Q(y)=Qser(n),A1=A1ser(n),A2=A2ser(n),A3=A3ser(n),eqQ),t,n));

KydJInRHNiIqJkkieUdGJCIiIiw6KiYiIiZGJylGJkYqRichIiIqJiIjREYnKUYmIiIlRidGLComLSZJInFHRiQ2IyIiJzYjRiZGJylGJiIiI0YnRicqJiIjVkYnKUYmIiIkRidGLComRjJGJ0YmRidGLCooRj1GJy1GMzYjRixGJ0YmRidGJyooRjlGJy0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGM0ZBRidGJkYnRiwqJiIjN0YnRjhGJ0YsRkBGJyomRjlGJ0ZDRidGLComIiNJRidGJkYnRiciIz5GJ0YnRjYtSSJPR0ZGNiNGJyIiKA==

co:=coeff(%,t,n-1)/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjb0dGKCw6KiYiIiYiIiIpSSJ5R0YoRjBGMSEiIiomIiNERjEpRjMiIiVGMUY0KiYtJkkicUdGKDYjIiInNiNGM0YxKUYzIiIjRjFGMSomIiNWRjEpRjMiIiRGMUY0KiZGOkYxRjNGMUY0KihGRUYxLUY7NiNGNEYxRjNGMUYxKihGQUYxLS1JIkRHRiU2I0Y7RklGMUYzRjFGNComIiM3RjFGQEYxRjRGSEYxKiZGQUYxRktGMUY0KiYiI0lGMUYzRjFGMSIjPkYxNyNGLg==

solve(co,q[n-1](y)):convert(%,parfrac,y);

LC4qJiIiJiIiIilJInlHNiIiIiRGJUYlKiYiI0lGJSlGJyIiI0YlRiUqJiIjdEYlRidGJUYlIiMmKUYlKiYsKComIiIlRiUtJkkicUdGKDYjIiInNiMhIiJGJUY7KiZGNEYlLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRig2I0Y2RjpGJUYlIiNPRiVGJSwmRidGJUYlRjtGO0YlKiYsKEY1RiUqJkYtRiVGPUYlRjsiIz5GJUYlRidGO0Yl

the coefficients cannot have poles solve([op(-1,%),op(-2,%)],[q[n-1](-1),D(q[n-1])(-1)]);

NyM3JC8tJkkicUc2IjYjIiInNiMhIiIiI1AvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyZJInFHNiI2IyIiJzYjISIiIiNH

factor(subs(op(%),co));

KihJInlHNiIiIiIsJkYjRiVGJSEiIkYlLCwqJiIiJkYlKUYjIiIkRiVGJyomIiNJRiUpRiMiIiNGJUYnLSZJInFHRiQ2IyIiJzYjRiNGJSomIiN0RiVGI0YlRiciIyYpRidGJQ==

q[n-1](y):=solve(%,q[n-1](y));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+LSZJInFHRig2IyIiJzYjSSJ5R0YoLCoqJiIiJiIiIilGMyIiJEY3RjcqJiIjSUY3KUYzIiIjRjdGNyomIiN0RjdGM0Y3RjciIyYpRjc3I0Y0

D(q[n-1])(-1):=subs(y=-1,diff(%,y));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+LS1JIkRHRiU2IyZJInFHRig2IyIiJzYjISIiIiNHNyNGNw==

Qser(n);

LCwiIiJGIyomLCYiIiNGI0kieUc2IkYjRiMpSSJ0R0YoRiZGI0YjKiYsKComRiZGIylGJ0YmRiNGIyomIiIpRiNGJ0YjRiMiIzZGI0YjKUYqIiIlRiNGIyomLCoqJiIiJkYjKUYnIiIkRiNGIyomIiNJRiNGLkYjRiMqJiIjdEYjRidGI0YjIiMmKUYjRiMpRioiIidGI0YjKiYtJkkicUdGKDYjIiIoNiNGJ0YjKUYqRkZGI0Yj

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwqSSJ4R0YoIiIiKiRGLyEiIkYwKiZGL0YwSSJ5R0YoRjBGMComRi9GMkY0RjJGMDcjRi4=

K:=expand(t*x*y*liste-x*y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCwqKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGMkYxRjZGMUYxKiZGMEYxRjZGMUYxKiZGM0YxRjZGMSEiIkYwRjE3I0Yu

Xsol:=[solve(K,x)];Ysol:=[solve(K,y)];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCosIyIiIiIiI0YyLCZJInlHRihGMiokKSwqKigiIiVGMilJInRHRihGM0YyKUY1IiIkRjIhIiIqKCIiKUYyRjtGMilGNUYzRjJGPyooRjpGMkY7RjJGNUYyRj8qJEZCRjJGMkYxRjJGMkYyRjxGP0Y1Rj8sJkYyRjJGNUYyRj9GMiwkKixGMUYyLCZGNUY/RjZGMkYyRjxGP0Y1Rj9GRUY/Rj83I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ0R0YoISIiKUkieEdGKEYzRjUsKiomRjRGMkY2RjJGNUY0RjVGN0YyKiQpLC4qJilGNEYzRjIpRjciIiVGMkYyKihGM0YyRj5GMkY2RjJGNSooRjNGMkY0RjIpRjciIiRGMkY1KiRGPkYyRjIqKEYzRjJGNEYyRjdGMkY1KiRGNkYyRjJGMUYyRjJGMkYyLCQqKkYxRjIsKkY5RjJGOkYyRjRGMkY3RjVGMkY0RjVGNkY1RjU3I0Yu

Jy:=y/t/(1+y)^2+t/y*(1+y)^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKeUdGKCwmKihJInlHRigiIiJJInRHRighIiIpLCZGMUYxRjBGMSIiI0YzRjEqKEYyRjFGMEYzRjRGMUYxNyNGLg==

factor(subs(y=Ysol[1],Jy)-subs(y=Ysol[2],Jy));

IiIh

Ix:=1/2*factor(expand(rationalize(subs(y=Ysol[1],Jy)+subs(y=Ysol[2],Jy)))); expand(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeEdGKCwkKiYpSSJ4R0YoIiIjISIiLCwqJkkidEdGKCIiIilGMSIiJUY3RjcqKEYyRjdGNkY3RjBGN0YzKiQpRjEiIiRGN0YzRjZGN0YxRjNGN0YzNyNGLg==

LCwqJkkidEc2IiIiIilJInhHRiUiIiNGJiEiIiomRilGJkYkRiZGJkYoRiYqJkYnRipGJEYmRioqJEYoRipGJg==

factor((Ix-Jy)/K/subs(x=1/x,K));

LCQqKEkidEc2IiEiIiksJiIiIkYpSSJ5R0YlRikiIiNGJkYqRiZGJg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiZJInRHRighIiIsJiIiIkYzSSJ5R0YoRjNGMUYxNyNGLg==

factor(subs(y=Ysol[1],Gy-x*y)-subs(y=Ysol[2],Gy-x*y));

IiIh

Fx:=1/2*factor(expand(rationalize(subs(y=Ysol[1],x*y-Gy)+subs(y=Ysol[2],x*y-Gy)))); expand(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKihJInhHRighIiIsJkkidEdGKCIiIkYwRjFGNEYzRjFGMTcjRi4=

LCYqJEkieEc2IiEiIkYmKiRJInRHRiVGJiIiIg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

For the calculations of invariants in Tables 1 and 2, see the Maple session "invariants-syst" (Section 4.1) To construct, or check the correctness of decoupling functions of Tables 3 and 4, see the Maple session "decoupling-syst" (Section 4.2) For the proof that there is no other decoupled model, see the Maple session "nodec" For the alternative to the invariant lemma (Section 4.4), see the Maple session "algebraic-models" (Kreweras model, and reverse Kreweras model) For the effective solution of algebraic models, see the Maple session "algebraic-models" (Section 4.5 and Appendix A)

JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH

stepset:=[[0,1],[-1,0],[1,0],[-1,-1]];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShzdGVwc2V0R0YoNyY3JCIiISIiIjckISIiRjA3JEYxRjA3JEYzRjM3I0Yu

liste:=convert(map(s->x^s[1]*y^s[2],stepset),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwqSSJ5R0YoIiIiKiRJInhHRighIiJGMEYyRjAqJkYyRjNGL0YzRjA3I0Yu

Number of walks of length n ending at (i,j) co:=proc(i,j,n) option remember: if i<0 or j<0 or n<0 then 0 elif n=0 and i=0 and j=0 then 1 elif n=0 then 0 else convert(map (s-> co(i-s[1],j-s[2],n-1),stepset),`+`): fi: end: co(0,0,2);

IiIi

The series Q(x,y;t) Qser:=proc(N) add(add(add(co(i,j,n)*x^i*y^j,i=0..n),j=0..n)*t^n,n=0..N): end: Qser(4);

LCwiIiJGIyomLCZJInhHNiJGI0kieUdGJ0YjRiNJInRHRidGI0YjKiYsKiokKUYmIiIjRiNGIyooRi5GI0YmRiNGKEYjRiMqJClGKEYuRiNGI0YjRiNGIylGKUYuRiNGIyomLDAqJClGJiIiJEYjRiMqKEY3RiNGLUYjRihGI0YjKihGN0YjRiZGI0YxRiNGIyokKUYoRjdGI0YjKiZGLkYjRiZGI0YjKiZGN0YjRihGI0YjRi5GI0YjKUYpRjdGI0YjKiYsOCokKUYmIiIlRiNGIyooRkNGI0Y2RiNGKEYjRiMqKCIiJ0YjRi1GI0YxRiNGIyooRkNGI0YmRiNGO0YjRiMqJClGKEZDRiNGIyomRjdGI0YtRiNGIyooIiIpRiNGJkYjRihGI0YjKiZGRkYjRjFGI0YjKiYiIiZGI0YmRiNGIyomRk9GI0YoRiNGI0YuRiNGIylGKUZDRiNGIw==

The series Q(0,y;t), for walks ending on the vertical axis Qvser:=proc(N) add(add(co(0,j,n)*y^j,j=0..n)*t^n,n=0..N): end: Qvser(5);

LC4iIiJGIyomSSJ5RzYiRiNJInRHRiZGI0YjKiYsJiokKUYlIiIjRiNGI0YjRiNGIylGJ0YsRiNGIyomLCgqJClGJSIiJEYjRiMqJkYyRiNGJUYjRiNGLEYjRiMpRidGMkYjRiMqJiwqKiQpRiUiIiVGI0YjKiYiIidGI0YrRiNGIyomIiImRiNGJUYjRiNGLEYjRiMpRidGOUYjRiMqJiwsKiQpRiVGPUYjRiMqJiIjNUYjRjFGI0YjKiYiIipGI0YrRiNGIyomRkRGI0YlRiNGIyIjOEYjRiMpRidGPUYjRiM=

The kernel K:=expand(x*y*(t*liste-1));subs(x=0,K);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCwqKEkidEdGKCIiIilJInhHRigiIiNGMUkieUdGKEYxRjEqKEYwRjFGM0YxKUY1RjRGMUYxKiZGMEYxRjVGMUYxKiZGM0YxRjVGMSEiIkYwRjE3I0Yu

LCYqJkkidEc2IiIiIkkieUdGJUYmRiZGJEYm

The discriminant tilde d(y) dy:=factor(discrim(K,x));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkeUdGKComSSJ5R0YoIiIiLCwqJilJInRHRigiIiNGMClGLyIiJEYwRjAqKCIiJUYwRjNGMEYvRjAhIiIqKEY1RjBGNEYwKUYvRjVGMEY6KiZGOUYwRjNGMEY6Ri9GMEYwNyNGLg==

Decoupling functtion G:=-1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJHR0YoLCQqJEkieUdGKCEiIkYxNyNGLg==

Connection between S(y) (or Q(0,y)) and the weak invariant w eqSw:=subs(x=0,K)*Q(y)-G=alpha/(w(y)-beta)+gamma;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcVN3R0YoLywmKiYsJiomSSJ0R0YoIiIiSSJ5R0YoRjRGNEYzRjRGNC1JIlFHRig2I0Y1RjRGNCokRjUhIiJGNCwmKiZJJmFscGhhR0YoRjQsJi1JIndHRihGOEY0SSViZXRhR0YoRjpGOkY0SSZnYW1tYUdGJkY0NyNGLg==

wyQ:=op(2,isolate(eqSw,w(y)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR3eVFHRigsJiomLCgqJiwmKiZJInRHRigiIiJJInlHRihGNUY1RjRGNUY1LUkiUUdGKDYjRjZGNSEiIiokRjZGOkY6SSZnYW1tYUdGJkY1RjpJJmFscGhhR0YoRjVGOkklYmV0YUdGKEY1NyNGLg==

series(wyQ,y,3);

KytJInlHNiJJJWJldGFHRiQiIiFJJmFscGhhR0YkIiIiLCQqJiwmKiZJInRHRiRGKC1JIlFHRiQ2I0YmRihGKEkmZ2FtbWFHJSpwcm90ZWN0ZWRHISIiRihGJ0YoRjMiIiMtSSJPRzYkRjJJKF9zeXNsaWJHRiQ2I0YoIiIk

DE for w(y) eqw:=4*dy*(diff(w(y),y))^2-(4*w(y)^3-g2*w(y)-g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcXdHRigsKioqIiIlIiIiSSJ5R0YoRjEsLComKUkidEdGKCIiI0YxKUYyIiIkRjFGMSooRjBGMUY1RjFGMkYxISIiKihGN0YxRjZGMSlGMkY3RjFGOyomRjBGMUY1RjFGO0YyRjFGMSktSSVkaWZmR0YmNiQtSSJ3R0YoNiNGMkYyRjdGMUYxKiZGMEYxKUZDRjlGMUY7KiZJI2cyR0YoRjFGQ0YxRjFJI2czR0YoRjE3I0Yu

subs(w(y)=wyQ,eqw);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsKioqIiIlIiIiSSJ5R0YoRi8sLComKUkidEdGKCIiI0YvKUYwIiIkRi9GLyooRi5GL0YzRi9GMEYvISIiKihGNUYvRjRGLylGMEY1Ri9GOSomRi5GL0YzRi9GOUYwRi9GLyktSSVkaWZmR0YmNiQsJiomLCgqJiwmKiZGNEYvRjBGL0YvRjRGL0YvLUkiUUdGKDYjRjBGL0Y5KiRGMEY5RjlJJmdhbW1hR0YmRi9GOUkmYWxwaGFHRihGL0Y5SSViZXRhR0YoRi9GMEY1Ri9GLyomRi5GLylGQUY3Ri9GOSomSSNnMkdGKEYvRkFGL0YvSSNnM0dGKEYvRis=

res:=numer(factor(eval(%))): nops(res);

IiQvIw==

DE for Q(0,y) edQ:=collect(res,[diff(Q(y), y),Q],factor);

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

indets(edQ);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkmYWxwaGFHRihJJWJldGFHRihJI2cyR0YoSSNnM0dGKEkidEdGKEkieUdGKC1JIlFHRig2I0YyLUklZGlmZkdGJjYkRjNGMkYr

Now we want to eliminate the unknown series, at the cost of introducing series Q_{0,i} factor(series(edQ,y,1));

KydJInlHNiIsKComIiIlIiIiKUklYmV0YUdGJCIiJEYoISIiKiZGKkYoSSNnMkdGJEYoRihJI2czR0YkRigiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYo

Expression of g3 g3s:=solve(coeff(%,y,0),g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnM3NHRigsJiomIiIlIiIiKUklYmV0YUdGKCIiJEYxRjEqJkYzRjFJI2cyR0YoRjEhIiI3I0Yu

edQ1:=op(4,factor(subs(g3=g3s,edQ))):nops(%);

IiMqKg==

factor(series(edQ1,y,1));

KydJInlHNiIsKCooIiM7IiIiSSZhbHBoYUdGJEYoKUkidEdGJCIiI0YoRigqJiIjN0YoKUklYmV0YUdGJEYsRihGKEkjZzJHRiQhIiIiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYo

Expression of g2 g2s:=solve(coeff(%,y,0),g2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnMnNHRigsJiooIiM7IiIiSSZhbHBoYUdGKEYxKUkidEdGKCIiI0YxRjEqJiIjN0YxKUklYmV0YUdGKEY1RjFGMTcjRi4=

We have thus eliminated g2 and g3 edQ2:=op(4,factor(subs(g2=g2s,edQ1)));

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

Now the series Q_{0,i} will start to come in factor(series(edQ2,y,1));

KydJInlHNiIsLCIiIkYmKigiIzdGJkkmZ2FtbWFHJSpwcm90ZWN0ZWRHRiYpSSJ0R0YkIiIjRiYhIiIqKEYoRiYtSSJRR0YkNiMiIiFGJilGLCIiJEYmRiYqJiIiJUYmRitGJkYuKiZGNUYmSSViZXRhR0YkRiZGLkYzLUkiT0c2JEYqSShfc3lzbGliR0YkNiNGJkYm

Expression of beta betas:=solve(coeff(%,y,0),beta);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZiZXRhc0dGKCwqKigiIiUiIiItSSJRR0YoNiMiIiFGMSlJInRHRigiIiRGMUYxKihGMEYxSSZnYW1tYUdGJkYxKUY3IiIjRjEhIiIqJiNGMEY4RjFGO0YxRj0jRjFGOEYxNyNGLg==

edQ3:=(factor(subs(beta=betas,edQ2))): nops(%);

IiN3

factor(series(edQ3,y,2));

KydJInlHNiIsNiooIiIpIiIiSSZnYW1tYUclKnByb3RlY3RlZEdGKClJInRHRiQiIiNGKCEiIiooIiM3RigpRilGLUYoRitGKEYuKihGMEYoKS1JIlFHRiQ2IyIiIUYtRigpRiwiIiVGKEYuKigiI0dGKEY0RigpRiwiIiRGKEYoKigiIz9GKC0tSSJERzYkRipJKF9zeXNsaWJHRiQ2I0Y1RjZGKEY8RihGKCoqIiNDRihGNEYoRilGKEY8RihGKCooRi1GKEYsRihGNEYoRi4qJkYtRihGKUYoRihJJmFscGhhR0YkRi4qJkYtRihGLEYoRi5GKC1JIk9HRkM2I0YoRi0=

Expression of alpha alphas:=solve(coeff(%,y,1),alpha);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbHBoYXNHRigsNCooIiM3IiIiKS1JIlFHRig2IyIiISIiI0YxKUkidEdGKCIiJUYxISIiKioiI0NGMUYzRjFJJmdhbW1hR0YmRjEpRjkiIiRGMUYxKigiIz9GMS0tSSJER0YlNiNGNEY1RjFGP0YxRjEqKCIjR0YxRjNGMUY/RjFGMSooRjBGMSlGPkY3RjEpRjlGN0YxRjsqKCIiKUYxRj5GMUZLRjFGOyooRjdGMUY5RjFGM0YxRjsqJkY3RjFGPkYxRjEqJkY3RjFGOUYxRjs3I0Yu

edQ4:=factor(subs(alpha=alphas,edQ3)):nops(%);

IiN3

factor(series(edQ4,y,3));

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

Now we do not have an expression of gamma, but a cubic equation for it eq1:=collect(coeff(%,y,2),gamma, factor);

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

We finally eliminate gamma factor(resultant(eq1,edQ4,gamma));nops(%);

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

IiIk

Here is the first order DE for Q(y) involving the series Q_{0,i} (here m=3) edQ:=collect(op(1,op(3,factor(resultant(eq1,edQ4,gamma)))),[diff(Q(y), y),Q(y)],distributed,factor);;

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

Checking this DE n:=10: normal(series(eval(subs(Q(y)=Qvser(n),Q(0)=subs(y=0,Qvser(n)), (D(Q))(0)=subs(y=0,diff(Qvser(n),y)),((D@@2)(Q))(0)=subs(y=0,diff(Qvser(n),y$2)),edQ)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Introducing the series Q_{0,i} (here denoted Q_i) edQ:=subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,edQ);

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

degree(edQ,Q0);degree(edQ,Q1);degree(edQ,Q2);

IiIk

IiIi

IiIi

dedQ:=factor(diff(edQ,y)): indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkjUTBHRihJI1ExR0YoSSNRMkdGKEkidEdGKEkieUdGKC1JIlFHRig2I0YxLUklZGlmZkdGJjYkRjJGMS1GNjYkRjItSSIkR0YmNiRGMSIiIzcjPCpGLUYuRi9GMEYxRjJGNS1GNjYkRjVGMQ==

We now want to obtain polynomial coefficients in t and y. We eliminate the Q_i one after the other, increasing the order of the DE res1:=factor(resultant(edQ,dedQ,Q2));nops(%);

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IiIm

Choosing and checking the correct factor (the first one cannot cancel because of the factors t) edQbis:=collect(op(5,res1),[diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0,Q1],factor);

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KUkjUTBHRihJI1ExR0YoSSJ0R0YoSSJ5R0YoLUkiUUdGKDYjRjAtSSVkaWZmR0YmNiRGMUYwLUY1NiRGMS1JIiRHRiY2JEYwIiIjNyM8KUYtRi5GL0YwRjFGNC1GNTYkRjRGMA==

degree(edQbis,Q1);

IiIi

Eliminating Q1 dedQbis:=factor(diff(edQbis,y)): factor(resultant(edQbis,dedQbis,Q1)): nops(%);

IiIk

edQter:=collect(%%,[diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0],factor);;

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KUkjUTBHRihJInRHRihJInlHRigtSSJRR0YoNiNGLy1JJWRpZmZHRiY2JEYwRi8tRjQ2JEYwLUkiJEdGJjYkRi8iIiMtRjQ2JEYwLUY5NiRGLyIiJDcjPClGLUYuRi9GMEYzLUY0NiRGM0YvLUY0NiRGQ0Yv

Eliminating Q0 dedQter:=op(4,factor(diff(edQter,y))):nops(%);

IiNd

res2:=factor(resultant(edQter,dedQter,Q0)):nops(res2);

IiIl

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

ISQ/Ig==

KiQpSSJ0RzYiIiIlIiIi

SSJ5RzYi

nops(op(4,res2));

IiRdIg==

edQpol:=collect(op(4,res2),[diff(Q(y), y, y, y, y),diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y)],factor);;

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

Let us check n:=20: normal(series(eval(subs(Q(y)=Qvser(n),edQpol)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0A=

We note that expanding the 1st order DE or the one we have just obtained gives the same information (relating Q3 to Q0, Q1 and Q2) convert(factor(subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,series(edQpol,y,1))),polynom);

LDYqKCIjTyIiIilJI1EwRzYiIiIjRiUpSSJ0R0YoIiIkRiUhIiIqKiIjc0YlRidGJUkjUTFHRihGJUYqRiVGLSooRiRGJSlGMEYpRiVGKkYlRi0qKCIjPUYlLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKDYkSSJER0Y5Riw2I0kiUUdGKDYjIiIhRiUpRitGKUYlRiUqKEYvRiVGMEYlRkJGJUYlKigiJCE9RiVJI1EyR0YoRiVGQkYlRiUqKEY0RiVGJ0YlRitGJUYlKihGNEYlRjBGJUYrRiVGJSomRjRGJUYwRiVGLSomRjRGJUZGRiVGLQ==

convert(normal(subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,series(edQ,y,4))),polynom);

KiYsNiooIiM3IiIiKUkjUTBHNiIiIiNGJilJInRHRikiIiRGJiEiIioqIiNDRiZGKEYmSSNRMUdGKUYmRitGJkYuKihGJUYmKUYxRipGJkYrRiZGLiooIiInRiYtLS1JI0BARzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YpNiRJIkRHRjpGLTYjSSJRR0YpNiMiIiFGJilGLEYqRiZGJiooRjBGJkYxRiZGQ0YmRiYqKCIjZ0YmSSNRMkdGKUYmRkNGJkYmKihGNUYmRihGJkYsRiZGJiooRjVGJkYxRiZGLEYmRiYqJkY1RiZGMUYmRi4qJkY1RiZGR0YmRi5GJilJInlHRilGLUYm

factor(%%/%);

LCQqJiIiJCIiIilJInlHNiJGJCEiIkYl

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

stepset:=[[0,1],[-1,1],[1,0],[-1,-1]];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShzdGVwc2V0R0YoNyY3JCIiISIiIjckISIiRjE3JEYxRjA3JEYzRjM3I0Yu

liste:=convert(map(s->x^s[1]*y^s[2],stepset),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwqSSJ5R0YoIiIiKiZJInhHRighIiJGL0YwRjBGMkYwKiZGMkYzRi9GM0YwNyNGLg==

Number of walks of length n ending at (i,j) co:=proc(i,j,n) option remember: if i<0 or j<0 or n<0 then 0 elif n=0 and i=0 and j=0 then 1 elif n=0 then 0 else convert(map (s-> co(i-s[1],j-s[2],n-1),stepset),`+`): fi: end: co(0,0,2);

IiIh

The series Q(x,y;t) Qser:=proc(N) add(add(add(co(i,j,n)*x^i*y^j,i=0..n),j=0..n)*t^n,n=0..N): end: Qser(4);

LCwiIiJGIyomLCZJInhHNiJGI0kieUdGJ0YjRiNJInRHRidGI0YjKiYsKiokKUYmIiIjRiNGIyooRi5GI0YmRiNGKEYjRiMqJClGKEYuRiNGI0YoRiNGIylGKUYuRiNGIyomLDAqJClGJiIiJEYjRiMqKEY3RiNGLUYjRihGI0YjKihGN0YjRiZGI0YxRiNGIyokKUYoRjdGI0YjRi9GIyomRjdGI0YxRiNGI0YuRiNGIylGKUY3RiNGIyomLDoqJClGJiIiJUYjRiMqKEZCRiNGNkYjRihGI0YjKigiIidGI0YtRiNGMUYjRiMqKEZCRiNGJkYjRjtGI0YjKiQpRihGQkYjRiNGOEYjKigiIilGI0YmRiNGMUYjRiMqJkZFRiNGO0YjRiMqJkYuRiNGMUYjRiMqJiIiJkYjRiZGI0YjKiZGTkYjRihGI0YjRi5GI0YjKUYpRkJGI0Yj

The series Q(0,y;t), for walks ending on the vertical axis Qvser:=proc(N) add(add(co(0,j,n)*y^j,j=0..n)*t^n,n=0..N): end: Qvser(5);

LC4iIiJGIyomSSJ5RzYiRiNJInRHRiZGI0YjKiYsJiokKUYlIiIjRiNGI0YlRiNGIylGJ0YsRiNGIyomLCgqJClGJSIiJEYjRiMqJkYyRiNGK0YjRiNGLEYjRiMpRidGMkYjRiMqJiwsKiQpRiUiIiVGI0YjKiYiIidGI0YxRiNGIyomRixGI0YrRiNGIyomIiImRiNGJUYjRiNGLEYjRiMpRidGOUYjRiMqJiwsKiQpRiVGPkYjRiMqJiIjNUYjRjhGI0YjKiZGRUYjRjFGI0YjKiYiIipGI0YrRiNGIyomIiM6RiNGJUYjRiNGIylGJ0Y+RiNGIw==

The kernel K:=expand(x*y*(t*liste-1));subs(x=0,K);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCwqKEkidEdGKCIiIilJInhHRigiIiNGMUkieUdGKEYxRjEqKEYwRjFGM0YxKUY1RjRGMUYxKiZGMEYxRjdGMUYxKiZGM0YxRjVGMSEiIkYwRjE3I0Yu

LCYqJkkidEc2IiIiIilJInlHRiUiIiNGJkYmRiRGJg==

The discriminant tilde d(y) dy:=factor(discrim(K,x));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkeUdGKComSSJ5R0YoIiIiLCwqJilJInRHRigiIiNGMClGLyIiJEYwRjAqKCIiJUYwRjNGMClGL0Y1RjAhIiIqKEY1RjBGNEYwRjpGMEY7KiZGOUYwRjNGMEY7Ri9GMEYwNyNGLg==

Decoupling functtion G:=-y-1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJHR0YoLCZJInlHRighIiIqJEYvRjBGMDcjRi4=

Connection between S(y) (or Q(0,y)) and the weak invariant w eqSw:=subs(x=0,K)*Q(y)-G=alpha/(w(y)-beta)+gamma;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcVN3R0YoLywoKiYsJiomSSJ0R0YoIiIiKUkieUdGKCIiI0Y0RjRGM0Y0RjQtSSJRR0YoNiNGNkY0RjRGNkY0KiRGNiEiIkY0LCYqJkkmYWxwaGFHRihGNCwmLUkid0dGKEY6RjRJJWJldGFHRihGPEY8RjRJJmdhbW1hR0YmRjQ3I0Yu

wyQ:=op(2,isolate(eqSw,w(y)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR3eVFHRigsJiomLCoqJiwmKiZJInRHRigiIiIpSSJ5R0YoIiIjRjVGNUY0RjVGNS1JIlFHRig2I0Y3RjUhIiJGN0Y8KiRGN0Y8RjxJJmdhbW1hR0YmRjVGPEkmYWxwaGFHRihGNUY8SSViZXRhR0YoRjU3I0Yu

series(wyQ,y,3);

KytJInlHNiJJJWJldGFHRiQiIiFJJmFscGhhR0YkIiIiLCQqJiwmKiZJInRHRiRGKC1JIlFHRiQ2I0YmRihGKEkmZ2FtbWFHJSpwcm90ZWN0ZWRHISIiRihGJ0YoRjMiIiMtSSJPRzYkRjJJKF9zeXNsaWJHRiQ2I0YoIiIk

DE for w(y) eqw:=4*dy*(diff(w(y),y))^2-(4*w(y)^3-g2*w(y)-g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcXdHRigsKioqIiIlIiIiSSJ5R0YoRjEsLComKUkidEdGKCIiI0YxKUYyIiIkRjFGMSooRjBGMUY1RjEpRjJGN0YxISIiKihGN0YxRjZGMUY7RjFGPComRjBGMUY1RjFGPEYyRjFGMSktSSVkaWZmR0YmNiQtSSJ3R0YoNiNGMkYyRjdGMUYxKiZGMEYxKUZDRjlGMUY8KiZJI2cyR0YoRjFGQ0YxRjFJI2czR0YoRjE3I0Yu

subs(w(y)=wyQ,eqw);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsKioqIiIlIiIiSSJ5R0YoRi8sLComKUkidEdGKCIiI0YvKUYwIiIkRi9GLyooRi5GL0YzRi8pRjBGNUYvISIiKihGNUYvRjRGL0Y5Ri9GOiomRi5GL0YzRi9GOkYwRi9GLyktSSVkaWZmR0YmNiQsJiomLCoqJiwmKiZGNEYvRjlGL0YvRjRGL0YvLUkiUUdGKDYjRjBGL0Y6RjBGOiokRjBGOkY6SSZnYW1tYUdGJkYvRjpJJmFscGhhR0YoRi9GOkklYmV0YUdGKEYvRjBGNUYvRi8qJkYuRi8pRkFGN0YvRjoqJkkjZzJHRihGL0ZBRi9GL0kjZzNHRihGL0Yr

res:=numer(factor(eval(%))): nops(res);

IiQtJA==

DE for Q(0,y) edQ:=collect(res,[diff(Q(y), y),Q],factor);

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

indets(edQ);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkmYWxwaGFHRihJJWJldGFHRihJI2cyR0YoSSNnM0dGKEkidEdGKEkieUdGKC1JIlFHRig2I0YyLUklZGlmZkdGJjYkRjNGMkYr

Now we want to eliminate the unknown series, at the cost of introducing series Q_{0,i} factor(series(edQ,y,1));

KydJInlHNiIsKComIiIlIiIiKUklYmV0YUdGJCIiJEYoISIiKiZGKkYoSSNnMkdGJEYoRihJI2czR0YkRigiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYo

Expression of g3 g3s:=solve(coeff(%,y,0),g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnM3NHRigsJiomIiIlIiIiKUklYmV0YUdGKCIiJEYxRjEqJkYzRjFJI2cyR0YoRjEhIiI3I0Yu

edQ1:=op(4,factor(subs(g3=g3s,edQ))):nops(%);

IiRQIg==

factor(series(edQ1,y,1));

KydJInlHNiIsKCooIiM7IiIiSSZhbHBoYUdGJEYoKUkidEdGJCIiI0YoRigqJiIjN0YoKUklYmV0YUdGJEYsRihGKEkjZzJHRiQhIiIiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYo

Expression of g2 g2s:=solve(coeff(%,y,0),g2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnMnNHRigsJiooIiM7IiIiSSZhbHBoYUdGKEYxKUkidEdGKCIiI0YxRjEqJiIjN0YxKUklYmV0YUdGKEY1RjFGMTcjRi4=

We have thus eliminated g2 and g3 edQ2:=op(4,factor(subs(g2=g2s,edQ1)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlZFEyR0YoLF53IiIiRi8qKCIiJUYvSSJ0R0YoRi8pSSJ5R0YoIiIkRi9GLyooIiIjRi8pRjJGN0YvKUY0RjFGLyEiIiooIiM7Ri9GOEYvRjNGL0YvKioiIilGLy1JIlFHRig2I0Y0Ri9GOEYvRjlGL0YvKipGMUYvRj9GL0YyRi9GM0YvRjoqLEY+Ri9GP0YvLUklZGlmZkdGJjYkRj9GNEYvKUYyRjVGLylGNEY+Ri9GOiosRjVGLylGP0Y3Ri9JJWJldGFHRihGL0Y4Ri8pRjQiIidGL0Y6KixGMUYvRj9GL0ZERi9GOEYvKUY0IiIoRi9GLyosIiM3Ri9GP0YvKUkmZ2FtbWFHRiZGN0YvRkdGL0Y5Ri9GLyosIiNDRi9GP0YvRlRGL0ZHRi8pRjQiIiZGL0Y6KixGTUYvRj9GL0ZLRi9GMkYvRldGL0Y6KixGUkYvRj9GL0ZLRi9GMkYvRjNGL0Y6KiZGOEYvRkxGL0YvKihGN0YvRjJGL0ZXRi9GOiooRjVGL0ZLRi9GOUYvRjoqKEZNRi9GS0YvKUY0RjdGL0Y6Ki5GTUYvRj9GL0ZLRi9GVEYvRjJGL0Y5Ri9GLyoqRjVGL0ZLRi9GU0YvRmluRi9GOioqRlJGL0ZTRi9GOEYvRjRGL0YvKihJJmFscGhhR0YoRi9GVEYvRmluRi9GLyoqRk1GL0ZLRi9GVEYvRjRGL0YvKipGN0YvKUZERjdGLylGMkYxRi9GSEYvRi8qKkZSRi9GYW9GL0Zib0YvRk9GL0Y6KipGUkYvKUY/RjVGLylGMkZYRi9GOUYvRi8qKEZhb0YvRmJvRi9GTEYvRi8qKkYxRi9GYW9GL0ZHRi9GT0YvRjoqKkZSRi9GYW9GL0Zib0YvRldGL0Y6KipGMUYvRmVvRi9GZm9GL0ZpbkYvRi8qKkY3Ri9GYW9GL0ZHRi9GV0YvRjoqKkY3Ri9GYW9GL0Y4Ri9GTEYvRi8qKiIjT0YvRkpGL0Zib0YvRjNGL0YvKipGMUYvRmFvRi9GYm9GL0YzRi9GOioqRj5GL0ZERi9GR0YvRldGL0Y6KipGXnBGL0Y/Ri9GR0YvRjlGL0YvKihGYW9GL0Y4Ri9GOUYvRi8qKkY3Ri9GREYvRkdGL0Y5Ri9GOioqRlJGL0ZKRi9GYm9GL0Y0Ri9GLyoqRj5GL0ZERi9GR0YvRjNGL0YvKioiI19GL0Y/Ri9GR0YvRmluRi9GLyoqRjFGL0ZERi9GOEYvRjNGL0YvKipGPkYvRkRGL0ZHRi9GNEYvRi8qKkY3Ri9GMkYvRkRGL0ZpbkYvRjoqKkYxRi8pRlRGNUYvRjhGL0ZpbkYvRjoqKEZSRi9GVEYvRjhGL0Y6KihGUkYvRj9GL0ZHRi9GLyoqRjdGL0Zhb0YvRkdGLylGNCIiKkYvRjoqKkYxRi9GSkYvRmJvRi9GT0YvRjoqKkY+Ri9GSkYvRkdGL0ZPRi9GOiooRmFvRi9GOEYvRkhGL0YvKipGN0YvRkRGL0ZHRi9GSEYvRi8qKiIjP0YvRkpGL0Zib0YvRldGL0YvKipGMUYvRj9GL0ZHRi9GT0YvRi8qKkY+Ri9GREYvRkdGL0ZPRi9GOioqRjFGL0ZKRi9GOEYvRkxGL0YvKipGMUYvRj9GL0ZHRi9GTEYvRjoqKkYxRi9GREYvRjhGL0ZPRi9GOioqRjFGL0Y/Ri9GR0YvRldGL0Y6KipGPkYvRj9GL0Y4Ri9GTEYvRjoqKkY3Ri9GREYvRjJGL0ZMRi9GLyoqRjFGL0Y/Ri9GMkYvRldGL0YvKipGUkYvRlNGL0Y4Ri9GM0YvRi8qKkZSRi9GVEYvRjhGL0Y5Ri9GOioqRk1GL0ZLRi9GVEYvRjNGL0YvKipGVkYvRlRGL0Y4Ri9GaW5GL0Y6KipGMUYvRmVvRi9GZm9GL0ZIRi9GLyooRmFvRi9GYm9GLylGNCIjNUYvRi8qKkYxRi9GYW9GL0Zib0YvRmBxRi9GOioqRlJGL0Zlb0YvRmZvRi9GTEYvRi8qKkYxRi9GSkYvRmJvRi9GSEYvRi8qLEYxRi9GP0YvRkRGL0Zib0YvRmBxRi9GLyosRjxGL0Y/Ri9GREYvRmJvRi9GSEYvRjoqLEZSRi9GSkYvRlRGL0Zib0YvRkxGL0Y6KiwiI1tGL0Y/Ri9GVEYvRkdGL0YzRi9GOiosRk1GL0Y/Ri9GS0YvRjJGL0Y0Ri9GOiosRjFGL0Y/Ri9GREYvRmJvRi9GT0YvRi8qLCIjS0YvRj9GL0ZERi9GYm9GL0ZMRi9GOiosRlZGL0ZKRi9GVEYvRmJvRi9GOUYvRjoqLEY+Ri9GP0YvRkRGL0ZHRi9GTEYvRjoqLEY8Ri9GP0YvRkRGL0Zib0YvRjlGL0Y6KixGTUYvRkpGL0ZLRi9GOEYvRjlGL0Y6KixGUkYvRkpGL0ZURi9GYm9GL0ZpbkYvRjoqLEYxRi9GP0YvRkRGL0Y4Ri9GV0YvRi8qLEZSRi9GP0YvRlNGL0ZHRi9GaW5GL0YvKixGNUYvRkpGL0ZLRi9GOEYvRmluRi9GOiosRlZGL0Y/Ri9GVEYvRkdGL0Y0Ri9GOioqRj9GL0Zeb0YvRjJGL0ZpbkYvRjoqKkY/Ri9GXm9GL0YyRi9GOUYvRjoqLkZNRi9GP0YvRktGL0ZURi9GMkYvRmluRi9GLyomRl5vRi9GM0YvRjoqJkY4Ri9GaW5GL0YvKiZGNEYvRl5vRi9GOiooRjxGL0Y4Ri9GNEYvRi8qJkY3Ri9GaW5GL0Y6KiRGOUYvRi8qJkY1Ri9GS0YvRjoqKEY3Ri9GNEYvRjJGL0Y6NyNGLg==

Now the series Q_{0,i} will start to come in factor(series(edQ2,y,1));

KydJInlHNiIsKiIiIkYmKigiIzdGJkkmZ2FtbWFHJSpwcm90ZWN0ZWRHRiYpSSJ0R0YkIiIjRiYhIiIqKEYoRiYtSSJRR0YkNiMiIiFGJilGLCIiJEYmRiYqJkY1RiZJJWJldGFHRiRGJkYuRjMtSSJPRzYkRipJKF9zeXNsaWJHRiQ2I0YmRiY=

Expression of beta betas:=solve(coeff(%,y,0),beta);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZiZXRhc0dGKCwoKigiIiUiIiItSSJRR0YoNiMiIiFGMSlJInRHRigiIiRGMUYxKihGMEYxSSZnYW1tYUdGJkYxKUY3IiIjRjEhIiIjRjFGOEYxNyNGLg==

edQ3:=(factor(subs(beta=betas,edQ2))): nops(%);

IiMlKg==

factor(series(edQ3,y,2));

KydJInlHNiIsNCooIiM3IiIiKUkmZ2FtbWFHJSpwcm90ZWN0ZWRHIiIjRigpSSJ0R0YkRixGKCEiIiooRidGKCktSSJRR0YkNiMiIiFGLEYoKUYuIiIlRihGLyooIiM/RigtLUkiREc2JEYrSShfc3lzbGliR0YkNiNGM0Y0RigpRi4iIiRGKEYoKioiI0NGKEYyRihGKkYoRkBGKEYoKihGLEYoRi5GKEYyRihGLyomRixGKEYqRihGKEkmYWxwaGFHRiRGLyomIiM7RihGLUYoRigqJkYsRihGLkYoRi9GKC1JIk9HRj02I0YoRiw=

Expression of alpha alphas:=solve(coeff(%,y,1),alpha);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbHBoYXNHRigsMiooIiM3IiIiKS1JIlFHRig2IyIiISIiI0YxKUkidEdGKCIiJUYxISIiKioiI0NGMUYzRjFJJmdhbW1hR0YmRjEpRjkiIiRGMUYxKigiIz9GMS0tSSJER0YlNiNGNEY1RjFGP0YxRjEqKEYwRjEpRj5GN0YxKUY5RjdGMUY7KihGN0YxRjlGMUYzRjFGOyomIiM7RjFGSUYxRjEqJkY3RjFGPkYxRjEqJkY3RjFGOUYxRjs3I0Yu

edQ4:=factor(subs(alpha=alphas,edQ3)):nops(%);

IiMjKg==

factor(series(edQ4,y,3));

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

Now we do not have an expression of gamma, but a cubic equation for it eq1:=collect(coeff(%,y,2),gamma, factor);

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

We finally eliminate gamma factor(resultant(eq1,edQ4,gamma));nops(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsJCooIiNrIiIiKUkidEdGKCIiKkYvKSxcdiosIiIlRi8tSSJRR0YoNiNJInlHRihGLy1JJWRpZmZHRiY2JEY3RjpGLylGMSIiJEYvKUY6RjJGLyEiIiosIiM3Ri8pRjciIiNGLy1GODYjIiIhRi8pRjFGNkYvKUY6IiInRi9GLyosIiIpRi9GN0YvRjtGLylGMUZFRi8pRjpGTUYvRi8qLEZDRi9GN0YvKUZGRkVGL0ZJRi8pRjpGNkYvRkEqLEY2Ri9GN0YvRjtGL0YxRi8pRjoiIihGL0ZBKiwiI0NGL0Y3Ri9GRkYvRj5GLylGOiIiJkYvRi8qLCIjP0YvLS1JIkRHRiU2I0Y4RkdGL0Y3Ri9GPkYvRlJGL0YvKixGRUYvRjdGL0ZGRi9GMUYvRlJGL0ZBKiZGMUYvRkpGL0ZBKihGRUYvRjFGL0ZSRi9GLyooRkVGL0Y7Ri9GSkYvRkEqKEZFRi9GN0YvRlhGL0ZBKihGTUYvRjdGLylGOkY/Ri9GLyooRkVGL0ZGRi9GYG9GL0ZBKihGNkYvRmZuRi8pRjpGRUYvRkEqKEZDRi9GN0YvRk5GL0ZBKihGRUYvRjtGL0Zjb0YvRi8qKEZFRi9GN0YvRjpGL0YvKihGRUYvRkZGL0Y6Ri9GQSoqRkVGLylGO0ZFRi9GPkYvRk9GL0ZBKipGQ0YvKUY3Rj9GL0ZJRi9GUkYvRkEqKkY2Ri9GaW9GL0ZORi9GVEYvRi8qKkY2Ri9GW3BGL0ZJRi9GY29GL0ZBKipGRUYvRmlvRi9GMUYvRkpGL0ZBKipGNkYvKUZGRj9GL0ZJRi9GY29GL0YvKioiI09GL0ZERi9GPkYvRmBvRi9GQSoqRjZGL0Zpb0YvRj5GL0Zgb0YvRi8qKkZNRi9GO0YvRk5GL0ZYRi9GLyoqRmVuRi9GN0YvRk5GL0ZSRi9GQSooRmlvRi9GMUYvRlJGL0ZBKipGQ0YvRkRGL0Y+Ri9GOkYvRkEqKkZDRi9GUUYvRj5GL0Y6Ri9GQSooRkRGL0YxRi9GY29GL0YvKioiIzlGLy0tLUkjQEBHRiU2JEZobkZFRmluRkdGL0ZORi9GY29GL0YvKihGUUYvRjFGL0Zjb0YvRi8qKkZicEYvRkZGL0ZORi9GY29GL0YvKipGZW5GL0ZmbkYvRk5GL0Y6Ri9GLyoqRk1GL0Y7Ri9GTkYvRjpGL0ZBKipGRUYvRkZGL0YxRi9GY29GL0YvKipGYnBGL0Y3Ri9GTkYvRmNvRi9GQSosRmVuRi9GZm5GL0Y3Ri9GPkYvRmNvRi9GLyosRmVuRi9GZm5GL0ZGRi9GPkYvRmNvRi9GQSosRldGL0Y3Ri9GRkYvRj5GL0Y6Ri9GLyosRkVGL0Y3Ri9GRkYvRjFGL0Zjb0YvRkEqLEY2Ri9GN0YvRjtGL0Y+Ri9GVEYvRkEqLEZXRi9GREYvRkZGL0ZJRi9GUkYvRi8qLEZNRi9GN0YvRjtGL0ZORi9GSkYvRi8qLEZDRi9GREYvRkZGL0ZJRi9GY29GL0YvKiwiIztGL0Y3Ri9GO0YvRj5GL0ZSRi9GLyosRkNGL0Y3Ri9GUUYvRklGL0Zjb0YvRkEqLEY2Ri9GN0YvRjtGL0YxRi9GWEYvRkEqLCIjW0YvRjdGL0ZGRi9GPkYvRmBvRi9GLyosRmByRi9GN0YvRjtGL0Y+Ri9GT0YvRi8qKEZDRi9GRkYvRk5GL0YvKipGQ0YvRmlvRi9GPkYvRlRGL0YvKihGaW9GL0Y+Ri9GSkYvRkEqKkZlbkYvRkRGL0Y+Ri9GWEYvRkEqKkZDRi9GaW9GL0Y+Ri9GWEYvRi8qKkZFRi9GaW9GL0ZORi9GWEYvRi8qKkZFRi9GO0YvRk5GL0ZSRi9GLyoqRk1GL0Y7Ri9GTkYvRmBvRi9GQSoqRjZGL0YxRi9GO0YvRmBvRi9GQSoqRjZGL0Zpb0YvRj5GL0ZARi9GLyoqRjZGL0ZERi9GPkYvRlRGL0YvKipGTUYvRjtGL0ZORi9GVEYvRi8qKkY2Ri9GN0YvRk5GL0ZKRi9GLyosIiNLRi9GN0YvRjtGL0Y+Ri9GSkYvRi8qKkY2Ri9GW3BGL0ZJRi9GT0YvRkEqKEZpb0YvRj5GLylGOiIjNUYvRkEqKkZDRi9GW3BGL0ZJRi9GSkYvRkEqKkY2Ri9GREYvRj5GL0ZPRi9GQSoqRkVGL0Zpb0YvRk5GL0ZARi9GLyoqRk1GL0ZERi9GTkYvRlRGL0YvKihGaW9GL0YxRi9GT0YvRkEqKkZFRi9GO0YvRk5GL0ZPRi9GQSoqRjZGL0Y3Ri9GTkYvRlRGL0ZBKipGP0YvRkRGL0YxRi9GSkYvRkEqKkY2Ri9GO0YvRjFGL0ZURi9GLyoqRjZGL0Y3Ri9GTkYvRlhGL0YvKipGTUYvRjdGL0YxRi9GSkYvRi8qKkZDRi9GUUYvRj5GL0Zgb0YvRkEqKkZFRi9GREYvRjFGL0ZSRi9GLyoqRkNGL0ZGRi9GTkYvRlJGL0YvKipGZW5GL0ZmbkYvRk5GL0Zgb0YvRi8qKkZoc0YvRjdGL0YxRi9GUkYvRkEqKkZFRi9GMUYvRjdGL0Zjb0YvRkEqJkZFRi9GWEYvRi8qJkZLRi9GYG9GL0ZBRj9GL0YvRis=

IiIk

Here is the first order DE for Q(y) involving the series Q_{0,i} (here m=3) edQ:=collect(op(1,op(3,factor(resultant(eq1,edQ4,gamma)))),[diff(Q(y), y),Q(y)],distributed,factor);;

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

Checking this DE n:=10: normal(series(eval(subs(Q(y)=Qvser(n),Q(0)=subs(y=0,Qvser(n)), (D(Q))(0)=subs(y=0,diff(Qvser(n),y)),((D@@2)(Q))(0)=subs(y=0,diff(Qvser(n),y$2)),edQ)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Introducing the series Q_{0,i} (here denoted Q_i) edQ:=subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,edQ);

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

degree(edQ,Q0);degree(edQ,Q1);degree(edQ,Q2);

IiIk

IiIi

IiIi

dedQ:=factor(diff(edQ,y)): indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkjUTBHRihJI1ExR0YoSSNRMkdGKEkidEdGKEkieUdGKC1JIlFHRig2I0YxLUklZGlmZkdGJjYkRjJGMS1GNjYkRjItSSIkR0YmNiRGMSIiIzcjPCpGLUYuRi9GMEYxRjJGNS1GNjYkRjVGMQ==

We now want to obtain polynomial coefficients in t and y. We eliminate the Q_i one after the other, increasing the order of the DE res1:=factor(resultant(edQ,dedQ,Q2));nops(%);

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

IiIm

Choosing and checking the correct factor (the first one cannot cancel because of the factors y and the CT 1) edQbis:=collect(op(5,res1),[diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0,Q1],factor);

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KUkjUTBHRihJI1ExR0YoSSJ0R0YoSSJ5R0YoLUkiUUdGKDYjRjAtSSVkaWZmR0YmNiRGMUYwLUY1NiRGMS1JIiRHRiY2JEYwIiIjNyM8KUYtRi5GL0YwRjFGNC1GNTYkRjRGMA==

degree(edQbis,Q1);

IiIi

Eliminating Q1 dedQbis:=factor(diff(edQbis,y)): factor(resultant(edQbis,dedQbis,Q1)): nops(%);

IiIk

edQter:=collect(%%,[diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0],factor);;

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KUkjUTBHRihJInRHRihJInlHRigtSSJRR0YoNiNGLy1JJWRpZmZHRiY2JEYwRi8tRjQ2JEYwLUkiJEdGJjYkRi8iIiMtRjQ2JEYwLUY5NiRGLyIiJDcjPClGLUYuRi9GMEYzLUY0NiRGM0YvLUY0NiRGQ0Yv

Eliminating Q0 dedQter:=op(4,factor(diff(edQter,y))):nops(%);

IiNk

res2:=factor(resultant(edQter,dedQter,Q0)):nops(res2);

IiIl

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

ISQ/Ig==

KiQpSSJ0RzYiIiIlIiIi

SSJ5RzYi

nops(op(4,res2));

IiQiPQ==

edQpol:=collect(op(4,res2),[diff(Q(y), y, y, y, y),diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y)],factor);;

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

Let us check n:=20: normal(series(eval(subs(Q(y)=Qvser(n),edQpol)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0A=

We note that expanding the 1st order DE or the one we have just obtained gives the same information (relating Q3 to Q0, Q1 and Q2) convert(factor(subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,series(edQpol,y,1))),polynom);

LDAqKCIjTyIiIilJInRHNiIiIiRGJSlJI1ExR0YoIiIjRiVGJSooIiM9RiUpRidGLEYlLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKDYkSSJER0Y0Rik2I0kiUUdGKDYjIiIhRiUhIiIqKCIjc0YlRitGJUYvRiVGPSooRi5GJUYrRiVGJ0YlRj0qJkYuRiVJI1EwR0YoRiVGJSomRi5GJUkjUTJHRihGJUYlRi5GPQ==

convert(normal(subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,series(edQ,y,4))),polynom);

KiYsMCooIiM3IiIiKUkidEc2IiIiJEYmKUkjUTFHRikiIiNGJkYmKigiIidGJilGKEYtRiYtLS1JI0BARzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YpNiRJIkRHRjVGKjYjSSJRR0YpNiMiIiFGJiEiIiooIiNDRiZGLEYmRjBGJkY+KihGL0YmRixGJkYoRiZGPiomRi9GJkkjUTBHRilGJkYmKiZGL0YmSSNRMkdGKUYmRiZGL0Y+RiYpSSJ5R0YpRipGJg==

factor(%%/%);

LCQqJiIiJCIiIilJInlHNiJGJCEiIkYl

JSFH

JSFH

stepset:=[[0,1],[0,-1],[-1,0],[1,1]];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShzdGVwc2V0R0YoNyY3JCIiISIiIjckRjAhIiI3JEYzRjA3JEYxRjE3I0Yu

liste:=convert(map(s->x^s[1]*y^s[2],stepset),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwqSSJ5R0YoIiIiKiRGLyEiIkYwKiRJInhHRihGMkYwKiZGNEYwRi9GMEYwNyNGLg==

Number of walks of length n ending at (i,j) co:=proc(i,j,n) option remember: if i<0 or j<0 or n<0 then 0 elif n=0 and i=0 and j=0 then 1 elif n=0 then 0 else convert(map (s-> co(i-s[1],j-s[2],n-1),stepset),`+`): fi: end: co(0,0,2);

IiIi

The series Q(x,y;t) Qser:=proc(N) add(add(add(co(i,j,n)*x^i*y^j,i=0..n),j=0..n)*t^n,n=0..N): end: Qser(4);

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

The series Q(0,y;t), for walks ending on the vertical axis Qvser:=proc(N) add(add(co(0,j,n)*y^j,j=0..n)*t^n,n=0..N): end: Qvser(5);

LC4iIiJGIyomSSJ5RzYiRiNJInRHRiZGI0YjKiYsKCokKUYlIiIjRiNGI0YlRiNGI0YjRiMpRidGLEYjRiMqJiwqKiQpRiUiIiRGI0YjKiZGMkYjRitGI0YjKiZGLEYjRiVGI0YjRixGI0YjKUYnRjJGI0YjKiYsLCokKUYlIiIlRiNGIyomIiInRiNGMUYjRiMqJiIiJkYjRitGI0YjKiYiIipGI0YlRiNGI0YsRiNGIylGJ0Y6RiNGIyomLC4qJClGJUY+RiNGIyomIiM1RiNGOUYjRiMqJiIjOUYjRjFGI0YjKiYiI0NGI0YrRiNGIyomIiM4RiNGJUYjRiNGTUYjRiMpRidGPkYjRiM=

The kernel K:=expand(x*y*(t*liste-1));subs(x=0,K);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCwqKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGM0YxRjVGMUYxKiZGMEYxRjNGMUYxKiZGMEYxRjZGMUYxKiZGM0YxRjZGMSEiIjcjRi4=

KiZJInlHNiIiIiJJInRHRiRGJQ==

The discriminant tilde d(y) dy:=factor(discrim(K,x));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkeUdGKCwwKiYpSSJ0R0YoIiIjIiIiKUkieUdGKCIiJUYzRjMqKEY2RjNGMEYzKUY1IiIkRjMhIiIqKEYyRjNGMEYzKUY1RjJGM0YzKihGMkYzRjFGM0Y4RjNGOiokRjBGM0YzKihGMkYzRjFGM0Y1RjNGOiokRjxGM0YzNyNGLg==

Decoupling function G:=-y-1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJHR0YoLCZJInlHRighIiIqJEYvRjBGMDcjRi4=

Connection between S(y) (or Q(0,y)) and the weak invariant w eqSw:=subs(x=0,K)*Q(y)-G=alpha/(w(y)-beta)+gamma;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcVN3R0YoLywoKihJInRHRigiIiItSSJRR0YoNiNJInlHRihGMkY2RjJGMkY2RjIqJEY2ISIiRjIsJiomSSZhbHBoYUdGKEYyLCYtSSJ3R0YoRjVGMkklYmV0YUdGKEY4RjhGMkkmZ2FtbWFHRiZGMjcjRi4=

wyQ:=op(2,isolate(eqSw,w(y)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR3eVFHRigsJiomLCoqKEkidEdGKCIiIi1JIlFHRig2I0kieUdGKEYzRjdGMyEiIkY3RjgqJEY3RjhGOEkmZ2FtbWFHRiZGM0Y4SSZhbHBoYUdGKEYzRjhJJWJldGFHRihGMzcjRi4=

series(wyQ,y,3);

KytJInlHNiJJJWJldGFHRiQiIiFJJmFscGhhR0YkIiIiKiZJJmdhbW1hRyUqcHJvdGVjdGVkR0YoRidGKCIiIy1JIk9HNiRGK0koX3N5c2xpYkdGJDYjRigiIiQ=

DE for w(y) eqw:=4*dy*(diff(w(y),y))^2-(4*w(y)^3-g2*w(y)-g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcXdHRigsKiooIiIlIiIiLDAqJilJInRHRigiIiNGMSlJInlHRihGMEYxRjEqKEYwRjFGNEYxKUY4IiIkRjEhIiIqKEY2RjFGNEYxKUY4RjZGMUYxKihGNkYxRjVGMUY6RjFGPCokRjRGMUYxKihGNkYxRjVGMUY4RjFGPCokRj5GMUYxRjEpLUklZGlmZkdGJjYkLUkid0dGKDYjRjhGOEY2RjFGMSomRjBGMSlGR0Y7RjFGPComSSNnMkdGKEYxRkdGMUYxSSNnM0dGKEYxNyNGLg==

subs(w(y)=wyQ,eqw);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsKiooIiIlIiIiLDAqJilJInRHRigiIiNGLylJInlHRihGLkYvRi8qKEYuRi9GMkYvKUY2IiIkRi8hIiIqKEY0Ri9GMkYvKUY2RjRGL0YvKihGNEYvRjNGL0Y4Ri9GOiokRjJGL0YvKihGNEYvRjNGL0Y2Ri9GOiokRjxGL0YvRi8pLUklZGlmZkdGJjYkLCYqJiwqKihGM0YvLUkiUUdGKDYjRjZGL0Y2Ri9GOkY2RjoqJEY2RjpGOkkmZ2FtbWFHRiZGL0Y6SSZhbHBoYUdGKEYvRjpJJWJldGFHRihGL0Y2RjRGL0YvKiZGLkYvKUZFRjlGL0Y6KiZJI2cyR0YoRi9GRUYvRi9JI2czR0YoRi9GKw==

res:=numer(factor(eval(%))): nops(res);

IiQ4Iw==

DE for Q(0,y) edQ:=collect(res,[diff(Q(y), y),Q],factor);

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

indets(edQ);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkmYWxwaGFHRihJJWJldGFHRihJI2cyR0YoSSNnM0dGKEkidEdGKEkieUdGKC1JIlFHRig2I0YyLUklZGlmZkdGJjYkRjNGMkYr

Now we want to eliminate the unknown series, at the cost of introducing series Q_{0,i} factor(series(edQ,y,1));

KydJInlHNiIsKiooIiIlIiIiKUkmYWxwaGFHRiQiIiNGKClJInRHRiRGK0YoRigqJkYnRigpSSViZXRhR0YkIiIkRighIiIqJkYwRihJI2cyR0YkRihGKEkjZzNHRiRGKCIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRig=

Expression of g3 g3s:=solve(coeff(%,y,0),g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnM3NHRigsKCooIiIlIiIiKUkmYWxwaGFHRigiIiNGMSlJInRHRihGNEYxISIiKiZGMEYxKUklYmV0YUdGKCIiJEYxRjEqJkY6RjFJI2cyR0YoRjFGNzcjRi4=

edQ1:=op(4,factor(subs(g3=g3s,edQ))):nops(%);

IiRKIg==

factor(series(edQ1,y,1));

KydJInlHNiIsKioqIiM7IiIiSSZhbHBoYUdGJEYoSSZnYW1tYUclKnByb3RlY3RlZEdGKClJInRHRiQiIiNGKCEiIiooIiIpRihGKUYoRi1GKEYoKiYiIzdGKClJJWJldGFHRiRGLkYoRihJI2cyR0YkRi8iIiEtSSJPRzYkRitJKF9zeXNsaWJHRiQ2I0YoRig=

Expression of g2 g2s:=solve(coeff(%,y,0),g2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnMnNHRigsKCoqIiM7IiIiSSZhbHBoYUdGKEYxSSZnYW1tYUdGJkYxKUkidEdGKCIiI0YxISIiKigiIilGMUYyRjFGNUYxRjEqJiIjN0YxKUklYmV0YUdGKEY2RjFGMTcjRi4=

We have thus eliminated g2 and g3 edQ2:=op(4,factor(subs(g2=g2s,edQ1)));

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

Now the series Q_{0,i} will start to come in factor(series(edQ2,y,1));

KydJInlHNiIsLiIiIiEiIiooIiInRiZJJmdhbW1hRyUqcHJvdGVjdGVkR0YmSSJ0R0YkRiZGJiooRilGJilGKiIiI0YmKUYsRi9GJkYnKihGKUYmLUkiUUdGJDYjIiIhRiYpRiwiIiRGJkYmKiYiIiVGJkYwRiZGJiomRjdGJkklYmV0YUdGJEYmRiZGNS1JIk9HNiRGK0koX3N5c2xpYkdGJDYjRiZGJg==

Expression of beta betas:=solve(coeff(%,y,0),beta);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZiZXRhc0dGKCwsKigiIiMiIiItSSJRR0YoNiMiIiFGMSlJInRHRigiIiRGMSEiIiooRjBGMSlJJmdhbW1hR0YmRjBGMSlGN0YwRjFGMSooRjBGMUY8RjFGN0YxRjkqJiMiIiVGOEYxRj1GMUY5I0YxRjhGMTcjRi4=

edQ3:=(factor(subs(beta=betas,edQ2))): nops(%);

IiMhKQ==

factor(series(edQ3,y,2));

KydJInlHNiIsNiooIiM1IiIiLUkiUUdGJDYjIiIhRigpSSJ0R0YkIiIjRighIiIqKCIiJUYoKUkmZ2FtbWFHJSpwcm90ZWN0ZWRHIiIkRihGLUYoRjAqKCIiJ0YoKUY0Ri9GKEYuRihGKCooIiIpRihGNEYoRi1GKEYoKihGO0YoLS1JIkRHNiRGNUkoX3N5c2xpYkdGJDYjRipGK0YoKUYuRjZGKEYoKioiIzdGKEYpRihGNEYoRkNGKEYoKiZGL0YoRjRGKEYwSSZhbHBoYUdGJEYoKiZGMkYoRi1GKEYoKiZGO0YoRi5GKEYwRigtSSJPR0ZANiNGKEYv

Expression of alpha alphas:=solve(coeff(%,y,1),alpha);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbHBoYXNHRigsNCoqIiM3IiIiLUkiUUdGKDYjIiIhRjFJJmdhbW1hR0YmRjEpSSJ0R0YoIiIkRjEhIiIqKCIiJUYxKUY2RjlGMSlGOCIiI0YxRjEqKCIiKUYxLS1JIkRHRiU2I0YzRjRGMUY3RjFGOiooIiM1RjFGMkYxRj5GMUYxKigiIidGMSlGNkY/RjFGOEYxRjoqKEZBRjFGNkYxRj5GMUY6KiZGPEYxRj5GMUY6KiZGP0YxRjZGMUYxKiZGQUYxRjhGMUYxNyNGLg==

edQ4:=factor(subs(alpha=alphas,edQ3)):nops(%);

IiNv

factor(series(edQ4,y,3));

KydJInlHNiIsQCooIiInIiIiLUkiUUdGJDYjIiIhRigpSSJ0R0YkIiIkRighIiIqKCIjOUYoLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YqRitGKClGLiIiI0YoRjAqKCIiKUYoSSZnYW1tYUdGN0YoRi5GKEYwKiYpRj4iIiVGKEY6RihGMCooRjtGKClGPkYvRihGLkYoRigqKkY9RihGM0YoRj5GKEYtRihGKCooRkFGKClGPkY7RihGOkYoRigqJEZGRihGMCooIiIoRigpRilGO0YoKUYuRkFGKEYwKigiIiZGKC0tLUkjQEBHRjY2JEY1RjtGOUYrRihGLUYoRigqKEZBRihGPkYoRjpGKEYoKipGJ0YoRilGKEZGRihGLUYoRigqKEZBRihGLkYoRilGKEYoKioiIzVGKEYpRihGPkYoRjpGKEYwRkFGKEY7LUkiT0dGNjYjRihGLw==

Now we do not have an expression of gamma, but a quartic equation for it eq1:=collect(coeff(%,y,2),gamma, factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTFHRigsNiomKUkmZ2FtbWFHRiYiIiUiIiIpSSJ0R0YoIiIjRjMhIiIqKEY2RjMpRjEiIiRGM0Y1RjNGMyomLCgqKCIiJ0YzLUkiUUdGKDYjIiIhRjMpRjVGOkYzRjMqJkYyRjNGNEYzRjNGM0Y3RjMpRjFGNkYzRjMqKkY2RjNGNUYzLCoqKEYyRjMtLUkiREdGJTYjRkBGQUYzRjRGM0YzKigiIiZGM0Y1RjNGP0YzRjcqJkY2RjNGNUYzRjNGMkY3RjNGMUYzRjMqKCIiKEYzKUY/RjZGMylGNUYyRjNGNyooRk5GMy0tLUkjQEBHRiU2JEZLRjZGTEZBRjNGQ0YzRjNGPUY3KigiIzlGM0ZJRjNGNEYzRjcqKEYyRjNGNUYzRj9GM0YzRjJGMzcjRi4=

We finally eliminate gamma factor(resultant(eq1,edQ4,gamma));nops(%);

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

IiIj

Here is the first order DE for Q(y) involving the series Q_{0,i} (here m=3) edQ:=collect(op(1,op(2,factor(resultant(eq1,edQ4,gamma)))),[diff(Q(y), y),Q(y)],distributed,factor);;

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

Checking this DE n:=10: normal(series(eval(subs(Q(y)=Qvser(n),Q(0)=subs(y=0,Qvser(n)), (D(Q))(0)=subs(y=0,diff(Qvser(n),y)),((D@@2)(Q))(0)=subs(y=0,diff(Qvser(n),y$2)),edQ)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Introducing the series Q_{0,i} (here denoted Q_i) edQ:=subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,edQ);

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

degree(edQ,Q0);degree(edQ,Q1);degree(edQ,Q2);

IiIj

IiIi

IiIi

dedQ:=factor(diff(edQ,y)): indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkjUTBHRihJI1ExR0YoSSNRMkdGKEkidEdGKEkieUdGKC1JIlFHRig2I0YxLUklZGlmZkdGJjYkRjJGMS1GNjYkRjItSSIkR0YmNiRGMSIiIzcjPCpGLUYuRi9GMEYxRjJGNS1GNjYkRjVGMQ==

We now want to obtain polynomial coefficients in t and y. We eliminate the Q_i one after the other, increasing the order of the DE res1:=factor(resultant(edQ,dedQ,Q2));nops(%);

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

IiIm

Choosing and checking the correct factor (the first one cannot cancel because of the factors y and the CT 1) edQbis:=collect(op(5,res1),[diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0,Q1],factor);

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KUkjUTBHRihJI1ExR0YoSSJ0R0YoSSJ5R0YoLUkiUUdGKDYjRjAtSSVkaWZmR0YmNiRGMUYwLUY1NiRGMS1JIiRHRiY2JEYwIiIjNyM8KUYtRi5GL0YwRjFGNC1GNTYkRjRGMA==

degree(edQbis,Q1);

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Eliminating Q1 dedQbis:=factor(diff(edQbis,y)): factor(resultant(edQbis,dedQbis,Q1)): nops(%);

IiIk

edQter:=collect(%%,[diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0],factor);;

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KUkjUTBHRihJInRHRihJInlHRigtSSJRR0YoNiNGLy1JJWRpZmZHRiY2JEYwRi8tRjQ2JEYwLUkiJEdGJjYkRi8iIiMtRjQ2JEYwLUY5NiRGLyIiJDcjPClGLUYuRi9GMEYzLUY0NiRGM0YvLUY0NiRGQ0Yv

Eliminating Q0 dedQter:=op(4,factor(diff(edQter,y))):nops(%);

IiNd

res2:=factor(resultant(edQter,dedQter,Q0)):nops(res2);

IiIl

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

IiND

KiQpSSJ0RzYiIiIlIiIi

SSJ5RzYi

nops(op(4,res2));

IiRNIg==

edQpol:=collect(op(4,res2),[diff(Q(y), y, y, y, y),diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y)],factor);;

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

Let us check n:=20: normal(series(eval(subs(Q(y)=Qvser(n),edQpol)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0A=

We note that expanding the 1st order DE or the one we have just obtained gives the same information (relating Q3 to Q0, Q1 and Q2) convert(factor(subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,series(edQpol,y,1))),polynom);

LDIqKiIjTyIiIkkjUTBHNiJGJUkjUTFHRidGJSlJInRHRiciIiRGJSEiIiooIiM9RiUpRiYiIiNGJSlGKkYwRiVGJSooIiInRiVGMUYlLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJzYkSSJER0Y4Ris2I0kiUUdGJzYjIiIhRiVGJSooRiRGJUYmRiVGMUYlRiwqKEYuRiVGJkYlRipGJUYlKigiI2FGJUkjUTJHRidGJUYqRiVGLComRi5GJUYoRiVGJSomRiRGJUYqRiVGLA==

convert(normal(subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,series(edQ,y,4))),polynom);

KiYsMioqIiM3IiIiSSNRMEc2IkYmSSNRMUdGKEYmKUkidEdGKCIiJEYmRiYqKCIiJ0YmKUYnIiIjRiYpRitGMEYmISIiKihGMEYmRjFGJi0tLUkjQEBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRig2JEkiREdGOEYsNiNJIlFHRig2IyIiIUYmRjIqKEYlRiZGJ0YmRjFGJkYmKihGLkYmRidGJkYrRiZGMiooIiM9RiZJI1EyR0YoRiZGK0YmRiYqJkYuRiZGKUYmRjIqJkYlRiZGK0YmRiZGJilJInlHRihGLEYm

factor(%%/%);

LCQqJiIiJCIiIilJInlHNiJGJCEiIkYp

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

stepset:=[[0,1],[1,0],[-1,0],[1,-1]];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShzdGVwc2V0R0YoNyY3JCIiISIiIjckRjFGMDckISIiRjA3JEYxRjQ3I0Yu

liste:=convert(map(s->x^s[1]*y^s[2],stepset),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwqSSJ5R0YoIiIiSSJ4R0YoRjAqJEYxISIiRjAqJkYxRjBGL0YzRjA3I0Yu

Number of walks of length n ending at (i,j) co:=proc(i,j,n) option remember: if i<0 or j<0 or n<0 then 0 elif n=0 and i=0 and j=0 then 1 elif n=0 then 0 else convert(map (s-> co(i-s[1],j-s[2],n-1),stepset),`+`): fi: end: co(0,0,2);

IiIi

The series Q(x,y;t) Qser:=proc(N) add(add(add(co(i,j,n)*x^i*y^j,i=0..n),j=0..n)*t^n,n=0..N): end: Qser(4);

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

The series Q(0,y;t), for walks ending on the vertical axis Qvser:=proc(N) add(add(co(0,j,n)*y^j,j=0..n)*t^n,n=0..N): end: Qvser(5);

LC4iIiJGIyomSSJ5RzYiRiNJInRHRiZGI0YjKiYsJiokKUYlIiIjRiNGI0YjRiNGIylGJ0YsRiNGIyomLCgqJClGJSIiJEYjRiMqJkYyRiNGJUYjRiNGI0YjRiMpRidGMkYjRiMqJiwqKiQpRiUiIiVGI0YjKiYiIidGI0YrRiNGI0YzRiNGLEYjRiMpRidGOUYjRiMqJiwsKiQpRiUiIiZGI0YjKiYiIzVGI0YxRiNGI0Y6RiMqJkZDRiNGJUYjRiMiIihGI0YjKUYnRkFGI0Yj

The kernel K:=expand(x*y*(t*liste-1));subs(x=0,K);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCwqKEkidEdGKCIiIilJInhHRigiIiNGMUkieUdGKEYxRjEqKEYwRjFGM0YxKUY1RjRGMUYxKiZGMEYxRjJGMUYxKiZGMEYxRjVGMUYxKiZGM0YxRjVGMSEiIjcjRi4=

KiZJInlHNiIiIiJJInRHRiRGJQ==

The discriminant tilde d(y) dy:=factor(discrim(K,x));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkeUdGKComLCwqJilJInRHRigiIiMiIiIpSSJ5R0YoIiIkRjRGNCooIiIlRjRGMUY0RjZGNCEiIiooRjNGNEYyRjQpRjZGM0Y0RjoqJkY5RjRGMUY0RjpGNkY0RjRGNkY0NyNGLg==

Decoupling function G:=-y^2+y/t+1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJHR0YoLCgqJClJInlHRigiIiMiIiIhIiIqJkYxRjNJInRHRihGNEYzKiRGMUY0RjM3I0Yu

Connection between S(y) (or Q(0,y)) and the weak invariant w eqSw:=subs(x=0,K)*Q(y)-G=alpha/(w(y)-beta)+gamma;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcVN3R0YoLywqKihJInlHRigiIiJJInRHRihGMi1JIlFHRig2I0YxRjJGMiokKUYxIiIjRjJGMiomRjFGMkYzISIiRjsqJEYxRjtGOywmKiZJJmFscGhhR0YoRjIsJi1JIndHRihGNkYySSViZXRhR0YoRjtGO0YySSZnYW1tYUdGJkYyNyNGLg==

wyQ:=op(2,isolate(eqSw,w(y)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR3eVFHRigsJiomLCwqKEkieUdGKCIiIkkidEdGKEYzLUkiUUdGKDYjRjJGMyEiIiokKUYyIiIjRjNGOComRjJGM0Y0RjhGMyokRjJGOEYzSSZnYW1tYUdGJkYzRjhJJmFscGhhR0YoRjNGOEklYmV0YUdGKEYzNyNGLg==

series(wyQ,y,3);

KytJInlHNiJJJWJldGFHRiQiIiEsJEkmYWxwaGFHRiQhIiIiIiIqJkkmZ2FtbWFHJSpwcm90ZWN0ZWRHRipGKEYqIiIjLUkiT0c2JEYtSShfc3lzbGliR0YkNiNGKiIiJA==

DE for w(y) eqw:=4*dy*(diff(w(y),y))^2-(4*w(y)^3-g2*w(y)-g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcXdHRigsKioqIiIlIiIiLCwqJilJInRHRigiIiNGMSlJInlHRigiIiRGMUYxKihGMEYxRjRGMUY4RjEhIiIqKEY2RjFGNUYxKUY4RjZGMUY7KiZGMEYxRjRGMUY7RjhGMUYxRjhGMSktSSVkaWZmR0YmNiQtSSJ3R0YoNiNGOEY4RjZGMUYxKiZGMEYxKUZDRjlGMUY7KiZJI2cyR0YoRjFGQ0YxRjFJI2czR0YoRjE3I0Yu

subs(w(y)=wyQ,eqw);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsKioqIiIlIiIiLCwqJilJInRHRigiIiNGLylJInlHRigiIiRGL0YvKihGLkYvRjJGL0Y2Ri8hIiIqKEY0Ri9GM0YvKUY2RjRGL0Y5KiZGLkYvRjJGL0Y5RjZGL0YvRjZGLyktSSVkaWZmR0YmNiQsJiomLCwqKEY2Ri9GM0YvLUkiUUdGKDYjRjZGL0Y5KiRGO0YvRjkqJkY2Ri9GM0Y5Ri8qJEY2RjlGL0kmZ2FtbWFHRiZGL0Y5SSZhbHBoYUdGKEYvRjlJJWJldGFHRihGL0Y2RjRGL0YvKiZGLkYvKUZBRjdGL0Y5KiZJI2cyR0YoRi9GQUYvRi9JI2czR0YoRi9GKw==

res:=numer(factor(eval(%))): nops(res);

IiRjJA==

DE for Q(0,y) edQ:=collect(res,[diff(Q(y), y),Q],factor);

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indets(edQ);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkmYWxwaGFHRihJJWJldGFHRihJI2cyR0YoSSNnM0dGKEkidEdGKEkieUdGKC1JIlFHRig2I0YyLUklZGlmZkdGJjYkRjNGMkYr

Now we want to eliminate the unknown series, at the cost of introducing series Q_{0,i} factor(series(edQ,y,1));

KydJInlHNiIsJComKUkidEdGJCIiJSIiIiwoKiZGKUYqKUklYmV0YUdGJCIiJEYqRioqJkYuRipJI2cyR0YkRiohIiJJI2czR0YkRjJGKkYyIiIhLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRipGKg==

Expression of g3 g3s:=solve(coeff(%,y,0),g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnM3NHRigqJkklYmV0YUdGKCIiIiwmKiYiIiVGMClGLyIiI0YwRjBJI2cyR0YoISIiRjA3I0Yu

edQ1:=op(5,factor(subs(g3=g3s,edQ))):nops(%);

IiRZIg==

factor(series(edQ1,y,1));

KydJInlHNiIqJilJInRHRiQiIiQiIiIsKCooIiM7RilJJmFscGhhR0YkRikpRiciIiNGKUYpKiYiIzdGKSlJJWJldGFHRiRGL0YpISIiSSNnMkdGJEYpRikiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKUYp

Expression of g2 g2s:=solve(coeff(%,y,0),g2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnMnNHRigsJiooIiM7IiIiSSZhbHBoYUdGKEYxKUkidEdGKCIiI0YxISIiKiYiIzdGMSlJJWJldGFHRihGNUYxRjE3I0Yu

We have thus eliminated g2 and g3 edQ2:=op(5,factor(subs(g2=g2s,edQ1)));

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

The series Q_{0,0} will only to come in later (compared to the first three models) factor(series(edQ2,y,1));

KydJInlHNiIsJComKUkidEdGJCIiIyIiIiwqKigiIzdGKkkmZ2FtbWFHJSpwcm90ZWN0ZWRHRipGJ0YqRioqJiIiJUYqRidGKiEiIiomIiIkRipJJWJldGFHRiRGKkYyRipGKkYqRjIiIiEtSSJPRzYkRi9JKF9zeXNsaWJHRiQ2I0YqRio=

Expression of beta betas:=solve(coeff(%,y,0),beta);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZiZXRhc0dGKCwoKigiIiUiIiJJJmdhbW1hR0YmRjEpSSJ0R0YoIiIjRjFGMSomI0YwIiIkRjFGM0YxISIiI0YxRjhGMTcjRi4=

edQ3:=op(3,factor(subs(beta=betas,edQ2))): nops(%);

IiNq

factor(series(edQ3,y,1));

KydJInlHNiIqJkkidEdGJCIiIiwuKigiIz9GJy1JIlFHRiQ2IyIiIUYnKUYmIiIkRidGJyooIiM3RicpSSZnYW1tYUclKnByb3RlY3RlZEciIiNGJylGJkY2RidGJyooIiIpRidGNEYnRjdGJyEiIiomRjZGJ0Y0RidGJ0kmYWxwaGFHRiRGOiomIiM9RidGJkYnRjpGJ0YuLUkiT0c2JEY1SShfc3lzbGliR0YkNiNGJ0Yn

Expression of alpha: now Q(0,0) comes in alphas:=solve(coeff(%,y,0),alpha);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbHBoYXNHRigsLCooIiM/IiIiLUkiUUdGKDYjIiIhRjEpSSJ0R0YoIiIkRjFGMSooIiM3RjEpSSZnYW1tYUdGJiIiI0YxKUY3Rj1GMUYxKigiIilGMUY8RjFGPkYxISIiKiZGPUYxRjxGMUYxKiYiIz1GMUY3RjFGQTcjRi4=

edQ4:=factor(subs(alpha=alphas,edQ3)):nops(%);

IiNc

factor(series(edQ4,y,2));

KydJInlHNiIsOCooIiIlIiIiKUkmZ2FtbWFHJSpwcm90ZWN0ZWRHIiIkRigpSSJ0R0YkRixGKCEiIiooRidGKClGKiIiI0YoRi1GKEYoKiZGMUYoRi5GKEYvKigiIztGKC1JIlFHRiQ2IyIiIUYoKUYuRidGKEYoKigiI0dGKC0tSSJERzYkRitJKF9zeXNsaWJHRiQ2I0Y3RjhGKEY6RihGKCooRidGKEY2RigpRi5GMkYoRi8qKiIjP0YoRjZGKEYqRihGOkYoRi8qJiIjRkYoRi1GKEYoRidGKCooIiM9RihGKkYoRkRGKEYoKiZGNUYoRkRGKEYvRigtSSJPR0ZANiNGKEYy

Now we do not have an expression of gamma, but a cubic equation for it eq1:=collect(coeff(%,y,1),gamma, factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTFHRigsNCooIiIlIiIiKUkmZ2FtbWFHRiYiIiRGMSlJInRHRihGNEYxISIiKipGNkYxLCYqJiIiI0YxRjZGMUYxRjFGN0YxLCZGOkYxRjFGMUYxKUYzRjtGMUYxKipGO0YxKUY2RjtGMSwmKigiIzVGMS1JIlFHRig2IyIiIUYxRj9GMUYxIiIqRjdGMUYzRjFGNyooIiM7RjFGQ0YxKUY2RjBGMUYxKigiI0dGMS0tSSJER0YlNiNGREZFRjFGSkYxRjEqKEYwRjFGQ0YxRj9GMUY3KiYiI0ZGMUY1RjFGMSomRklGMUY/RjFGN0YwRjE3I0Yu

We finally eliminate gamma factor(resultant(eq1,edQ4,gamma));nops(%);

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

IiIk

Here is the first order DE for Q(y) involving the series Q_{0,i} (here m=3) edQ:=collect(op(1,op(3,factor(resultant(eq1,edQ4,gamma)))),[diff(Q(y), y),Q(y)],distributed,factor);;

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

Checking this DE n:=10: normal(series(eval(subs(Q(y)=Qvser(n),Q(0)=subs(y=0,Qvser(n)), (D(Q))(0)=subs(y=0,diff(Qvser(n),y)),((D@@2)(Q))(0)=subs(y=0,diff(Qvser(n),y$2)),edQ)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

We expand the DE in y to see a connection between Q_{0,2} and Q0, Q1 factor(series(edQ,y,3));

KydJInlHNiIsJCooIiInIiIiSSJ0R0YkRigsLiooIiIjRigpLUkiUUdGJDYjIiIhRixGKClGKSIiJEYoRigqKEYzRigtLS1JI0BARzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiRJIkRHRjlGLDYjRi9GMEYoKUYpRixGKEYoKigiIiVGKC0tRj1GPkYwRihGP0YoRigqKEYzRihGLkYoRilGKCEiIkZCRkUqJkZBRihGKUYoRihGKEYoRiwtSSJPR0Y5NiNGKEYz

subs(((D@@2)(Q))(0)=2*Q2,(D(Q))(0)=Q1,Q(0)=Q0,coeff(%,y,2));

LCQqKCIiJyIiIkkidEc2IkYlLC4qKCIiI0YlKUkjUTBHRidGKkYlKUYmIiIkRiVGJSooIiIlRiVJI1ExR0YnRiUpRiZGKkYlRiUqKEYkRiVJI1EyR0YnRiVGMkYlRiUqKEYuRiVGLEYlRiZGJSEiIkYxRjYqJkYwRiVGJkYlRiVGJUYl

op(3,%);

LC4qKCIiIyIiIilJI1EwRzYiRiRGJSlJInRHRigiIiRGJUYlKigiIiVGJUkjUTFHRihGJSlGKkYkRiVGJSooIiInRiVJI1EyR0YoRiVGL0YlRiUqKEYrRiVGJ0YlRipGJSEiIkYuRjQqJkYtRiVGKkYlRiU=

Introducing the series Q_{0,i} (here denoted Q_i) edQ:=subs(Q(0)=Q0, (D(Q))(0)=Q1,edQ);

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

degree(edQ,Q0);degree(edQ,Q1);

IiIi

IiIi

dedQ:=factor(diff(edQ,y)): indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KUkjUTBHRihJI1ExR0YoSSJ0R0YoSSJ5R0YoLUkiUUdGKDYjRjAtSSVkaWZmR0YmNiRGMUYwLUY1NiRGMS1JIiRHRiY2JEYwIiIjNyM8KUYtRi5GL0YwRjFGNC1GNTYkRjRGMA==

We now want to obtain polynomial coefficients in t and y. We eliminate the Q_i one after the other, increasing the order of the DE res1:=factor(resultant(edQ,dedQ,Q1));nops(%);

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

IiIl

Choosing and checking the correct factor (the first one cannot cancel) edQbis:=collect(op(4,res1),[diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0,Q1],factor);

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KEkjUTBHRihJInRHRihJInlHRigtSSJRR0YoNiNGLy1JJWRpZmZHRiY2JEYwRi8tRjQ2JEYwLUkiJEdGJjYkRi8iIiM3IzwoRi1GLkYvRjBGMy1GNDYkRjNGLw==

degree(edQbis,Q0);

IiIi

Eliminating Q0 dedQbis:=factor(diff(edQbis,y)): factor(resultant(edQbis,dedQbis,Q0)); nops(%);

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

IiIj

edQpol:=collect(op(2,%%),[diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0],factor);;

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

Let us check n:=20: normal(series(eval(subs(Q(y)=Qvser(n),edQpol)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0A=

We note that expanding the 1st order DE or the one we have just obtained gives the same information (relating Q2 to Q0, Q1) convert(factor(subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,series(edQpol,y,1))),polynom);

LC4qKCIjNyIiIilJI1EwRzYiIiIjRiUpSSJ0R0YoIiIkRiUhIiIqKCIjQ0YlSSNRMUdGKEYlKUYrRilGJUYtKigiI09GJUkjUTJHRihGJUYxRiVGLSooIiM9RiVGJ0YlRitGJUYlKiYiIidGJUYwRiVGJSomRi9GJUYrRiVGLQ==

convert(normal(subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,series(edQ,y,3))),polynom);

KiYsLiooIiM3IiIiKUkjUTBHNiIiIiNGJilJInRHRikiIiVGJkYmKigiI0NGJkkjUTFHRilGJilGLCIiJEYmRiYqKCIjT0YmSSNRMkdGKUYmRjFGJkYmKigiIz1GJkYoRiYpRixGKkYmISIiKigiIidGJkYwRiZGLEYmRjkqJkYvRiZGOEYmRiZGJilJInlHRilGKkYm

factor(%%/%);

LCQqJkkidEc2IiEiIilJInlHRiUiIiNGJkYm

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

stepset:=[[0,1],[-1,0],[1,0],[1,1],[-1,-1]];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShzdGVwc2V0R0YoNyc3JCIiISIiIjckISIiRjA3JEYxRjA3JEYxRjE3JEYzRjM3I0Yu

liste:=convert(map(s->x^s[1]*y^s[2],stepset),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwsSSJ5R0YoIiIiKiRJInhHRighIiJGMEYyRjAqJkYyRjBGL0YwRjAqJkYyRjNGL0YzRjA3I0Yu

Number of walks of length n ending at (i,j) co:=proc(i,j,n) option remember: if i<0 or j<0 or n<0 then 0 elif n=0 and i=0 and j=0 then 1 elif n=0 then 0 else convert(map (s-> co(i-s[1],j-s[2],n-1),stepset),`+`): fi: end: co(0,0,2);

IiIj

The series Q(x,y;t) Qser:=proc(N) add(add(add(co(i,j,n)*x^i*y^j,i=0..n),j=0..n)*t^n,n=0..N): end: Qser(4);

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

The series Q(0,y;t), for walks ending on the vertical axis Qvser:=proc(N) add(add(co(0,j,n)*y^j,j=0..n)*t^n,n=0..N): end: Qvser(5);

LC4iIiJGIyomSSJ0RzYiRiNJInlHRiZGI0YjKiYsKCokKUYnIiIjRiNGI0YnRiNGLEYjRiMpRiVGLEYjRiMqJiwqKiQpRiciIiRGI0YjKiZGMkYjRitGI0YjKiYiIidGI0YnRiNGI0YsRiNGIylGJUYyRiNGIyomLCwqJClGJyIiJUYjRiMqJkY1RiNGMUYjRiMqJiIjOUYjRitGI0YjKiYiIzhGI0YnRiNGIyIjNkYjRiMpRiVGO0YjRiMqJiwuKiQpRiciIiZGI0YjKiYiIzVGI0Y6RiNGIyomIiNJRiNGMUYjRiMqJiIjXEYjRitGI0YjKiYiI2RGI0YnRiNGIyIjRkYjRiMpRiVGR0YjRiM=

The kernel K:=expand(x*y*(t*liste-1));subs(x=0,K);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGMkYxRjZGMUYxKihGMEYxRjNGMUY1RjFGMSomRjBGMUY2RjFGMSomRjNGMUY2RjEhIiJGMEYxNyNGLg==

LCYqJkkidEc2IiIiIkkieUdGJUYmRiZGJEYm

The discriminant tilde d(y) dy:=factor(discrim(K,x));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkeUdGKComLC4qJilJInRHRigiIiMiIiIpSSJ5R0YoIiIkRjRGNCooIiIlRjRGMUY0KUY2RjNGNCEiIiooIiIpRjRGMUY0RjZGNEY7KihGM0Y0RjJGNEY6RjRGOyomRjlGNEYxRjRGO0Y2RjRGNEY2RjQ3I0Yu

Decoupling function G:=-(1+t)/t/(1+y)-y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJHR0YoLCYqKCwmIiIiRjFJInRHRihGMUYxRjIhIiIsJkkieUdGKEYxRjFGMUYzRjNGNUYzNyNGLg==

Connection between S(y) (or Q(0,y)) and the weak invariant w eqSw:=subs(x=0,K)*Q(y)-G=alpha/(w(y)-beta)+gamma;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcVN3R0YoLywoKiYsJiomSSJ0R0YoIiIiSSJ5R0YoRjRGNEYzRjRGNC1JIlFHRig2I0Y1RjRGNCooLCZGNEY0RjNGNEY0RjMhIiIsJkY1RjRGNEY0RjtGNEY1RjQsJiomSSZhbHBoYUdGKEY0LCYtSSJ3R0YoRjhGNEklYmV0YUdGKEY7RjtGNEkmZ2FtbWFHRiZGNDcjRi4=

wyQ:=op(2,isolate(eqSw,w(y)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR3eVFHRigsJiomLCoqJiwmKiZJInRHRigiIiJJInlHRihGNUY1RjRGNUY1LUkiUUdGKDYjRjZGNSEiIiooLCZGNUY1RjRGNUY1RjRGOiwmRjZGNUY1RjVGOkY6RjZGOkkmZ2FtbWFHRiZGNUY6SSZhbHBoYUdGKEY1RjpJJWJldGFHRihGNTcjRi4=

series(wyQ,y,3);

KytJInlHNiIsJiomLCgqJi1JIlFHRiQ2IyIiISIiIkkidEdGJEYtISIiKiYsJkYtRi1GLkYtRi1GLkYvRi9JJmdhbW1hRyUqcHJvdGVjdGVkR0YtRi9JJmFscGhhR0YkRi1GL0klYmV0YUdGJEYtRiwsJCosRi5GLSwqKiZGKUYtKUYuIiIjRi1GLSomRjJGLUYuRi1GL0YuRi1GLUYtRi8sKiomRi5GLS0tSSJERzYkRjNJKF9zeXNsaWJHRiQ2I0YqRitGLUYvRihGL0YwRi1GLUYvRi1GJ0YvRjRGLUYvRi0sJCooLCYqKEYuRi1GOEYvLCgqKCNGLUY7Ri1GLkYtLS0tSSNAQEdGQjYkRkFGO0ZERitGLUYvRj5GL0YwRi9GLUYtKipGLkYtLChGOUYtKiZGP0YtRjpGLUYtRi1GL0YtKUY4RjtGL0Y9Ri1GL0YtRidGL0Y0Ri1GL0Y7LUkiT0dGQjYjRi0iIiQ=

DE for w(y) eqw:=4*dy*(diff(w(y),y))^2-(4*w(y)^3-g2*w(y)-g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcXdHRigsKioqIiIlIiIiLC4qJilJInRHRigiIiNGMSlJInlHRigiIiRGMUYxKihGMEYxRjRGMSlGOEY2RjEhIiIqKCIiKUYxRjRGMUY4RjFGPCooRjZGMUY1RjFGO0YxRjwqJkYwRjFGNEYxRjxGOEYxRjFGOEYxKS1JJWRpZmZHRiY2JC1JIndHRig2I0Y4RjhGNkYxRjEqJkYwRjEpRkVGOUYxRjwqJkkjZzJHRihGMUZFRjFGMUkjZzNHRihGMTcjRi4=

subs(w(y)=wyQ,eqw);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsKioqIiIlIiIiLC4qJilJInRHRigiIiNGLylJInlHRigiIiRGL0YvKihGLkYvRjJGLylGNkY0Ri8hIiIqKCIiKUYvRjJGL0Y2Ri9GOiooRjRGL0YzRi9GOUYvRjoqJkYuRi9GMkYvRjpGNkYvRi9GNkYvKS1JJWRpZmZHRiY2JCwmKiYsKiomLCYqJkYzRi9GNkYvRi9GM0YvRi8tSSJRR0YoNiNGNkYvRjoqKCwmRi9GL0YzRi9GL0YzRjosJkY2Ri9GL0YvRjpGOkY2RjpJJmdhbW1hR0YmRi9GOkkmYWxwaGFHRihGL0Y6SSViZXRhR0YoRi9GNkY0Ri9GLyomRi5GLylGQ0Y3Ri9GOiomSSNnMkdGKEYvRkNGL0YvSSNnM0dGKEYvRis=

res:=numer(factor(eval(%))): nops(res);

IiVqNQ==

DE for Q(0,y) edQ:=collect(res,[diff(Q(y), y),Q],factor): indets(edQ);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkmYWxwaGFHRihJJWJldGFHRihJI2cyR0YoSSNnM0dGKEkidEdGKEkieUdGKC1JIlFHRig2I0YyLUklZGlmZkdGJjYkRjNGMkYr

Now we want to eliminate the unknown series, at the cost of introducing series Q_{0,i} factor(series(edQ,y=-1,1));

KycsJkkieUc2IiIiIkYmRiYqJiksJkYmRiZJInRHRiVGJiIiJUYmLCoqKEYrRiYpSSZhbHBoYUdGJSIiI0YmKUYqRjBGJkYmKiZGK0YmKUklYmV0YUdGJSIiJEYmISIiKiZGNEYmSSNnMkdGJUYmRiZJI2czR0YlRiZGJiIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YmRiY=

Expression of g3 g3s:=solve(coeff(%,y+1,0),g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnM3NHRigsKCooIiIlIiIiKUkmYWxwaGFHRigiIiNGMSlJInRHRihGNEYxISIiKiZGMEYxKUklYmV0YUdGKCIiJEYxRjEqJkY6RjFJI2cyR0YoRjFGNzcjRi4=

edQ1:=op(5,factor(subs(g3=g3s,edQ))):nops(%);

IiQnXA==

factor(series(edQ1,y=-1,1));

KycsJkkieUc2IiIiIkYmRiYsJComKSwmRiZGJkkidEdGJUYmIiIkRiYsKioqIiM7RiZJJmFscGhhR0YlRiZJJmdhbW1hRyUqcHJvdGVjdGVkR0YmKUYrIiIjRiZGJiooIiIpRiZGMEYmRitGJiEiIiomIiM3RiYpSSViZXRhR0YlRjRGJkY3SSNnMkdGJUYmRiZGNyIiIS1JIk9HNiRGMkkoX3N5c2xpYkdGJTYjRiZGJg==

Expression of g2 g2s:=solve(coeff(%,y+1,0),g2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnMnNHRigsKCoqIiM7IiIiSSZhbHBoYUdGKEYxSSZnYW1tYUdGJkYxKUkidEdGKCIiI0YxISIiKigiIilGMUYyRjFGNUYxRjEqJiIjN0YxKUklYmV0YUdGKEY2RjFGMTcjRi4=

We have thus eliminated g2 and g3 edQ2:=op(5,factor(subs(g2=g2s,edQ1)));

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

Now the series diff(Q,y$i) will start to come in factor(series(edQ2,y=-1,1));

KycsJkkieUc2IiIiIkYmRiYqJiksJkYmRiZJInRHRiVGJiIiI0YmLDIqKCIiJ0YmLUkiUUdGJTYjISIiRiYpRioiIiRGJkYmKihGLkYmKUkmZ2FtbWFHJSpwcm90ZWN0ZWRHRitGJilGKkYrRiZGMiooRi5GJkYvRiZGOUYmRiYqKEYuRiZGN0YmRipGJkYmKiZGK0YmRjlGJkYmKiZGNEYmSSViZXRhR0YlRiZGJiomRi5GJkYqRiZGJkYmRjJGJiIiIS1JIk9HNiRGOEkoX3N5c2xpYkdGJTYjRiZGJg==

Expression of beta betas:=solve(coeff(%,y+1,0),beta);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZiZXRhc0dGKCwwKigiIiMiIiItSSJRR0YoNiMhIiJGMSlJInRHRigiIiRGMUY1KihGMEYxKUkmZ2FtbWFHRiZGMEYxKUY3RjBGMUYxKihGMEYxRjJGMUY8RjFGNSooRjBGMUY7RjFGN0YxRjUqJiNGMEY4RjFGPEYxRjUqJkYwRjFGN0YxRjUjRjFGOEYxNyNGLg==

edQ3:=op(3,factor(subs(beta=betas,edQ2))): nops(%);

IiRYIw==

factor(series(edQ3,y=-1,2));

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

Expression of alpha alphas:=solve(coeff(%,y+1,1),alpha);

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

edQ4:=factor(subs(alpha=alphas,edQ3)):nops(%);

IiRLIw==

factor(series(edQ4,y=-1,3));

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

Now we do not have an expression of gamma, but a quartic equation for it eq1:=collect(coeff(%,y+1,2),gamma, factor);

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

We finally eliminate gamma factor(resultant(eq1,edQ4,gamma));nops(%);

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

IiIj

Here is the first order DE for Q(y) involving the series Q_{0,i} (here m=3) edQ:=collect(op(1,op(2,factor(resultant(eq1,edQ4,gamma)))),[diff(Q(y), y),Q(y)],distributed,factor);;

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

Checking this DE n:=10: normal(series(eval(subs(Q(y)=Qvser(n),Q(-1)=subs(y=-1,Qvser(n)), (D(Q))(-1)=subs(y=-1,diff(Qvser(n),y)),((D@@2)(Q))(-1)=subs(y=-1,diff(Qvser(n),y$2)),edQ)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

Introducing the series Q_{0,i} (here denoted Q_i) edQ:=subs(Q(-1)=Q0, (D(Q))(-1)=Q1, ((D@@2)(Q))(-1)=2*Q2,edQ);

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

degree(edQ,Q0);degree(edQ,Q1);degree(edQ,Q2);

IiIj

IiIi

IiIi

dedQ:=factor(diff(edQ,y)): indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkjUTBHRihJI1ExR0YoSSNRMkdGKEkidEdGKEkieUdGKC1JIlFHRig2I0YxLUklZGlmZkdGJjYkRjJGMS1GNjYkRjItSSIkR0YmNiRGMSIiIzcjPCpGLUYuRi9GMEYxRjJGNS1GNjYkRjVGMQ==

We now want to obtain polynomial coefficients in t and y. We eliminate the Q_i one after the other, increasing the order of the DE res1:=factor(resultant(edQ,dedQ,Q2));nops(%);

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

IiIm

Choosing and checking the correct factor (the first one cannot cancel because of the factors y and the CT 1) edQbis:=collect(op(5,res1),[diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0,Q1],factor);

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KUkjUTBHRihJI1ExR0YoSSJ0R0YoSSJ5R0YoLUkiUUdGKDYjRjAtSSVkaWZmR0YmNiRGMUYwLUY1NiRGMS1JIiRHRiY2JEYwIiIjNyM8KUYtRi5GL0YwRjFGNC1GNTYkRjRGMA==

degree(edQbis,Q1);

IiIi

Eliminating Q1 dedQbis:=factor(diff(edQbis,y)): factor(resultant(edQbis,dedQbis,Q1)): nops(%);

IiIk

edQter:=collect(op(3,%%),[diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0],factor);;

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KUkjUTBHRihJInRHRihJInlHRigtSSJRR0YoNiNGLy1JJWRpZmZHRiY2JEYwRi8tRjQ2JEYwLUkiJEdGJjYkRi8iIiMtRjQ2JEYwLUY5NiRGLyIiJDcjPClGLUYuRi9GMEYzLUY0NiRGM0YvLUY0NiRGQ0Yv

Eliminating Q0 dedQter:=op(2,factor(diff(edQter,y))):nops(%);

IiQ3Ig==

res2:=factor(resultant(edQter,dedQter,Q0)):nops(res2);

IiIl

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

IiIn

LCYiIiJGI0kidEc2IkYj

LCZJInlHNiIiIiJGJUYl

nops(op(4,res2));

IiQtJA==

edQpol:=collect(op(4,res2),[diff(Q(y), y, y, y, y),diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y)],factor);;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdlZFFwb2xHRigsOiomLCgqLClJInRHRigiIiMiIiJJInlHRihGNSksJkY2RjVGNUY1IiIlRjUsLiomRjJGNSlGNiIiJEY1RjUqKEY5RjVGMkY1KUY2RjRGNSEiIiooIiIpRjVGMkY1RjZGNUZAKihGNEY1RjNGNUY/RjVGQComRjlGNUYyRjVGQEY2RjVGNS1JJWRpZmZHRiY2JC1JIlFHRig2I0Y2RjZGNUY1KixGMkY1RjZGNSlGOEY9RjVGOkY1RkhGNUY1KipGNkY1RjhGNSwoKiZGM0Y1Rj9GNUY1KihGNEY1RjNGNUY2RjVGNUY1RkBGNUY6RjVGNUY1LUZGNiRGSC1JIiRHRiY2JEY2RjlGNUY1KiYsSiosRjJGNUY2RjVGN0Y1RjpGNS1GRjYkRkgtRlQ2JEY2RjRGNUZAKipGMkY1RkxGNSw0KigiIzdGNUYyRjUpRjZGOUY1RjUqKCIjR0Y1RjJGNUY8RjVGQCooIiMnKUY1RjJGNUY/RjVGQCooIiM+RjVGM0Y1RjxGNUZAKigiI2VGNUYyRjVGNkY1RkAqKCIjOkY1RjNGNUY/RjVGQComIiM1RjVGMkY1RkAqJiIiKEY1Rj9GNUY1KiYiIiZGNUY2RjVGNUY1RkVGNUY1KipGMkY1KUY4RjRGNSw0KigiIzlGNUYyRjVGW29GNUY1KigiI09GNUYyRjVGPEY1RkAqKCIkLSJGNUYyRjVGP0Y1RkAqKCIjQkY1RjNGNUY8RjVGQCooIiNtRjVGMkY1RjZGNUZARmRvRkBGZm9GQComIiIqRjVGP0Y1RjVGam9GNUY1RkhGNUY1KihGYHBGNSlGNiIiJ0Y1KUYzRj1GNUY1KihGQkY1KUY2RltwRjVGXnFGNUZAKigiJHciRjVGXnFGNUZbb0Y1RkAqKEZmcEY1RmBxRjVGMkY1RkAqKCIkaSNGNUZecUY1RjxGNUZAKigiI3hGNUZbb0Y1RjJGNUZAKigiJEUiRjVGXnFGNUY/RjVGQCooIiM9RjVGMkY1RjxGNUY1KihGanBGNUYzRjVGW29GNUY1KihGam5GNUZecUY1RjZGNUZAKigiJD0iRjVGMkY1Rj9GNUY1KigiI11GNUY8RjVGM0Y1RjUqKCIjdUY1RjJGNUY2RjVGNSooRmZwRjVGM0Y1Rj9GNUY1RmZvRjUqJiIjNkY1Rj9GNUZARmpvRkBGNS1GRjYkRkgtRlQ2JEY2Rj1GNUY1KixGPUY1RjJGNUZMRjUsNCooRj1GNUYyRjVGW29GNUY1KihGQkY1RjJGNUY8RjVGQCooIiNBRjVGMkY1Rj9GNUZAKihGW3BGNUYzRjVGPEY1RkAqKEZgcEY1RjJGNUY2RjVGQCooRj1GNUYzRjVGP0Y1RkAqJkY0RjVGMkY1RkAqJkY0RjVGP0Y1RjVGNkY1RjUpRllGNEY1RkAqJixGKixGPUY1RjJGNUZdcEY1LDYqKEZdcUY1RjJGNUZbb0Y1RjUqKEY5RjVGMkY1RjxGNUY1KigiI0NGNUYyRjVGP0Y1RkBGYXNGQCooIiNJRjVGMkY1RjZGNUZARmRvRkBGZm9GQCooRl1xRjVGM0Y1RjZGNUZAKiZGPUY1RjZGNUY1RjVGNUY1RkVGNUY1KixGPUY1RjJGNUZdcEY1LDIqKCIjO0Y1RjJGNUY8RjVGNSooRmJwRjVGMkY1Rj9GNUZAKigiI2dGNUYyRjVGNkY1RkAqKCIjQEY1RjNGNUY/RjVGQComRltyRjVGMkY1RkBGYXRGQComRl1xRjVGNkY1RjVGNUY1RjVGSEY1RjUqKCIjW0Y1RmBxRjVGXnFGNUY1KihGam5GNUZecUY1RltvRjVGQCooIiQ/JUY1Rl5xRjVGPEY1RkAqKCIjakY1RltvRjVGMkY1RkAqKCIkVSRGNUZecUY1Rj9GNUZAKigiJDsjRjVGMkY1RjxGNUZARl1yRkAqKCIkIT1GNUYyRjVGP0Y1RjUqKEZbckY1RjxGNUYzRjVGNSomRl50RjVGXnFGNUY1KigiJHcjRjVGMkY1RjZGNUY1KigiJFEiRjVGM0Y1Rj9GNUY1KiYiI3lGNUYyRjVGNSooRmpuRjVGM0Y1RjZGNUY1KiZGYHRGNUY2RjVGQEY9RkBGNUZZRjVGNSomLCYqKkZqbkY1KUYzRl1xRjUpRjhGXXFGNUZIRjVGQCoqRl1xRjUpRjNGOUY1RjdGNSwsRkNGNUZQRjUqJkY0RjVGM0Y1RjVGNkZARjVGNUY1RkBGNSlGRUY9RjVGNSomLCgqKkZicEY1Rmh2RjUpRjhGW3BGNSlGSEY0RjVGQCosRltyRjVGW3dGNUY3RjUsKCooRjlGNUYzRjVGNkY1RjVGXXdGNUY1RkBGNUZIRjVGQCoqRl1xRjVGMkY1RjhGNSw4KihGNEY1RjJGNUZbb0Y1RjUqKEZdcUY1RjJGNUY8RjVGNSooRmZ0RjVGMkY1Rj9GNUY1KihGNEY1RjNGNUY8RjVGQCooRl50RjVGMkY1RjZGNUY1KiZGXXFGNUYyRjVGNSooRmpwRjVGM0Y1RjZGNUY1KiRGP0Y1RjUqJkZbcEY1RjNGNUZARmJ0RjVGOUZARjVGQEY1KUZFRjRGNUY1KiYsQioqRmJwRjVGaHZGNUY3RjUpRkhGPUY1RkAqLEZbckY1Rlt3RjVGXXBGNSwsKihGXXFGNUYzRjVGP0Y1RjUqKEZnb0Y1RjNGNUY2RjVGNUZdd0Y1RjZGQEY9RkBGNUZjd0Y1RkAqKkZdcUY1RjJGNSw6KihGQkY1RjJGNUZbb0Y1RjUqKCIjX0Y1RjJGNUY8RjVGNUZeb0Y1Rlx4RjVGaHRGNSooRmpuRjVGM0Y1Rj9GNUZAKiZGam5GNUYyRjVGNUZjdkZARmB4RkAqJkZnb0Y1RjNGNUZAKiZGOUY1RjZGNUY1RjVGQEY1RkhGNUZAKihGXnRGNUZecUY1RltvRjVGNSooRml0RjVGXnFGNUY8RjVGQCooIiRDJEY1Rl5xRjVGP0Y1RkAqKEZgdEY1RjJGNUY8RjVGQCooRmJwRjVGXnFGNUY2RjVGQCooIiQ5IkY1RjJGNUY/RjVGQComRml0RjVGXnFGNUY1KigiJGsjRjVGMkY1RjZGNUY1Rml4RjUqJiIkSyJGNUYyRjVGNSooRmJ2RjVGM0Y1RjZGNUY1KiZGW3JGNUYzRjVGQEZbckZARjVGRUY1RjUqKkZqbkY1Rmh2RjVGTEY1KUZIRjlGNUZAKixGXXFGNUZbd0Y1RjhGNSwsKihGQkY1RjNGNUY/RjVGNSooRmBwRjVGM0Y1RjZGNUY1Rl13RjVGNkZARltwRkBGNUZmeEY1RkAqKkZdcUY1RjJGNSwyKihGZ29GNUYyRjVGPEY1RjUqKEZdb0Y1RjJGNUY/RjVGNUZic0Y1RkNGQEZkc0Y1KihGZnRGNUYzRjVGNkY1RkBGYXhGQEY9RjVGNUZjd0Y1RkAqKkZdcUY1RjNGNSwyRlx0RjUqKEZgcEY1RjJGNUY/RjVGNUZBRjVGT0ZARmRzRkBGanhGQComRl1xRjVGM0Y1RkBGPUY1RjVGSEY1RkAqKEZqbkY1RjNGNSwqRk9GNUZQRjVGXXdGQEY9RkBGNUZANyMsOiomRjBGNS1GRjYkLUZGNiQtRkY2JEZFRjZGNkY2RjVGNSomLEoqLEYyRjVGNkY1RjdGNUY6RjVGW1xsRjVGQEZnbkY1RlxwRjVGW3FGNUZfcUZARmFxRkBGY3FGQEZkcUZARmZxRkBGaHFGQEZqcUY1RlxyRjVGXXJGQEZeckY1RmByRjVGYnJGNUZkckY1RmZvRjVGZXJGQEZqb0ZARjVGaVtsRjVGNSosRj1GNUYyRjVGTEY1RlxzRjUpRltcbEY0RjVGQComRmhzRjVGW1xsRjVGNUZldkY1Rl93RjVGY3hGNUZjekZARmV6RkBGaXpGQEZeW2xGQEZiW2xGQA==

Let us check n:=20: normal(series(eval(subs(Q(y)=Qvser(n),edQpol)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0A=

We note that expanding the 1st order DE or the one we have just obtained gives the same information (relating Q3 to Q0, Q1 and Q2) convert(factor(subs(Q(-1)=Q0, (D(Q))(-1)=Q1, ((D@@2)(Q))(-1)=2*Q2,series(edQpol,y=-1,1))),polynom);

LCQqKCIiJyIiIiwmRiVGJUkidEc2IkYlRiUsQCooRiRGJSlJI1EwR0YoIiIjRiUpRiciIiRGJUYlKipGJEYlRixGJUkjUTFHRihGJUYuRiUhIiIqKEYvRiVGK0YlKUYnRi1GJUYlKipGJEYlRixGJUYxRiVGNEYlRjIqJi0tLUkjQEBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRig2JEkiREdGO0YvNiNJIlFHRig2I0YyRiVGNEYlRiUqKCIjQ0YlRjFGJUY0RiVGJSooIiM9RiVGNEYlSSNRMkdGKEYlRjIqKEYtRiVGN0YlRidGJUYlKihGL0YlRixGJUYnRiVGJSooIiM3RiVGJ0YlRjFGJUYlKigiI0ZGJUYnRiVGR0YlRjJGN0YlKiZGL0YlRjFGJUYlKiYiIipGJUZHRiVGMiomRiRGJUYnRiVGMkYlRjI=

convert(normal(subs(Q(-1)=Q0, (D(Q))(-1)=Q1, ((D@@2)(Q))(-1)=2*Q2,series(edQ,y=-1,4))),polynom);

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

factor(%%/%);

LCQqJiIiJCIiIiksJkkieUc2IkYlRiVGJUYkISIiRiU=

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

stepset:=[[0,1],[0,-1],[-1,0],[1,1],[-1,-1]];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShzdGVwc2V0R0YoNyc3JCIiISIiIjckRjAhIiI3JEYzRjA3JEYxRjE3JEYzRjM3I0Yu

liste:=convert(map(s->x^s[1]*y^s[2],stepset),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwsSSJ5R0YoIiIiKiRGLyEiIkYwKiRJInhHRihGMkYwKiZGNEYwRi9GMEYwKiZGNEYyRi9GMkYwNyNGLg==

Number of walks of length n ending at (i,j) co:=proc(i,j,n) option remember: if i<0 or j<0 or n<0 then 0 elif n=0 and i=0 and j=0 then 1 elif n=0 then 0 else convert(map (s-> co(i-s[1],j-s[2],n-1),stepset),`+`): fi: end: co(0,0,2);

IiIj

The series Q(x,y;t) Qser:=proc(N) add(add(add(co(i,j,n)*x^i*y^j,i=0..n),j=0..n)*t^n,n=0..N): end: Qser(4);

LCwiIiJGIyomLCYqJkkieEc2IkYjSSJ5R0YoRiNGI0YpRiNGI0kidEdGKEYjRiMqJiwuKiYpRiciIiNGIylGKUYvRiNGIyooRi9GI0YnRiNGMEYjRiMqJEYwRiNGI0YnRiNGKUYjRi9GI0YjKUYqRi9GI0YjKiYsNiomKUYnIiIkRiMpRilGOEYjRiMqKEY4RiNGLkYjRjlGI0YjKihGOEYjRidGI0Y5RiNGIyooRi9GI0YuRiNGKUYjRiNGMUYjKiRGOUYjRiMqKCIiJ0YjRidGI0YpRiNGIyomRjhGI0YwRiNGIyomIiImRiNGKUYjRiNGL0YjRiMpRipGOEYjRiMqJixEKiYpRiciIiVGIylGKUZIRiNGIyooRkhGI0Y3RiNGSUYjRiMqKEY/RiNGLkYjRklGI0YjKihGOEYjRjdGI0YwRiNGI0Y6RiMqKEZIRiNGJ0YjRklGI0YjKigiIzdGI0YuRiNGMEYjRiMqKCIiKUYjRidGI0Y5RiNGIyokRklGI0YjKigiIzxGI0YnRiNGMEYjRiMqJkY/RiNGOUYjRiMqJkYvRiNGLkYjRiNGPkYjKiYiIzZGI0YwRiNGIyomRlFGI0YnRiNGIyomIiM4RiNGKUYjRiNGWEYjRiMpRipGSEYjRiM=

The series Q(0,y;t), for walks ending on the vertical axis Qvser:=proc(N) add(add(co(0,j,n)*y^j,j=0..n)*t^n,n=0..N): end: Qvser(5);

LC4iIiJGIyomSSJ0RzYiRiNJInlHRiZGI0YjKiYsKCokKUYnIiIjRiNGI0YnRiNGLEYjRiMpRiVGLEYjRiMqJiwqKiQpRiciIiRGI0YjKiZGMkYjRitGI0YjKiYiIiZGI0YnRiNGI0YsRiNGIylGJUYyRiNGIyomLCwqJClGJyIiJUYjRiMqJiIiJ0YjRjFGI0YjKiYiIzZGI0YrRiNGIyomIiM4RiNGJ0YjRiNGP0YjRiMpRiVGO0YjRiMqJiwuKiQpRidGNUYjRiMqJiIjNUYjRjpGI0YjKiYiI0NGI0YxRiNGIyomIiNXRiNGK0YjRiMqJiIjWEYjRidGI0YjIiNGRiNGIylGJUY1RiNGIw==

The kernel K:=expand(x*y*(t*liste-1));subs(x=0,K);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGM0YxRjVGMUYxKiZGMEYxRjNGMUYxKiZGMEYxRjZGMUYxKiZGM0YxRjZGMSEiIkYwRjE3I0Yu

LCYqJkkidEc2IiIiIkkieUdGJUYmRiZGJEYm

The discriminant tilde d(y) dy:=factor(discrim(K,x));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkeUdGKCwwKiYpSSJ0R0YoIiIjIiIiKUkieUdGKCIiJUYzRjMqKEY2RjNGMEYzKUY1IiIkRjMhIiIqKEYyRjNGMEYzKUY1RjJGM0Y6KihGMkYzRjFGM0Y4RjNGOiokRjBGM0YzKihGMkYzRjFGM0Y1RjNGOiokRjxGM0YzNyNGLg==

Decoupling function G:=-1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJHR0YoLCQqJEkieUdGKCEiIkYxNyNGLg==

Connection between S(y) (or Q(0,y)) and the weak invariant w eqSw:=subs(x=0,K)*Q(y)-G=alpha/(w(y)-beta)+gamma;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcVN3R0YoLywmKiYsJiomSSJ0R0YoIiIiSSJ5R0YoRjRGNEYzRjRGNC1JIlFHRig2I0Y1RjRGNCokRjUhIiJGNCwmKiZJJmFscGhhR0YoRjQsJi1JIndHRihGOEY0SSViZXRhR0YoRjpGOkY0SSZnYW1tYUdGJkY0NyNGLg==

wyQ:=op(2,isolate(eqSw,w(y)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR3eVFHRigsJiomLCgqJiwmKiZJInRHRigiIiJJInlHRihGNUY1RjRGNUY1LUkiUUdGKDYjRjZGNSEiIiokRjZGOkY6SSZnYW1tYUdGJkY1RjpJJmFscGhhR0YoRjVGOkklYmV0YUdGKEY1NyNGLg==

series(wyQ,y,3);

KytJInlHNiJJJWJldGFHRiQiIiFJJmFscGhhR0YkIiIiLCQqJiwmKiYtSSJRR0YkNiNGJkYoSSJ0R0YkRihGKEkmZ2FtbWFHJSpwcm90ZWN0ZWRHISIiRihGJ0YoRjMiIiMtSSJPRzYkRjJJKF9zeXNsaWJHRiQ2I0YoIiIk

DE for w(y) eqw:=4*dy*(diff(w(y),y))^2-(4*w(y)^3-g2*w(y)-g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcXdHRigsKiooIiIlIiIiLDAqJilJInRHRigiIiNGMSlJInlHRihGMEYxRjEqKEYwRjFGNEYxKUY4IiIkRjEhIiIqKEY2RjFGNEYxKUY4RjZGMUY8KihGNkYxRjVGMUY6RjFGPCokRjRGMUYxKihGNkYxRjVGMUY4RjFGPCokRj5GMUYxRjEpLUklZGlmZkdGJjYkLUkid0dGKDYjRjhGOEY2RjFGMSomRjBGMSlGR0Y7RjFGPComSSNnMkdGKEYxRkdGMUYxSSNnM0dGKEYxNyNGLg==

subs(w(y)=wyQ,eqw);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsKiooIiIlIiIiLDAqJilJInRHRigiIiNGLylJInlHRihGLkYvRi8qKEYuRi9GMkYvKUY2IiIkRi8hIiIqKEY0Ri9GMkYvKUY2RjRGL0Y6KihGNEYvRjNGL0Y4Ri9GOiokRjJGL0YvKihGNEYvRjNGL0Y2Ri9GOiokRjxGL0YvRi8pLUklZGlmZkdGJjYkLCYqJiwoKiYsJiomRjNGL0Y2Ri9GL0YzRi9GLy1JIlFHRig2I0Y2Ri9GOiokRjZGOkY6SSZnYW1tYUdGJkYvRjpJJmFscGhhR0YoRi9GOkklYmV0YUdGKEYvRjZGNEYvRi8qJkYuRi8pRkVGOUYvRjoqJkkjZzJHRihGL0ZFRi9GL0kjZzNHRihGL0Yr

res:=numer(factor(eval(%))): nops(res);

IiQ+Iw==

DE for Q(0,y) edQ:=collect(res,[diff(Q(y), y),Q],factor);

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

indets(edQ);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkmYWxwaGFHRihJJWJldGFHRihJI2cyR0YoSSNnM0dGKEkidEdGKEkieUdGKC1JIlFHRig2I0YyLUklZGlmZkdGJjYkRjNGMkYr

Now we want to eliminate the unknown series, at the cost of introducing series Q_{0,i} factor(series(edQ,y,1));

KydJInlHNiIsKiooIiIlIiIiKUkmYWxwaGFHRiQiIiNGKClJInRHRiRGK0YoRigqJkYnRigpSSViZXRhR0YkIiIkRighIiIqJkYwRihJI2cyR0YkRihGKEkjZzNHRiRGKCIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRig=

Expression of g3 g3s:=solve(coeff(%,y,0),g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnM3NHRigsKCooIiIlIiIiKUkmYWxwaGFHRigiIiNGMSlJInRHRihGNEYxISIiKiZGMEYxKUklYmV0YUdGKCIiJEYxRjEqJkY6RjFJI2cyR0YoRjFGNzcjRi4=

edQ1:=op(4,factor(subs(g3=g3s,edQ))):nops(%);

IiRYIg==

factor(series(edQ1,y,1));

KydJInlHNiIsLCoqIiM7IiIiSSZhbHBoYUdGJEYoSSZnYW1tYUclKnByb3RlY3RlZEdGKClJInRHRiQiIiNGKCEiIioqRidGKC1JIlFHRiQ2IyIiIUYoRilGKClGLSIiJEYoRigqJiIjN0YoKUklYmV0YUdGJEYuRihGKEkjZzJHRiRGLyooIiIpRihGKUYoRi1GKEYoRjQtSSJPRzYkRitJKF9zeXNsaWJHRiQ2I0YoRig=

Expression of g2 g2s:=solve(coeff(%,y,0),g2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnMnNHRigsKioqIiM7IiIiLUkiUUdGKDYjIiIhRjFJJmFscGhhR0YoRjEpSSJ0R0YoIiIkRjFGMSoqRjBGMUY2RjFJJmdhbW1hR0YmRjEpRjgiIiNGMSEiIiooIiIpRjFGNkYxRjhGMUYxKiYiIzdGMSlJJWJldGFHRihGPUYxRjE3I0Yu

We have thus eliminated g2 and g3 edQ2:=op(3,factor(subs(g2=g2s,edQ1)));

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

Now the series Q_{0,i} will start to come in factor(series(edQ2,y,2));

KydJInlHNiIsNiomIiIkIiIiSSViZXRhR0YkRighIiIqKiIjN0YoLUkiUUdGJDYjIiIhRihJJmdhbW1hRyUqcHJvdGVjdGVkR0YoKUkidEdGJEYnRihGKiooIiInRigpRjEiIiNGKClGNEY4RihGKCooRjZGKClGNCIiJUYoKUYtRjhGKEYoKihGNkYoRi1GKEY5RihGKCooRjZGKEYxRihGNEYoRioqKEY2RihGM0YoLS1JIkRHNiRGMkkoX3N5c2xpYkdGJDYjRi5GL0YoRipGKEYoKiZGOEYoRjlGKEYqKihGNkYoRi1GKEYzRihGKkYoLUkiT0dGRDYjRihGOA==

Expression of beta betas:=solve(coeff(%,y,1),beta);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZiZXRhc0dGKCw0KigiIiMiIiIpSSJ0R0YoIiIlRjEpLUkiUUdGKDYjIiIhRjBGMUYxKipGNEYxRjZGMUkmZ2FtbWFHRiZGMSlGMyIiJEYxISIiKihGMEYxRjZGMUY8RjFGPiooRjBGMUY8RjEtLUkiREdGJTYjRjdGOEYxRj4qKEYwRjEpRjtGMEYxKUYzRjBGMUYxKihGMEYxRjZGMUZHRjFGMSooRjBGMUY7RjFGM0YxRj4qJiNGMEY9RjFGR0YxRj4jRjFGPUYxNyNGLg==

edQ3:=(factor(subs(beta=betas,edQ2))): nops(%);

IiRcIg==

factor(series(edQ3,y,3));

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

Expression of alpha alphas:=solve(coeff(%,y,2),alpha);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbHBoYXNHRigsTCooIiIlIiIiKS1JIlFHRig2IyIiISIiJEYxKUkidEdGKCIiJkYxISIiKioiIzdGMSlGMyIiI0YxSSZnYW1tYUdGJkYxKUY5RjBGMUYxKihGPUYxRkFGMUY+RjFGMSoqRj1GMUYzRjEtLUkiREdGJTYjRjRGNUYxRkFGMUYxKipGPUYxRjNGMSlGQEY/RjEpRjlGN0YxRjsqKCIiJ0YxRj5GMUZKRjFGOyoqRj1GMUYzRjFGQEYxRkpGMUY7KipGPUYxRkRGMUZARjFGSkYxRjsqKEYwRjEpRkBGN0YxKUY5Rj9GMUYxKihGMEYxRkpGMS0tLUkjQEBHRiU2JEZGRj9GR0Y1RjFGOyoqRj1GMUYzRjFGQEYxRlFGMUYxKihGMEYxRjNGMUZKRjFGMSooIiIpRjFGSkYxRkRGMUY7KigiIzVGMUYzRjFGUUYxRjEqKEZnbkYxRkRGMUZRRjFGMSooRkxGMUZJRjFGOUYxRjsqKEYwRjFGQEYxRlFGMUY7KihGP0YxRjNGMUY5RjFGOyomRjBGMUZRRjFGOyomRj9GMUZARjFGMSomRj9GMUY5RjFGOzcjRi4=

edQ4:=factor(subs(alpha=alphas,edQ3)):nops(%);

IiRiIg==

factor(series(edQ4,y,4));

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

Now we do not have an expression of gamma, but a quartic equation for it eq1:=collect(coeff(%,y,3),gamma, factor);

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

We finally eliminate gamma numer(factor(resultant(eq1,edQ4,gamma)));nops(%);

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

IiIj

Here is the first order DE for Q(y) involving the series Q_{0,i} (here m=3) edQ:=collect(op(1,op(2,numer(factor(resultant(eq1,edQ4,gamma))))),[diff(Q(y), y),Q(y)],distributed,factor);;

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Checking this DE n:=20: normal(series(eval(subs(Q(y)=Qvser(n),Q(0)=subs(y=0,Qvser(n)), (D(Q))(0)=subs(y=0,diff(Qvser(n),y)),((D@@2)(Q))(0)=subs(y=0,diff(Qvser(n),y$2)),((D@@3)(Q))(0)=subs(y=0,diff(Qvser(n),y$3)),edQ)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0A=

Introducing the series Q_{0,i} (here denoted Q_i) edQ:=subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2, ((D@@3)(Q))(0)=6*Q3,edQ);

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degree(edQ,Q0);degree(edQ,Q1);degree(edQ,Q2);degree(edQ,Q3);

IiIl

IiIj

IiIi

IiIi

dedQ:=factor(diff(edQ,y)): indets(%);

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We now want to obtain polynomial coefficients in t and y. We eliminate the Q_i one after the other, increasing the order of the DE res1:=factor(resultant(edQ,dedQ,Q3));nops(%);

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IiIm

Choosing and checking the correct factor (the first one cannot cancel because of the factors y and the CT 1) edQbis:=collect(op(5,res1),[diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0,Q1],factor);

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkjUTBHRihJI1ExR0YoSSNRMkdGKEkidEdGKEkieUdGKC1JIlFHRig2I0YxLUklZGlmZkdGJjYkRjJGMS1GNjYkRjItSSIkR0YmNiRGMSIiIzcjPCpGLUYuRi9GMEYxRjJGNS1GNjYkRjVGMQ==

degree(edQbis,Q2);

IiIi

Eliminating Q2 dedQbis:=factor(diff(edQbis,y)): factor(resultant(edQbis,dedQbis,Q2)): nops(%);

IiIl

edQter:=collect(op(4,%%),[diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0],factor);;

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkjUTBHRihJI1ExR0YoSSJ0R0YoSSJ5R0YoLUkiUUdGKDYjRjAtSSVkaWZmR0YmNiRGMUYwLUY1NiRGMS1JIiRHRiY2JEYwIiIjLUY1NiRGMS1GOjYkRjAiIiQ3IzwqRi1GLkYvRjBGMUY0LUY1NiRGNEYwLUY1NiRGREYw

Eliminating Q1 dedQter:=(factor(diff(edQter,y))):nops(%);

IiREIg==

res2:=factor(resultant(edQter,dedQter,Q1)):nops(res2);

IiIk

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

IiIn

KiQpSSJ0RzYiIiIjIiIi

nops(op(3,res2));

IiRKIw==

edQquat:=collect(op(3,res2),[diff(Q(y), y, y, y, y),diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0],factor);;

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkjUTBHRihJInRHRihJInlHRigtSSJRR0YoNiNGLy1JJWRpZmZHRiY2JEYwRi8tRjQ2JEYwLUkiJEdGJjYkRi8iIiMtRjQ2JEYwLUY5NiRGLyIiJC1GNDYkRjAtRjk2JEYvIiIlNyM8KkYtRi5GL0YwRjMtRjQ2JEYzRi8tRjQ2JEZIRi8tRjQ2JEZKRi8=

Eliminating Q0 dedQquat:=(factor(diff(edQquat,y))):nops(%);

IiIj

res2:=factor(resultant(edQquat,dedQquat,Q0)):nops(res2);

IiIm

op(1,res2);op(2,res2);op(3,res2);op(4,res2);

ISM3

KiQpSSJ5RzYiIiIkIiIi

KiQpSSJ0RzYiIiIjIiIi

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMqJCksKiooLUklZGlmZkdGJjYkLUkiUUdGKDYjSSJ5R0YoRjYiIiJJInRHRihGNylGNiIiJEY3RjcqKEYzRjdGOEY3KUY2IiIjRjdGNyooRjBGN0Y4RjdGPEY3RjdGNyEiIkY9RjdGKw==

nops(op(5,res2));

IiQtJw==

edQpol:=collect(op(5,res2),[diff(Q(y), y, y, y, y, y),diff(Q(y), y, y, y, y),diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y)],factor);;

-_I,TypesettingG6$%*protectedGI(_syslibG6"I,mprintslashGF(6$7#>I'edQpolGF(,D*&,,*,)I"yGF(""&""")I"tGF(""#F5),&F3F5F5F5""$F5,0*&F6F5)F3""%F5F5*(F?F5F6F5)F3F;F5!""*(F8F5F6F5)F3F8F5FB*(F8F5F7F5FAF5FB*$F6F5F5*(F8F5F7F5F3F5FB*$FDF5F5F5)-I%diffGF&6$-I"QGF(6#F3F3F8F5FB*&,&*.F8F5F2F5F6F5)F:F8F5F<F5FMF5FB*,F8F5FAF5F7F5FSF5F<F5F5F5FJF5F5*,F2F5F6F5F:F5F<F5)FMF8F5FB*.F8F5FAF5F7F5F:F5F<F5FMF5F5*(F3F5F:F5F<F5FBF5-FK6$FM-I"$GF&6$F3F4F5F5*&,D*&,(*.F;F5F2F5F6F5F9F5F<F5FJF5F5*.F;F5F2F5F6F5FSF5F<F5FMF5F5*,F;F5FAF5F7F5FSF5F<F5FBF5-FK6$FM-Ffn6$F3F8F5F5*,F2F5F6F5FSF5,8*("#8F5F6F5F>F5F5*("#CF5F6F5FAF5FB*("#aF5F6F5FDF5FB*("#>F5F7F5FAF5FB*("#9F5F6F5F3F5FB*("#@F5F7F5FDF5FBFFFB*(F4F5F7F5F3F5FB*&""'F5FDF5F5*&""(F5F7F5FB*&FepF5F3F5F5F5FIF5FB*&,&*.F8F5F2F5F6F5F:F5,8*("#;F5F6F5F>F5F5*("#OF5F6F5FAF5FB*("#gF5F6F5FDF5FB*("#DF5F7F5FAF5FBF]pFBF_pFB*&F8F5F6F5F5*("#6F5F7F5F3F5FB*&""*F5FDF5F5FdpFBFfpF5F5FMF5FB*,F8F5F7F5FDF5F:F5,:*(F\pF5F6F5F2F5F5*("#XF5F6F5F>F5FB*("#yF5F6F5FAF5FB*("#JF5F7F5F>F5FB*("#?F5F6F5FDF5FB*("#FF5F7F5FAF5FB*(F4F5F6F5F3F5F5*("#<F5F7F5FDF5FB*&"#7F5FAF5F5*&F;F5F6F5F5*(FfoF5F7F5F3F5FB*&"#5F5FDF5F5F5F5F5FJF5F5**F2F5F6F5,8*(F\pF5F6F5F>F5F5*("#[F5F6F5FAF5FB*("#mF5F6F5FDF5FB*(F`rF5F7F5FAF5FBF]pFBF_pFB*&F4F5F6F5F5*(FgrF5F7F5F3F5FB*&FirF5FDF5F5FdpFBFfpF5F5FVF5FB*,F8F5F7F5FDF5,:*("#AF5F6F5F2F5F5*("#dF5F6F5F>F5FB*("#%)F5F6F5FAF5FB*("#PF5F7F5F>F5FBFarFBFcrFB*("")F5F6F5F3F5F5*("#BF5F7F5FDF5FB*&"#:F5FAF5F5FjrF5F[sFBF\sF5F5FMF5F5*(FbqF5F2F5F6F5FB*(FdsF5F>F5F6F5F5*("$-"F5F6F5FAF5F5*("#VF5F7F5F>F5F5*("#EF5F6F5FDF5F5*("#LF5FAF5F7F5F5*(FeqF5F6F5F3F5FB*("#HF5F7F5FDF5F5*&"#=F5FAF5FB*&FcpF5F6F5FB*(F\pF5F7F5F3F5F5*&FfoF5FDF5FBF5-FK6$FM-Ffn6$F3F?F5F5*&,(*,F2F5F6F5F9F5F<F5FJF5F5*,F2F5F6F5FSF5F<F5FMF5F5**FAF5F7F5FSF5F<F5FBF5)-FK6$FM-Ffn6$F3F;F8F5F5*&,D*.F;F5F2F5F6F5F9F5F<F5)F_oF8F5FB*&,(*.FcpF5F2F5F6F5FSF5,8*(FepF5F6F5F>F5F5*(F\qF5F6F5FAF5FBF_uFB*(FeqF5F7F5FAF5FB*(FcpF5F6F5F3F5FB*(FgqF5F7F5FDF5FBFFF5FapFB*&F?F5FDF5F5*&F;F5F7F5FB*&F;F5F3F5F5F5FJF5F5*.FguF5F2F5F6F5F:F5,8*(F;F5F6F5F>F5F5*(FdtF5F6F5FAF5FB*(F]sF5F6F5FDF5FB*(F4F5F7F5FAF5FB*(F8F5F6F5F3F5FB*(F;F5F7F5FDF5FBFFF5*(F;F5F7F5F3F5FB*&F8F5FDF5F5F7FBF3F5F5FMF5F5*,FcpF5F7F5FDF5F:F5,:*(FeqF5F6F5F2F5F5*("#IF5F6F5F>F5FB*("#UF5F6F5FAF5FB*(F\pF5F7F5F>F5FBF]xFB*(FfoF5F7F5FAF5FBFerF5*(FfoF5F7F5FDF5FB*&FdtF5FAF5F5FcqF5*(FepF5F7F5F3F5FB*&F4F5FDF5F5F5FBF5F_oF5F5*.F?F5F2F5F6F5F:F5,8*(F4F5F6F5F>F5F5*(FbrF5F6F5FAF5F5*(FguF5F6F5FDF5FB*("#GF5F6F5F3F5FB*(FguF5F7F5FDF5FB*&FepF5F6F5FB*(F?F5F7F5F3F5FB*&F;F5FDF5FB*&F8F5F7F5F5*&F8F5F3F5F5F8F5F5FIF5FB*&,&*,F?F5F2F5F6F5,:*(FfyF5F6F5F>F5F5F@F5*("$3"F5F6F5FDF5FBFcrFB*("#uF5F6F5F3F5FB*("#jF5F7F5FDF5FB*&F^pF5F6F5FBFgsFBFjyF5*&F4F5F7F5FB*&FfoF5F3F5F5F?F5F5FMF5FB**F?F5F7F5F3F5,@*(F^uF5F6F5)F3FcpF5F5*(FdtF5F6F5F2F5FB*("$*=F5F6F5F>F5FB*(FbsF5F7F5F2F5FB*("$S"F5F6F5FAF5FB*("$0"F5F7F5F>F5FB*(FeuF5F6F5FDF5FB*("#TF5F7F5FAF5FB*&FgqF5F>F5F5FdwFB*(FeqF5F7F5FDF5FB*&FbqF5FAF5F5FjrFBFaxFBF\sF5F5F5F5FJF5F5*,F?F5F2F5F6F5,4*(F`uF5F6F5FAF5F5FioFB*(FixF5F6F5F3F5FB*(FbuF5F7F5FDF5FB*&F?F5F6F5FB*(FirF5F7F5F3F5FBFdpFB*&FgqF5F3F5F5F8F5F5FVF5FB*,F?F5F7F5F3F5,:*("#tF5F6F5F2F5F5*("$f"F5F6F5F>F5FB*("$]"F5F6F5FAF5FB*("#**F5F7F5F>F5FBFarFB*(F\rF5F7F5FAF5FB*(F;F5F6F5F3F5F5FduFB*&FgxF5FAF5F5FjrFBFaxFBF\sF5F5FMF5F5*("$+#F5F>F5F6F5FB*("$c%F5F6F5FAF5F5*("$/&F5F6F5FDF5F5*("$)GF5FAF5F7F5F5*("#))F5F6F5F3F5F5*("$c"F5F7F5FDF5F5*&FhoF5F6F5FB*("$7"F5F7F5F3F5F5*&"#'*F5FDF5FB*&F^qF5F7F5F5*&"#WF5F3F5FBF5FevF5F5*.FgqF5F2F5F6F5FSF5FjwF5)F_oF;F5FB*&,(*.FgqF5F2F5F6F5F:F5,4F[xF5*(FirF5F6F5FAF5F5*(F^pF5F6F5FDF5FB*(F\qF5F6F5F3F5FB*(FirF5F7F5FDF5FBFfsFBFbxFBF\zF5F5F5F5FJF5F5*,FgqF5F2F5F6F5,:F`sF5*(FfyF5F6F5FAF5FB*("#qF5F6F5FDF5FB*(F`uF5F7F5FAF5FBFeyFB*(FgxF5F7F5FDF5FBFFFB*(F^pF5F7F5F3F5FB*&FdtF5FDF5F5*&FcpF5F7F5FB*&FdtF5F3F5F5F5F5F5FMF5F5**FgqF5F7F5F3F5,>*(FbuF5F6F5F^[lF5F5*(FbsF5F6F5F2F5FB*("$["F5F6F5F>F5FBFb[lFB*("#!)F5F6F5FAF5FB*(FdsF5F7F5F>F5FB*(FepF5F6F5FDF5FB*("#QF5F7F5FAF5FB*&F\qF5F>F5F5FgyFB*&FbrF5FAF5F5FcqFBFGFBF_yF5F5FBF5F\wF5F5*&,<*,FirF5F2F5F6F5,:F=F5F\xF5F]_lFB*&F7F5FAF5F5*(FirF5F6F5F3F5FBFewFBFjrFBFaxF5FbxFBF7F5F\zF5F5F5F5FIF5F5*&,&*,FirF5F2F5F6F5,2F\xF5*("#_F5F6F5FDF5FB*(F\pF5F7F5FDF5FBFhuF5*(F]sF5F7F5F3F5F5FdpFBFj_lF5F5FBF5FMF5F5*(FirF5F7F5,<*(FftF5F6F5F^[lF5F5*("$4"F5F6F5F2F5FB*(F`qF5F6F5F>F5FB*("#\F5F7F5F2F5FB*(F4F5F7F5F>F5FB*(F;F5F6F5FDF5FB*(F\qF5F7F5FAF5FB*&FbrF5F>F5F5*(FhtF5F6F5F3F5FBF`xF5*&F4F5FAF5F5FhuFBFaxF5F5FBF5FJF5F5*,FirF5F2F5F6F5,.F]_lF5*(FhoF5F6F5F3F5FBFhuFBF[sFBF7FBF8F5F5FVF5FB**FirF5F7F5,6*(F^blF5F6F5F2F5F5*("#!*F5F6F5F>F5FB*(FgxF5F6F5FAF5FB*("#]F5F7F5F>F5FBF^xFBF`]lF5*(FcpF5F7F5FDF5FB*&F]sF5FAF5F5FhuF5FaxFBF5FMF5F5*("$+'F5F6F5FAF5FB*("%_6F5F6F5FDF5F5*("$#zF5F6F5F3F5F5*("$'pF5F7F5FDF5F5*&"#sF5F6F5F5*("$G#F5F7F5F3F5F5*&FczF5F7F5F5*&"$o"F5F3F5FBF^qFBF5F_oF5F5*,FirF5)F7FcpF5F2F5)F:F4F5)FJF4F5F5*&,&*,F`qF5FbdlF5F2F5)F:F?F5FMF5F5**F`qF5)F7F4F5FAF5FhdlF5FBF5)FJF?F5F5*&,(*,"$?"F5FbdlF5F2F5F9F5FVF5F5*,"$S#F5FjdlF5FAF5F9F5FMF5FB**FirF5F6F5F3F5,<*&F6F5)F3FepF5F5*(FeqF5F6F5F^[lF5F5*(FeuF5F6F5F2F5FB*(F8F5F7F5F^[lF5F5F[xFB*(F^pF5F7F5F2F5FB*(F]sF5F6F5FAF5F5*(FdtF5F7F5F>F5F5*&F;F5F2F5FB*(FgxF5F6F5FDF5F5*&F;F5F>F5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Let us check n:=30: normal(series(eval(subs(Q(y)=Qvser(n),edQpol)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0o=

We note that expanding the 1st order DE or the one we have just obtained gives the same information (relating Q3 to Q0, Q1 and Q2) convert(factor(subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,((D@@3)(Q))(0)=6*Q3,series(edQpol,y,1))),polynom);

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

convert(normal(subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,((D@@3)(Q))(0)=6*Q3,series(edQ,y,5))),polynom);

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

factor(%%/%);

LCQqJiIiJSIiIilJInlHNiJGJCEiIkYp

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

stepset:=[[0,1],[-1,0],[1,0],[-1,-1],[-1,1]];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShzdGVwc2V0R0YoNyc3JCIiISIiIjckISIiRjA3JEYxRjA3JEYzRjM3JEYzRjE3I0Yu

liste:=convert(map(s->x^s[1]*y^s[2],stepset),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwsSSJ5R0YoIiIiKiRJInhHRighIiJGMEYyRjAqJkYyRjNGL0YzRjAqJkYyRjNGL0YwRjA3I0Yu

Number of walks of length n ending at (i,j) co:=proc(i,j,n) option remember: if i<0 or j<0 or n<0 then 0 elif n=0 and i=0 and j=0 then 1 elif n=0 then 0 else convert(map (s-> co(i-s[1],j-s[2],n-1),stepset),`+`): fi: end: co(0,0,2);

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The series Q(x,y;t) Qser:=proc(N) add(add(add(co(i,j,n)*x^i*y^j,i=0..n),j=0..n)*t^n,n=0..N): end: Qser(4);

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

The series Q(0,y;t), for walks ending on the vertical axis Qvser:=proc(N) add(add(co(0,j,n)*y^j,j=0..n)*t^n,n=0..N): end: Qvser(5);

LC4iIiJGIyomSSJ0RzYiRiNJInlHRiZGI0YjKiYsKCokKUYnIiIjRiNGI0YnRiNGI0YjRiMpRiVGLEYjRiMqJiwqKiQpRiciIiRGI0YjKiZGMkYjRitGI0YjKiZGMkYjRidGI0YjRixGI0YjKUYlRjJGI0YjKiYsLCokKUYnIiIlRiNGIyomIiInRiNGMUYjRiMqJiIiKUYjRitGI0YjKiYiIipGI0YnRiNGI0Y6RiNGIylGJUY6RiNGIyomLC4qJClGJyIiJkYjRiMqJiIjNUYjRjlGI0YjKiYiIz9GI0YxRiNGIyomIiNIRiNGK0YjRiMqJiIjREYjRidGI0YjIiM4RiNGIylGJUZGRiNGIw==

The kernel K:=expand(x*y*(t*liste-1));subs(x=0,K);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMUkieUdGKEYxRjEqKEYwRjFGM0YxKUY1RjRGMUYxKiZGMEYxRjdGMUYxKiZGMEYxRjVGMUYxKiZGM0YxRjVGMSEiIkYwRjE3I0Yu

LCgqJkkidEc2IiIiIilJInlHRiUiIiNGJkYmKiZGJEYmRihGJkYmRiRGJg==

The discriminant tilde d(y) dy:=factor(discrim(K,x));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkeUdGKComSSJ5R0YoIiIiLC4qJilJInRHRigiIiNGMClGLyIiJEYwRjAqKCIiJUYwRjNGMClGL0Y1RjAhIiIqKEY5RjBGM0YwRi9GMEY7KihGNUYwRjRGMEY6RjBGOyomRjlGMEYzRjBGO0YvRjBGMDcjRi4=

Decoupling function G:=-y-1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJHR0YoLCZJInlHRighIiIqJEYvRjBGMDcjRi4=

Connection between S(y) (or Q(0,y)) and the weak invariant w eqSw:=subs(x=0,K)*Q(y)-G=alpha/(w(y)-beta)+gamma;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcVN3R0YoLywoKiYsKComSSJ0R0YoIiIiKUkieUdGKCIiI0Y0RjQqJkYzRjRGNkY0RjRGM0Y0RjQtSSJRR0YoNiNGNkY0RjRGNkY0KiRGNiEiIkY0LCYqJkkmYWxwaGFHRihGNCwmLUkid0dGKEY7RjRJJWJldGFHRihGPUY9RjRJJmdhbW1hR0YmRjQ3I0Yu

wyQ:=op(2,isolate(eqSw,w(y)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR3eVFHRigsJiomLCoqJiwoKiZJInRHRigiIiIpSSJ5R0YoIiIjRjVGNSomRjRGNUY3RjVGNUY0RjVGNS1JIlFHRig2I0Y3RjUhIiJGN0Y9KiRGN0Y9Rj1JJmdhbW1hR0YmRjVGPUkmYWxwaGFHRihGNUY9SSViZXRhR0YoRjU3I0Yu

series(wyQ,y,3);

KytJInlHNiJJJWJldGFHRiQiIiFJJmFscGhhR0YkIiIiLCQqJiwmKiYtSSJRR0YkNiNGJkYoSSJ0R0YkRihGKEkmZ2FtbWFHJSpwcm90ZWN0ZWRHISIiRihGJ0YoRjMiIiMtSSJPRzYkRjJJKF9zeXNsaWJHRiQ2I0YoIiIk

DE for w(y) eqw:=4*dy*(diff(w(y),y))^2-(4*w(y)^3-g2*w(y)-g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcXdHRigsKioqIiIlIiIiSSJ5R0YoRjEsLiomKUkidEdGKCIiI0YxKUYyIiIkRjFGMSooRjBGMUY1RjEpRjJGN0YxISIiKihGMEYxRjVGMUYyRjFGPCooRjdGMUY2RjFGO0YxRjwqJkYwRjFGNUYxRjxGMkYxRjEpLUklZGlmZkdGJjYkLUkid0dGKDYjRjJGMkY3RjFGMSomRjBGMSlGREY5RjFGPComSSNnMkdGKEYxRkRGMUYxSSNnM0dGKEYxNyNGLg==

subs(w(y)=wyQ,eqw);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsKioqIiIlIiIiSSJ5R0YoRi8sLiomKUkidEdGKCIiI0YvKUYwIiIkRi9GLyooRi5GL0YzRi8pRjBGNUYvISIiKihGLkYvRjNGL0YwRi9GOiooRjVGL0Y0Ri9GOUYvRjoqJkYuRi9GM0YvRjpGMEYvRi8pLUklZGlmZkdGJjYkLCYqJiwqKiYsKComRjRGL0Y5Ri9GLyomRjRGL0YwRi9GL0Y0Ri9GLy1JIlFHRig2I0YwRi9GOkYwRjoqJEYwRjpGOkkmZ2FtbWFHRiZGL0Y6SSZhbHBoYUdGKEYvRjpJJWJldGFHRihGL0YwRjVGL0YvKiZGLkYvKUZCRjdGL0Y6KiZJI2cyR0YoRi9GQkYvRi9JI2czR0YoRi9GKw==

res:=numer(factor(eval(%))): nops(res);

IiRlJQ==

DE for Q(0,y) edQ:=collect(res,[diff(Q(y), y),Q],factor);

-_I,TypesettingG6$%*protectedGI(_syslibG6"I,mprintslashGF(6$7#>I$edQGF(,hv*0""%""")I&alphaGF(""#F1)I"tGF(F4F1)I"yGF(""&F1),(*$)F8F4F1F1F8F1F1F1F4F1,.*&F5F1)F8""$F1F1*(F0F1F5F1F=F1!""*(F0F1F5F1F8F1FC*(F4F1F6F1F=F1FC*&F0F1F5F1FCF8F1F1)-I%diffGF&6$-I"QGF(6#F8F8F4F1F1**"#;F1F2F1F5F1F7F1F1**FOF1F2F1F6F1F7F1F1**"#[F1)I%betaGF(FAF1)I&gammaGF&F4F1)F8F0F1FC**"#CF1F2F1FTF1FWF1FC**FRF1FSF1FVF1F@F1F1**"#OF1F3F1)FTF4F1F@F1FC**FAF1F3F1I#g2GF(F1F@F1F1*("")F1F2F1FWF1FC**F6F1F8F1,\[l**FOF1F2F1F5F1)F8F[oF1F1**"#cF1F2F1F5F1)F8""(F1FC**FOF1FSF1)FVFAF1F7F1F1**FRF1FSF1FUF1)F8""'F1FC**FRF1FSF1FVF1FboF1F1*(FOF1FSF1F_oF1FC**"$7"F1F2F1F5F1FgoF1FC**"#KF1F2F1F6F1FboF1FC*,FfnF1F3F1FgnF1FUF1F7F1FC*,"#sF1F3F1FgnF1FVF1FgoF1F1**FfnF1F3F1FgnF1FboF1FC**FOF1FSF1FeoF1FWF1F1**FRF1FSF1FUF1F7F1FC**FRF1FSF1FVF1FgoF1F1*(FOF1FSF1FboF1FC*,F0F1FTF1FinF1FeoF1F7F1FC*,"#7F1FTF1FinF1FUF1FgoF1F1*,FipF1FTF1FinF1FVF1FboF1FC**F0F1FTF1FinF1F_oF1F1*,FYF1F2F1FTF1FVF1F7F1F1**FYF1F2F1FTF1FgoF1FC**"#SF1F2F1F5F1F7F1FC**FOF1F2F1F6F1FgoF1FC*,FfnF1F3F1FgnF1FUF1FWF1FC*,FapF1F3F1FgnF1FVF1F7F1F1**FfnF1F3F1FgnF1FgoF1FC*,FAF1F3F1FinF1FUF1F7F1F1*,FhoF1F3F1FinF1FVF1FgoF1FC**FAF1F3F1FinF1FboF1F1**FOF1FSF1FeoF1F@F1F1**"#'*F1FSF1FUF1FWF1FC**"$W"F1FSF1FVF1F7F1F1*("#kF1FSF1FgoF1FC*,F0F1FTF1FinF1FeoF1FWF1FC*,FipF1FTF1FinF1FUF1F7F1F1*,FipF1FTF1FinF1FVF1FgoF1FC**F0F1FTF1FinF1FboF1F1**F0F1I#g3GF(F1FeoF1F7F1FC**FipF1FcrF1FUF1FgoF1F1**FipF1FcrF1FVF1FboF1FC*(F0F1FcrF1F_oF1F1*(F0F1)F3FAF1F7F1FC*,FYF1F2F1FTF1FVF1FWF1F1**FYF1F2F1FTF1F7F1FC**F]rF1F2F1F5F1FWF1F1**F^pF1F2F1F6F1F7F1F1*(FOF1F2F1FgoF1F1*,FfnF1F3F1FgnF1FUF1F@F1FC*,F[rF1F3F1FgnF1FVF1FWF1F1**"$3"F1F3F1FgnF1F7F1FC*,FAF1F3F1FinF1FUF1FWF1F1*,FhoF1F3F1FinF1FVF1F7F1FC**FAF1F3F1FinF1FgoF1F1**FRF1FSF1FUF1F@F1FC**FiqF1FSF1FVF1FWF1F1*(FRF1FSF1F7F1FC*,F0F1FTF1FinF1FeoF1F@F1FC*,FYF1FTF1FinF1FUF1FWF1F1*,FfnF1FTF1FinF1FVF1F7F1FC**FOF1FTF1FinF1FgoF1F1**F0F1FcrF1FeoF1FWF1FC**FipF1FcrF1FUF1F7F1F1**FipF1FcrF1FVF1FgoF1FC*(F0F1FcrF1FboF1F1*(F0F1FhrF1FWF1FC*,FYF1F2F1FTF1FVF1F@F1F1**FRF1F2F1FTF1FWF1FC**FiqF1F2F1F5F1F@F1F1**FOF1F2F1F6F1FWF1F1*(F[oF1F2F1F7F1F1*,FapF1F3F1FgnF1FVF1F@F1F1**FapF1F3F1FgnF1FWF1FC*,FAF1F3F1FinF1FUF1F@F1F1*,FipF1F3F1FinF1FVF1FWF1FC**""*F1F3F1FinF1F7F1F1**FRF1FSF1FUF1F=F1FC**F[rF1FSF1FVF1F@F1F1*(FiqF1FSF1FWF1FC*,FipF1FTF1FinF1FUF1F@F1F1*,FYF1FTF1FinF1FVF1FWF1FC**FipF1FTF1FinF1F7F1F1**F0F1FcrF1FeoF1F@F1FC**FYF1FcrF1FUF1FWF1F1**FfnF1FcrF1FVF1F7F1FC*(FOF1FcrF1FgoF1F1*(F0F1FhrF1F@F1FC**FYF1F2F1FTF1F@F1FC**F^pF1F2F1F5F1F=F1F1*(FOF1F2F1FWF1FC*,FapF1F3F1FgnF1FVF1F=F1F1**FasF1F3F1FgnF1F@F1FC*,FhoF1F3F1FinF1FVF1F@F1FC**FhoF1F3F1FinF1FWF1F1**FRF1FSF1FVF1F=F1F1*(FRF1FSF1F@F1FC*,FipF1FTF1FinF1FUF1F=F1F1*,FfnF1FTF1FinF1FVF1F@F1FC**FYF1FTF1FinF1FWF1F1**FipF1FcrF1FUF1F@F1F1**FYF1FcrF1FVF1FWF1FC*(FipF1FcrF1F7F1F1**FYF1F2F1FTF1F=F1FC*(F[oF1F2F1F@F1FC**FfnF1F3F1FgnF1F=F1FC*,FhoF1F3F1FinF1FVF1F=F1FC**F[uF1F3F1FinF1F@F1F1**FRF1FSF1FVF1F8F1F1*(F]rF1FSF1F=F1FC*,FipF1FTF1FinF1FVF1F=F1FC**FipF1FTF1FinF1F@F1F1**FipF1FcrF1FUF1F=F1F1**FfnF1FcrF1FVF1F@F1FC*(FYF1FcrF1FWF1F1**FfnF1F3F1FgnF1F8F1FC**FAF1F3F1FinF1F=F1F1*(FOF1FSF1F8F1FC*,FipF1FTF1FinF1FVF1F8F1FC**FOF1FTF1FinF1F=F1F1**FipF1FcrF1FVF1F=F1FC*(FipF1FcrF1F@F1F1**FAF1F3F1FinF1F8F1F1*&FOF1FSF1FC**F0F1FTF1FinF1F8F1F1**FipF1FcrF1FVF1F8F1FC*(FOF1FcrF1F=F1F1*(F0F1FTF1FinF1F1*(F0F1FcrF1F8F1F1*&F0F1FcrF1F1F1FKF1F1*,FAF1FUF1F3F1FinF1F@F1F1*,FYF1FVF1F2F1FTF1F@F1F1*,FhoF1FUF1FTF1FinF1F=F1F1*,FfnF1FVF1F3F1FgnF1F=F1F1*,FAF1FVF1F3F1FinF1F=F1FC*,F0F1FVF1FTF1FinF1F8F1FC**)FVF0F1FTF1FinF1FWF1F1*,FipF1FeoF1F3F1FgnF1FWF1F1**FeoF1F3F1FinF1FWF1FC*,FipF1FUF1F2F1FTF1FWF1FC*,F0F1FeoF1FTF1FinF1F@F1FC*,FfnF1FUF1F3F1FgnF1F@F1FC**FOF1F2F1F5F1F8F1FC**FOF1F2F1F5F1F@F1F1**F[oF1F2F1F6F1F@F1FC**FipF1F2F1FTF1F=F1FC**FipF1F3F1FgnF1F8F1FC*(F3F1FinF1F8F1F1**F0F1FhxF1FSF1FWF1FC*(FhxF1FcrF1FWF1F1**FOF1FeoF1FSF1F@F1F1**F0F1FVF1FhrF1FWF1F1**F0F1FeoF1FcrF1F@F1FC**FYF1FUF1FSF1F=F1FC**FhoF1FUF1FcrF1F=F1F1**FOF1FVF1FSF1F8F1F1**F0F1FVF1FcrF1F8F1FC*,FipF1FTF1FinF1FUF1FWF1F1*,FapF1F3F1FgnF1FVF1FWF1F1**F5F1F=F1,\xF^oFC**FRF1F2F1F5F1FboF1F1**FYF1FSF1FUF1FgoF1F1FioFC*(FYF1FSF1F_oF1F1**"$C"F1F2F1F5F1FgoF1F1F]pF1*,FfnF1F3F1FgnF1FVF1FgoF1FCFbpF1FdpF1**FiqF1FSF1FVF1FgoF1FC*(FRF1FSF1FboF1F1*,FhoF1FTF1FinF1FUF1FgoF1FCFjpF1**FhoF1FTF1FinF1F_oF1FC**FipF1F2F1FTF1FgoF1F1**F[rF1F2F1F5F1F7F1F1**F^pF1F2F1F6F1FgoF1F1FbqFC**FapF1F3F1FgnF1FgoF1F1*,FAF1F3F1FinF1FVF1FgoF1F1FfqFC**FapF1FSF1FUF1FWF1F1**"$#>F1FSF1FVF1F7F1FC*("$?"F1FSF1FgoF1F1F_rFC*,FYF1FTF1FinF1FVF1FgoF1F1**FipF1FTF1FinF1FboF1FC**FhoF1FcrF1FUF1FgoF1FCFerF1*(FhoF1FcrF1F_oF1FCFjrF1**"#!)F1F2F1F5F1FWF1F1**F[oF1F2F1F6F1F7F1F1F]sFC*,FasF1F3F1FgnF1FVF1FWF1FC**F[rF1F3F1FgnF1F7F1F1FcsF1**FhoF1F3F1FinF1FgoF1FCFesF1**Fb[lF1FSF1FVF1FWF1FC*(F[rF1FSF1F7F1F1*,"#=F1FTF1FinF1FUF1FWF1FC*,FRF1FTF1FinF1FVF1F7F1F1**"#IF1FTF1FinF1FgoF1FCF]tFC**FYF1FcrF1FVF1FgoF1F1*(FipF1FcrF1FboF1FC**FfnF1F2F1FTF1FWF1F1F_yF1*(FOF1F2F1F7F1FCFftFC**F[rF1F3F1FgnF1FWF1F1*,F[uF1F3F1FinF1FVF1FWF1F1**FipF1F3F1FinF1F7F1FC**FYF1FSF1FUF1F=F1F1**Fb[lF1FSF1FVF1F@F1FC*(Fb[lF1FSF1FWF1F1F_uFC*,FRF1FTF1FinF1FVF1FWF1F1**FfnF1FTF1FinF1F7F1FC**Fb\lF1FcrF1FUF1FWF1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indets(edQ);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkmYWxwaGFHRihJJWJldGFHRihJI2cyR0YoSSNnM0dGKEkidEdGKEkieUdGKC1JIlFHRig2I0YyLUklZGlmZkdGJjYkRjNGMkYr

Now we want to eliminate the unknown series, at the cost of introducing series Q_{0,i} factor(series(edQ,y,1));

KydJInlHNiIsKComIiIlIiIiKUklYmV0YUdGJCIiJEYoISIiKiZGKkYoSSNnMkdGJEYoRihJI2czR0YkRigiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYo

Expression of g3 g3s:=solve(coeff(%,y,0),g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnM3NHRigsJiomIiIlIiIiKUklYmV0YUdGKCIiJEYxRjEqJkYzRjFJI2cyR0YoRjEhIiI3I0Yu

edQ1:=op(4,factor(subs(g3=g3s,edQ))):nops(%);

IiQuIw==

factor(series(edQ1,y,1));

KydJInlHNiIsKCooIiM7IiIiSSZhbHBoYUdGJEYoKUkidEdGJCIiI0YoRigqJiIjN0YoKUklYmV0YUdGJEYsRihGKEkjZzJHRiQhIiIiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYo

Expression of g2 g2s:=solve(coeff(%,y,0),g2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnMnNHRigsJiooIiM7IiIiSSZhbHBoYUdGKEYxKUkidEdGKCIiI0YxRjEqJiIjN0YxKUklYmV0YUdGKEY1RjFGMTcjRi4=

We have thus eliminated g2 and g3 edQ2:=op(4,factor(subs(g2=g2s,edQ1)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlZFEyR0YoLGZcbCIiIkYvKi4iIidGLy1JIlFHRig2I0kieUdGKEYvSSZnYW1tYUdGJkYvSSViZXRhR0YoRi9JInRHRihGLylGNSIiJUYvRi8qLkYxRi9GMkYvRjZGL0Y3Ri9GOEYvKUY1IiIkRi9GLyosIiM1Ri9GMkYvLUklZGlmZkdGJjYkRjJGNUYvKUY4RjpGLylGNSIiKUYvISIiKixGMUYvKUYyIiIjRi9GN0YvKUY4RklGLylGNSIiJkYvRkYqLCIiKkYvRkhGL0Y3Ri9GSkYvRjlGL0ZGKixGMUYvRkhGL0Y3Ri9GSkYvRjxGL0ZGKipGMkYvSSZhbHBoYUdGKEYvRjhGL0Y5Ri9GRiosIiM3Ri9GMkYvRjdGL0Y4Ri9GPEYvRkYqLEYxRi9GMkYvRjdGL0Y4Ri8pRjVGSUYvRkYqLCIjQ0YvRjJGL0Y2Ri8pRjhGPUYvRlVGL0ZGKixGV0YvRjJGL0Y2Ri9GWEYvRjVGL0ZGKipGU0YvKUY2RklGL0ZKRi9GNUYvRi8qKiIjV0YvRkhGL0ZDRi9GPEYvRi8qJkZRRi9GPEYvRkYqKkYyRi9GUUYvRjhGL0ZVRi9GRiosRjFGL0YyRi9GN0YvRjhGL0Y1Ri9GRiosRlNGL0ZIRi9GNkYvRkNGLylGNUYxRi9GRiosRldGL0ZIRi9GNkYvRkNGL0ZLRi9GRiosRj1GL0ZIRi9GN0YvRkpGL0Zcb0YvRkYqLCIjT0YvRkhGL0Y2Ri9GQ0YvRjlGL0ZGKixGV0YvRkhGL0Y2Ri9GQ0YvRjxGL0ZGKixGU0YvRjJGL0ZlbkYvRlhGL0Y5Ri9GLyosRldGL0YyRi9GNkYvRlhGL0ZLRi9GRiosRlNGL0ZIRi9GNkYvRkNGL0ZVRi9GRiosRjFGL0YyRi9GN0YvRjhGL0ZLRi9GRiosRj1GL0ZIRi9GN0YvRkpGL0ZVRi9GRiosRjFGL0YyRi9GN0YvRjhGL0Y5Ri9GRioqRjpGL0ZIRi9GSkYvRktGL0YvKipGPUYvRjdGL0ZlbkYvRlVGL0ZGKipGMUYvRjdGL0Y2Ri9GPEYvRi8qKEZRRi9GNkYvRlVGL0YvKipGMUYvRjdGL0Y2Ri9GNUYvRi8qKkY6Ri8pRjJGPUYvKUY4RkxGL0ZERi9GLyooKUZARklGL0ZDRi8pRjVGP0YvRi8qKkY6Ri9GSEYvRkNGL0ZERi9GLyosRjpGL0YyRi9GQEYvRkNGLylGNUZORi9GLyosRjFGL0YyRi9GQEYvRkpGL0Zcb0YvRi8qLEZTRi9GMkYvRkBGL0ZYRi8pRjUiIihGL0ZGKipGSUYvRmFwRi9GWEYvRmVwRi9GRioqRkVGL0ZIRi9GWEYvRmhwRi9GRiooRmFwRi9GSkYvRkRGL0YvKipGOkYvRkhGL0ZKRi9GXG9GL0YvKi5GMUYvRjJGL0Y3Ri9GNkYvRjhGL0ZVRi9GLyooRj1GL0Y3Ri9GOUYvRkYqKEYxRi9GN0YvRlVGL0ZGKixGU0YvRjJGL0ZlbkYvRlhGL0Y8Ri9GLyosRlNGL0YyRi9GZW5GL0ZYRi9GVUYvRi8qLEY6Ri9GMkYvRkBGL0ZKRi9GaHBGL0YvKixGRUYvRjJGL0ZARi9GWEYvRkRGL0ZGKiZGUUYvRjVGL0ZGKipGRUYvRjJGL0ZKRi9GXG9GL0ZGKihGSEYvRkpGL0Y5Ri9GLyoqRklGL0ZARi9GOEYvRlxvRi9GLyoqRkVGL0ZIRi9GWEYvRlxvRi9GRioqRjpGL0YyRi9GWEYvRmhwRi9GLyoqRklGL0ZhcEYvRkpGL0ZocEYvRi8qKkY6Ri9GQEYvRkpGL0ZocEYvRkYqKkY6Ri9GYXBGL0ZYRi9GREYvRkYqKkZJRi9GQEYvRlhGL0ZERi9GLyoqRlNGL0ZecEYvRl9wRi9GaHBGL0YvKipGSUYvRmFwRi9GQ0YvRmVwRi9GRioqRjpGL0YyRi9GOEYvRktGL0YvKioiI0dGL0ZecEYvRl9wRi9GS0YvRi8qKkY6Ri9GXnBGL0ZfcEYvRlVGL0YvKipGOkYvKUY2Rj1GL0ZKRi9GVUYvRkYqKkZTRi9GZW5GL0ZKRi9GPEYvRi8qKkYyRi9GUUYvRjhGL0Y8Ri9GRiooRlNGL0Y2Ri9GSkYvRkYqKkY6Ri9GMkYvRkpGL0Y8Ri9GLyoqRklGL0ZARi9GOEYvRjxGL0ZGKipGSUYvRjJGL0Y4Ri9GVUYvRkYqKkZXRi9GNkYvRkpGL0ZVRi9GRioqRklGL0ZhcEYvRlhGL0ZLRi9GRioqRklGL0ZARi9GWEYvRjlGL0ZGKiwiI01GL0YyRi9GQEYvRkNGL0ZocEYvRkYqLEZTRi9GMkYvRkBGL0ZYRi9GXG9GL0ZGKixGMUYvRjJGL0ZARi9GSkYvRktGL0YvKihGOUYvRkpGL0ZhcEYvRi8qKkY6Ri9GYXBGL0ZDRi9GPEYvRkYqKkZTRi9GSEYvRkNGL0Y1Ri9GLyoqRjpGL0YyRi9GOEYvRjxGL0ZGKioiI0ZGL0ZhcEYvRkNGL0Zcb0YvRkYqKkZjckYvRkNGL0Y5Ri9GSEYvRi8qKiIjO0YvRkBGL0ZYRi9GXG9GL0ZGKipGXHRGL0YyRi9GWEYvRktGL0ZGKipGRUYvRjJGL0ZKRi9GOUYvRi8qKkZcdEYvRkBGL0ZYRi9GPEYvRi8qKiIjZ0YvRjJGL0ZYRi9GVUYvRi8qKkZTRi9GNkYvRkpGL0Y5Ri9GRiosIiNpRi9GMkYvRkBGL0ZDRi9GXG9GL0ZGKiwiI0tGL0YyRi9GQEYvRkNGL0Y5Ri9GRiosRjpGL0YyRi9GQEYvRlhGL0ZLRi9GRiosRkVGL0YyRi9GQEYvRkNGL0Y8Ri9GRiosRklGL0YyRi9GQEYvRkpGL0Y5Ri9GLyosIiNbRi9GMkYvRjZGL0ZYRi9GPEYvRkYqLCIjY0YvRjJGL0ZARi9GQ0YvRktGL0ZGKiZGSkYvRlxvRi9GLyooRklGL0Y4Ri9GS0YvRkYqKEZcdEYvRkpGL0Y8Ri9GLyosRldGL0YyRi9GNkYvRlhGL0Y5Ri9GRiooRjFGL0Y5Ri9GSkYvRkYqKEY6Ri9GPEYvRjhGL0YvKipGV0YvRl5wRi9GX3BGL0Zcb0YvRi8qKkZORi9GYXBGL0ZDRi9GREYvRkYqKkZpcEYvRkhGL0ZDRi9GXG9GL0ZGKipGMUYvRmFwRi9GWEYvRmhwRi9GRioqRklGL0ZIRi9GWEYvRktGL0ZGKipGPUYvRmFwRi9GSkYvRlxvRi9GLyooRk5GL0ZKRi9GVUYvRi8qKEZTRi9GMkYvRlhGL0YvKihGSUYvRjhGL0Y1Ri9GRioqRlNGL0ZecEYvRl9wRi9GPEYvRi8qJkY9Ri9GN0YvRkYqJkZJRi9GVUYvRkYqKEZcdEYvRkpGL0Y1Ri9GLyomRjpGL0ZKRi9GRiokRjlGL0YvKioiI0FGL0ZhcEYvRkNGL0ZocEYvRkYqKkZXRi9GXnBGL0ZfcEYvRjlGL0YvKipGU0YvRkhGL0ZDRi9GS0YvRi8qKkZXRi9GYXBGL0ZDRi9GS0YvRkYqKkY6Ri9GYXBGL0ZYRi9GXG9GL0ZGKipGMUYvRkBGL0ZYRi9GaHBGL0ZGKipGSUYvRjJGL0ZYRi9GXG9GL0ZGKipGU0YvRmFwRi9GQ0YvRjlGL0ZGKipGSUYvRmFwRi9GSkYvRktGL0YvKioiIz1GL0ZARi9GWEYvRktGL0ZGKipGOkYvRkBGL0ZKRi9GXG9GL0ZGKipGV0YvRkhGL0ZDRi9GVUYvRi8qKiIjRUYvRjJGL0ZYRi9GOUYvRi8qKkY6Ri9GMkYvRkpGL0ZLRi9GRioqIiNTRi9GMkYvRlhGL0Y8Ri9GLyoqRjpGL0ZARi9GSkYvRjlGL0YvKipGSUYvRkBGL0Y4Ri9GS0YvRi8qKkZJRi9GMkYvRjhGL0Y5Ri9GLyoqRlx0Ri9GQEYvRlhGL0ZVRi9GLyoqRjpGL0ZARi9GSkYvRjxGL0YvKioiIz9GL0YyRi9GWEYvRjVGL0YvKipGRUYvRkBGL0ZYRi9GNUYvRi8qKkZJRi9GQEYvRjhGL0ZVRi9GRjcjRi4=

Now the series Q_{0,i} will start to come in factor(series(edQ2,y,1));

KydJInlHNiIsLCIiIkYmKigiIzdGJkkmZ2FtbWFHJSpwcm90ZWN0ZWRHRiYpSSJ0R0YkIiIjRiYhIiIqKEYoRiYtSSJRR0YkNiMiIiFGJilGLCIiJEYmRiYqJkY1RiZJJWJldGFHRiRGJkYuKiYiIiVGJkYrRiZGLkYzLUkiT0c2JEYqSShfc3lzbGliR0YkNiNGJkYm

Expression of beta betas:=solve(coeff(%,y,0),beta);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZiZXRhc0dGKCwqKigiIiUiIiItSSJRR0YoNiMiIiFGMSlJInRHRigiIiRGMUYxKihGMEYxSSZnYW1tYUdGJkYxKUY3IiIjRjEhIiIqJiNGMEY4RjFGO0YxRj0jRjFGOEYxNyNGLg==

edQ3:=(factor(subs(beta=betas,edQ2))): nops(%);

IiRSIg==

factor(series(edQ3,y,2));

KydJInlHNiIsOCoqIiNDIiIiLUkiUUdGJDYjIiIhRihJJmdhbW1hRyUqcHJvdGVjdGVkR0YoKUkidEdGJCIiJEYoRigqKCIjN0YoKUYtIiIjRigpRjBGNUYoISIiKihGM0YoKUYwIiIlRigpRilGNUYoRjdJJmFscGhhR0YkRjcqJkY1RihGLUYoRigqKCIjP0YoRi9GKC0tSSJERzYkRi5JKF9zeXNsaWJHRiQ2I0YqRitGKEYoKiZGNUYoRjBGKEY3KigiIilGKEYtRihGNkYoRjcqJiIjO0YoRjZGKEYoKigiI0dGKEYpRihGL0YoRigqKEY1RihGKUYoRjBGKEY3RigtSSJPR0ZDNiNGKEY1

Expression of alpha alphas:=solve(coeff(%,y,1),alpha);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbHBoYXNHRigsNiooIiM3IiIiKUkidEdGKCIiJUYxKS1JIlFHRig2IyIiISIiI0YxISIiKioiI0NGMUY2RjFJJmdhbW1hR0YmRjEpRjMiIiRGMUYxKigiI0dGMUY2RjFGP0YxRjEqKCIjP0YxRj9GMS0tSSJER0YlNiNGN0Y4RjFGMSooRjBGMSlGPkY6RjEpRjNGOkYxRjsqKCIiKUYxRj5GMUZLRjFGOyooRjpGMUY2RjFGM0YxRjsqJiIjO0YxRktGMUYxKiZGOkYxRj5GMUYxKiZGOkYxRjNGMUY7NyNGLg==

edQ4:=factor(subs(alpha=alphas,edQ3)):nops(%);

IiRNIg==

factor(series(edQ4,y,3));

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

Now we do not have an expression of gamma, but a cubic equation for it eq1:=collect(coeff(%,y,2),gamma, factor);

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

We finally eliminate gamma factor(resultant(eq1,edQ4,gamma));nops(%);

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

IiIk

Here is the first order DE for Q(y) involving the series Q_{0,i} (here m=3) edQ:=collect(op(1,op(3,factor(resultant(eq1,edQ4,gamma)))),[diff(Q(y), y),Q(y)],distributed,factor);;

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

Checking this DE n:=20: normal(series(eval(subs(Q(y)=Qvser(n),Q(0)=subs(y=0,Qvser(n)), (D(Q))(0)=subs(y=0,diff(Qvser(n),y)),((D@@2)(Q))(0)=subs(y=0,diff(Qvser(n),y$2)),edQ)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0A=

Introducing the series Q_{0,i} (here denoted Q_i) edQ:=subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,edQ);

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degree(edQ,Q0);degree(edQ,Q1);degree(edQ,Q2);

IiIk

IiIi

IiIi

dedQ:=factor(diff(edQ,y)): indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkjUTBHRihJI1ExR0YoSSNRMkdGKEkidEdGKEkieUdGKC1JIlFHRig2I0YxLUklZGlmZkdGJjYkRjJGMS1GNjYkRjItSSIkR0YmNiRGMSIiIzcjPCpGLUYuRi9GMEYxRjJGNS1GNjYkRjVGMQ==

We now want to obtain polynomial coefficients in t and y. We eliminate the Q_i one after the other, increasing the order of the DE res1:=factor(resultant(edQ,dedQ,Q2));nops(%);

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

IiIm

Choosing and checking the correct factor (the first one cannot cancel because of the factors y and the CT 1) edQbis:=collect(op(5,res1),[diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0,Q1],factor);

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KUkjUTBHRihJI1ExR0YoSSJ0R0YoSSJ5R0YoLUkiUUdGKDYjRjAtSSVkaWZmR0YmNiRGMUYwLUY1NiRGMS1JIiRHRiY2JEYwIiIjNyM8KUYtRi5GL0YwRjFGNC1GNTYkRjRGMA==

degree(edQbis,Q1);

IiIi

Eliminating Q1 dedQbis:=factor(diff(edQbis,y)): factor(resultant(edQbis,dedQbis,Q1)): nops(%);

IiIk

edQter:=collect(%%,[diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0],factor);;

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KUkjUTBHRihJInRHRihJInlHRigtSSJRR0YoNiNGLy1JJWRpZmZHRiY2JEYwRi8tRjQ2JEYwLUkiJEdGJjYkRi8iIiMtRjQ2JEYwLUY5NiRGLyIiJDcjPClGLUYuRi9GMEYzLUY0NiRGM0YvLUY0NiRGQ0Yv

Eliminating Q0 dedQter:=op(4,factor(diff(edQter,y))):nops(%);

IiMiKQ==

res2:=factor(resultant(edQter,dedQter,Q0)):nops(res2);

IiIl

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

IiQ/Ig==

KiQpSSJ0RzYiIiIlIiIi

SSJ5RzYi

nops(op(4,res2));

IiR3Iw==

edQpol:=collect(op(4,res2),[diff(Q(y), y, y, y, y),diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y)],factor);;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdlZFFwb2xHRigsPiomLCgqLClJInlHRigiIiQiIiJJInRHRihGNSksKCokKUYzIiIjRjVGNUYzRjVGNUY1RjtGNSwuKiYpRjZGO0Y1RjJGNUY1KigiIiVGNUY+RjVGOkY1ISIiKihGQEY1Rj5GNUYzRjVGQSooRjtGNUY2RjVGOkY1RkEqJkZARjVGPkY1RkFGM0Y1RjUtSSVkaWZmR0YmNiQtSSJRR0YoNiNGM0YzRjVGNSouRjJGNUY2RjUsJiomRjtGNUYzRjVGNUY1RjVGNUY4RjVGPEY1RkhGNUY1KixGM0Y1LCZGM0Y1RjVGQUY1LCZGM0Y1RjVGNUY1RjhGNUY8RjVGNUY1LUZGNiRGSC1JIiRHRiY2JEYzRkBGNUY1KiYsTCosRjJGNUY2RjVGN0Y1RjxGNS1GRjYkRkgtRlQ2JEYzRjtGNUZBKixGNkY1RjpGNUY4RjUsOiooIiM5RjVGPkY1KUYzIiImRjVGNSooIiNNRjVGPkY1KUYzRkBGNUZBKigiI2tGNUY+RjVGMkY1RkEqKCIjQkY1RjZGNUZfb0Y1RkEqKCIjJSlGNUY+RjVGOkY1RkEqKCIjPkY1RjZGNUYyRjVGQSooIiNRRjVGPkY1RjNGNUZBKigiIzpGNUY2RjVGOkY1RkEqJiIiKkY1RjJGNUY1KiYiIzVGNUY+RjVGQSomIiIoRjVGOkY1RjUqJkZcb0Y1RjNGNUY1RjVGRUY1RjUqKkY2RjVGOkY1LEAqKEZeb0Y1Rj5GNSlGMyIiJ0Y1RjUqKCIjc0Y1Rj5GNUZbb0Y1RkEqKCIkNSNGNUY+RjVGX29GNUZBKigiI2VGNUY2RjVGW29GNUZBKigiJCFHRjVGPkY1RjJGNUZBKigiI3RGNUY2RjVGX29GNUZBKigiJCU9RjVGPkY1RjpGNUZBKigiI1xGNUY2RjVGMkY1RkEqJiIjQ0Y1Rl9vRjVGNSooRl1xRjVGPkY1RjNGNUZBRmpvRkEqJiIjSEY1RjJGNUY1Rl5wRkEqJiIjPEY1RjpGNUY1RmJwRjVGNUZIRjVGNSooIiM9RjVGPkY1KUYzRmFwRjVGNSooIiNbRjVGPkY1RmZwRjVGQSooIiQzIkY1RltvRjVGPkY1RkEqKCIjSkY1RjZGNUZmcEY1RkEqKCIjYUY1Rl9vRjVGPkY1RkEqKEZjb0Y1RjZGNUZbb0Y1RkEqKCIjY0Y1Rj5GNUYyRjVGNSooIiM/RjVGNkY1Rl9vRjVGNSomIiM4RjVGW29GNUY1KigiJEEiRjVGPkY1RjpGNUY1KigiI0ZGNUYyRjVGNkY1RjUqJkZdcEY1Rl9vRjVGNSooIiNpRjVGPkY1RjNGNUY1KihGZ29GNUY2RjVGOkY1RjUqJkZfcEY1RjJGNUZBKiZGXnJGNUY+RjVGNSomIiM2RjVGOkY1RkEqJkZhcEY1RjNGNUZBRjUtRkY2JEZILUZUNiRGM0Y0RjVGNSouRjRGNUY2RjVGOkY1RjhGNSw6KihGQEY1Rj5GNUZbb0Y1RjUqKEZqc0Y1Rj5GNUZfb0Y1RkEqKEZcc0Y1Rj5GNUYyRjVGQSooRmFwRjVGNkY1Rl9vRjVGQSooRmdxRjVGPkY1RjpGNUZBKihGXG9GNUY2RjVGMkY1RkEqKEZfcEY1Rj5GNUYzRjVGQSooRjRGNUY2RjVGOkY1RkEqJkY0RjVGMkY1RjUqJkY7RjVGPkY1RkEqJkY7RjVGOkY1RjVGM0Y1RjUpRllGO0Y1RkEqJixIKixGNEY1RjZGNUY6RjUsQiooRkBGNUY+RjVGZnBGNUY1KigiIzdGNUY+RjVGW29GNUY1KihGXnJGNUY+RjVGX29GNUY1KigiIztGNUY+RjVGMkY1RkEqKEZdcEY1RjZGNUZfb0Y1RkEqKCIjSUY1Rj5GNUY6RjVGQSooRmJzRjVGNkY1RjJGNUZBKiZGNEY1Rl9vRjVGQSooRmp1RjVGPkY1RjNGNUZBRmpvRkEqJEYyRjVGQSomRmdwRjVGPkY1RkEqKEZncEY1RjZGNUYzRjVGQSomRmdwRjVGOkY1RjUqJkY0RjVGM0Y1RjVGNUY1RjVGRUY1RjUqLEY0RjVGNkY1RjpGNSw+KihGYXJGNUY+RjVGW29GNUY1KihGZW9GNUY+RjVGX29GNUZBKigiJDMjRjVGPkY1RjJGNUZBKigiI29GNUY2RjVGX29GNUZBKigiJG8iRjVGPkY1RjpGNUZBKigiI3FGNUY2RjVGMkY1RkEqKEZqdkY1Rj5GNUYzRjVGQSooIiNARjVGNkY1RjpGNUZBKiYiI0FGNUYyRjVGNUZfdkZBRmB2RkEqJkZhd0Y1RjpGNUY1KiZGQEY1RjNGNUY1RjVGNUY1RkhGNUY1KigiIyEqRjVGPkY1RmZwRjVGNSooIiQ7I0Y1RltvRjVGPkY1RkEqKCIkMyVGNUZfb0Y1Rj5GNUZBKigiJEsiRjVGNkY1RltvRjVGQSooRmlwRjVGPkY1RjJGNUY1KigiI2pGNUY2RjVGX29GNUZBKigiJHMkRjVGPkY1RjpGNUY1KigiJGkiRjVGMkY1RjZGNUY1KiYiI1hGNUZfb0Y1RjUqKCIkZyRGNUY+RjVGM0Y1RjUqKCIjKipGNUY2RjVGOkY1RjUqJkZeckY1RjJGNUY1KiZGZ3dGNUY+RjVGNSooIiNVRjVGNkY1RjNGNUY1KiYiI2dGNUY6RjVGQSomRmp1RjVGM0Y1RkFGXXBGQUY1RllGNUY1KixGZHVGNSlGNkZARjVGOkY1KUY4RjRGNSlGRUY0RjVGNSomLCYqLiIjT0Y1RmN5RjVGOkY1RkxGNUY3RjVGSEY1RjUqKEZncEY1RjZGNSxEKihGQEY1Rj5GNUZfckY1RjUqKCIjR0Y1Rj5GNUZmcEY1RjVGZXZGNSooRkBGNUY2RjVGZnBGNUY1KigiIilGNUY+RjVGX29GNUZBKihGYXBGNUY2RjVGW29GNUZBKihGZ3FGNUY+RjVGMkY1RkEqKEZqdUY1RjZGNUZfb0Y1RkEqJkZcb0Y1RltvRjVGQUZmdEZBKihGYXpGNUY2RjVGMkY1RkEqJkY7RjVGX29GNUZBKihGZHVGNUY+RjVGM0Y1RkEqJkY2RjVGOkY1RkEqJkZncEY1RjJGNUY1Rl92RkFGOUY1RjVGNUY1KUZFRjtGNUY1KiYsRCouRml5RjVGY3lGNUY6RjVGOEY1KUZMRjtGNSlGSEY7RjVGNSoqRmdwRjVGNkY1LD4qKCIjU0Y1Rj5GNUZmcEY1RjVGYnRGQSooRmlvRjVGPkY1Rl9vRjVGQSooRml5RjVGNkY1RltvRjVGQSooRmRbbEY1Rj5GNUYyRjVGQSooIiNSRjVGNkY1Rl9vRjVGQSooRmlvRjVGPkY1RjpGNUZBKihGXnJGNUY2RjVGMkY1RkEqJkZcb0Y1Rl9vRjVGNSooRml5RjVGPkY1RjNGNUZBRkNGQUZqekY1KiZGZHVGNUY+RjVGQUZcdUY1RjVGSEY1RjUqKCIkYyJGNUZbb0Y1Rj5GNUY1KigiJF8jRjVGX29GNUY+RjVGQSooIiRXJUY1Rj5GNUYyRjVGQSooIiR1IkY1RjZGNUZfb0Y1RkEqKEZoeEY1Rj5GNUY6RjVGNSooRmFyRjVGMkY1RjZGNUZBKigiJD8lRjVGPkY1RjNGNUY1KigiJClHRjVGNkY1RjpGNUY1KiZGXnlGNUYyRjVGNSomIiRXIkY1Rj5GNUY1KigiJC0iRjVGNkY1RjNGNUY1RmF2RjUqJkZeckY1RjZGNUY1KiYiI3lGNUYzRjVGQUZeckZBRjVGRUY1RjUqLEZkdUY1RmN5RjVGOkY1KUZMRjRGNSlGSEY0RjVGNSoqRmdwRjVGNkY1LDYqKEZndUY1Rj5GNUZbb0Y1RjVGZXVGNSooRmdwRjVGPkY1RjJGNUY1KihGX3BGNUY2RjVGX29GNUZBKihGXHNGNUY+RjVGOkY1RkEqKEZfcEY1RjZGNUYyRjVGQSooRmdxRjVGPkY1RjNGNUZBRml6RkFGX3ZGQUY5RjVGNUZgW2xGNUY1KiYsOEZmdkY1RmN6RkEqKEZdeEY1Rj5GNUY6RjVGQSooRmB5RjVGNkY1RjJGNUZBKihGaXBGNUY+RjVGM0Y1RjUqKEZkdUY1RjZGNUY6RjVGQSomRml5RjVGPkY1RjUqKCIkPyJGNUY2RjVGM0Y1RjVGYXZGNUZiXWxGNUZeckZBRjVGSEY1RjUqKEZkdUY1RjJGNUY2RjVGNSooRmdxRjVGNkY1RjNGNUZBRmF2RkFGXnJGNTcjLD4qJkYwRjUtRkY2JC1GRjYkLUZGNiRGRUYzRjNGM0Y1RjUqJixMKixGMkY1RjZGNUY3RjVGPEY1RmJfbEY1RkFGZ25GNUZjcEY1Rl1yRjVGYHJGQUZickZBRmRyRkFGZnJGQUZockZBRmlyRjVGW3NGNUZdc0Y1Rl9zRjVGYXNGNUZjc0Y1RmRzRjVGZnNGNUZnc0ZBRmhzRjVGaXNGQUZbdEZBRjVGYF9sRjVGNSouRjRGNUY2RjVGOkY1RjhGNUZhdEY1KUZiX2xGO0Y1RkEqJkZfdUY1RmJfbEY1RjVGYnlGNUZmeUY1RlxbbEY1RmVdbEY1RmhdbEY1RmBebEY1RmlebEY1RmpebEZBRmF2RkFGXnJGNQ==

Let us check n:=30: normal(series(eval(subs(Q(y)=Qvser(n),edQpol)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0o=

We note that expanding the 1st order DE or the one we have just obtained gives the same information (relating Q3 to Q0, Q1 and Q2) convert(factor(subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,series(edQpol,y,1))),polynom);

LDwqKCIjTyIiIilJI1EwRzYiIiIjRiUpSSJ0R0YoIiIkRiUhIiIqKiIjc0YlRidGJUkjUTFHRihGJUYqRiVGLSooRiRGJSlGMEYpRiVGKkYlRi0qKCIjPUYlLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKDYkSSJER0Y5Riw2I0kiUUdGKDYjIiIhRiUpRitGKUYlRiUqKEYkRiVGJ0YlRkJGJUYlKigiJFciRiVGMEYlRkJGJUYlKigiJCE9RiVGQkYlSSNRMkdGKEYlRiUqKEY0RiVGJ0YlRitGJUYlKihGNEYlRitGJUYwRiVGJSomRjRGJUYnRiVGLSomRjRGJUYwRiVGLSomRjRGJUZIRiVGLUY0RiU=

convert(normal(subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,series(edQ,y,4))),polynom);

KiYsPCooIiM3IiIiKUkjUTBHNiIiIiNGJilJInRHRikiIiRGJkYmKioiI0NGJkYoRiZJI1ExR0YpRiZGK0YmRiYqKEYlRiYpRjBGKkYmRitGJkYmKigiIidGJi0tLUkjQEBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRik2JEkiREdGOUYtNiNJIlFHRik2IyIiIUYmKUYsRipGJiEiIiooRiVGJkYoRiZGQkYmRkMqKCIjW0YmRjBGJkZCRiZGQyooIiNnRiZGQkYmSSNRMkdGKUYmRkMqKEY0RiZGKEYmRixGJkZDKihGNEYmRixGJkYwRiZGQyomRjRGJkYoRiZGJiomRjRGJkYwRiZGJiomRjRGJkZJRiZGJkY0RkNGJilJInlHRilGLUYm

factor(%%/%);

LCQqJiIiJCIiIilJInlHNiJGJCEiIkYp

JSFH

JSFH

stepset:=[[0,1],[0,-1],[-1,0],[1,0],[1,1]];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShzdGVwc2V0R0YoNyc3JCIiISIiIjckRjAhIiI3JEYzRjA3JEYxRjA3JEYxRjE3I0Yu

liste:=convert(map(s->x^s[1]*y^s[2],stepset),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwsSSJ5R0YoIiIiKiRGLyEiIkYwKiRJInhHRihGMkYwRjRGMComRjRGMEYvRjBGMDcjRi4=

Number of walks of length n ending at (i,j) co:=proc(i,j,n) option remember: if i<0 or j<0 or n<0 then 0 elif n=0 and i=0 and j=0 then 1 elif n=0 then 0 else convert(map (s-> co(i-s[1],j-s[2],n-1),stepset),`+`): fi: end: co(0,0,2);

IiIj

The series Q(x,y;t) Qser:=proc(N) add(add(add(co(i,j,n)*x^i*y^j,i=0..n),j=0..n)*t^n,n=0..N): end: Qser(4);

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

The series Q(0,y;t), for walks ending on the vertical axis Qvser:=proc(N) add(add(co(0,j,n)*y^j,j=0..n)*t^n,n=0..N): end: Qvser(5);

LC4iIiJGIyomSSJ0RzYiRiNJInlHRiZGI0YjKiYsKCokKUYnIiIjRiNGI0YnRiNGLEYjRiMpRiVGLEYjRiMqJiwqKiQpRiciIiRGI0YjKiZGMkYjRitGI0YjKiYiIiZGI0YnRiNGI0YsRiNGIylGJUYyRiNGIyomLCwqJClGJyIiJUYjRiMqJiIiJ0YjRjFGI0YjKiYiIzZGI0YrRiNGIyomIiM4RiNGJ0YjRiMiIzVGI0YjKUYlRjtGI0YjKiYsLiokKUYnRjVGI0YjKiZGQkYjRjpGI0YjKiYiI0NGI0YxRiNGIyomIiNXRiNGK0YjRiMqJiIjVkYjRidGI0YjIiNFRiNGIylGJUY1RiNGIw==

The kernel K:=expand(x*y*(t*liste-1));subs(x=0,K);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGMkYxRjZGMUYxKihGMEYxRjNGMUY1RjFGMSomRjBGMUYzRjFGMSomRjBGMUY2RjFGMSomRjNGMUY2RjEhIiI3I0Yu

KiZJInRHNiIiIiJJInlHRiRGJQ==

The discriminant tilde d(y) dy:=factor(discrim(K,x));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkeUdGKCwwKiYpSSJ0R0YoIiIjIiIiKUkieUdGKCIiJUYzRjMqKEY2RjNGMEYzKUY1IiIkRjMhIiIqKEYyRjNGMEYzKUY1RjJGM0Y6KihGMkYzRjFGM0Y4RjNGOiokRjBGM0YzKihGMkYzRjFGM0Y1RjNGOiokRjxGM0YzNyNGLg==

Decoupling function G:=-y-1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJHR0YoLCZJInlHRighIiIqJEYvRjBGMDcjRi4=

Connection between S(y) (or Q(0,y)) and the weak invariant w eqSw:=subs(x=0,K)*Q(y)-G=alpha/(w(y)-beta)+gamma;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcVN3R0YoLywoKihJInlHRigiIiJJInRHRihGMi1JIlFHRig2I0YxRjJGMkYxRjIqJEYxISIiRjIsJiomSSZhbHBoYUdGKEYyLCYtSSJ3R0YoRjZGMkklYmV0YUdGKEY4RjhGMkkmZ2FtbWFHRiZGMjcjRi4=

wyQ:=op(2,isolate(eqSw,w(y)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR3eVFHRigsJiomLCoqKEkieUdGKCIiIkkidEdGKEYzLUkiUUdGKDYjRjJGMyEiIkYyRjgqJEYyRjhGOEkmZ2FtbWFHRiZGM0Y4SSZhbHBoYUdGKEYzRjhJJWJldGFHRihGMzcjRi4=

series(wyQ,y,3);

KytJInlHNiJJJWJldGFHRiQiIiFJJmFscGhhR0YkIiIiKiZJJmdhbW1hRyUqcHJvdGVjdGVkR0YoRidGKCIiIy1JIk9HNiRGK0koX3N5c2xpYkdGJDYjRigiIiQ=

DE for w(y) eqw:=4*dy*(diff(w(y),y))^2-(4*w(y)^3-g2*w(y)-g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcXdHRigsKiooIiIlIiIiLDAqJilJInRHRigiIiNGMSlJInlHRihGMEYxRjEqKEYwRjFGNEYxKUY4IiIkRjEhIiIqKEY2RjFGNEYxKUY4RjZGMUY8KihGNkYxRjVGMUY6RjFGPCokRjRGMUYxKihGNkYxRjVGMUY4RjFGPCokRj5GMUYxRjEpLUklZGlmZkdGJjYkLUkid0dGKDYjRjhGOEY2RjFGMSomRjBGMSlGR0Y7RjFGPComSSNnMkdGKEYxRkdGMUYxSSNnM0dGKEYxNyNGLg==

subs(w(y)=wyQ,eqw);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsKiooIiIlIiIiLDAqJilJInRHRigiIiNGLylJInlHRihGLkYvRi8qKEYuRi9GMkYvKUY2IiIkRi8hIiIqKEY0Ri9GMkYvKUY2RjRGL0Y6KihGNEYvRjNGL0Y4Ri9GOiokRjJGL0YvKihGNEYvRjNGL0Y2Ri9GOiokRjxGL0YvRi8pLUklZGlmZkdGJjYkLCYqJiwqKihGNkYvRjNGLy1JIlFHRig2I0Y2Ri9GOkY2RjoqJEY2RjpGOkkmZ2FtbWFHRiZGL0Y6SSZhbHBoYUdGKEYvRjpJJWJldGFHRihGL0Y2RjRGL0YvKiZGLkYvKUZFRjlGL0Y6KiZJI2cyR0YoRi9GRUYvRi9JI2czR0YoRi9GKw==

res:=numer(factor(eval(%))): nops(res);

IiQ6Iw==

DE for Q(0,y) edQ:=collect(res,[diff(Q(y), y),Q],factor);

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

indets(edQ);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkmYWxwaGFHRihJJWJldGFHRihJI2cyR0YoSSNnM0dGKEkidEdGKEkieUdGKC1JIlFHRig2I0YyLUklZGlmZkdGJjYkRjNGMkYr

Now we want to eliminate the unknown series, at the cost of introducing series Q_{0,i} factor(series(edQ,y,1));

KydJInlHNiIsKiooIiIlIiIiKUkmYWxwaGFHRiQiIiNGKClJInRHRiRGK0YoRigqJkYnRigpSSViZXRhR0YkIiIkRighIiIqJkYwRihJI2cyR0YkRihGKEkjZzNHRiRGKCIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRig=

Expression of g3 g3s:=solve(coeff(%,y,0),g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnM3NHRigsKCooIiIlIiIiKUkmYWxwaGFHRigiIiNGMSlJInRHRihGNEYxISIiKiZGMEYxKUklYmV0YUdGKCIiJEYxRjEqJkY6RjFJI2cyR0YoRjFGNzcjRi4=

edQ1:=op(4,factor(subs(g3=g3s,edQ))):nops(%);

IiRJIg==

factor(series(edQ1,y,1));

KydJInlHNiIsKioqIiM7IiIiSSZhbHBoYUdGJEYoSSZnYW1tYUclKnByb3RlY3RlZEdGKClJInRHRiQiIiNGKCEiIiooIiIpRihGKUYoRi1GKEYoKiYiIzdGKClJJWJldGFHRiRGLkYoRihJI2cyR0YkRi8iIiEtSSJPRzYkRitJKF9zeXNsaWJHRiQ2I0YoRig=

Expression of g2 g2s:=solve(coeff(%,y,0),g2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnMnNHRigsKCoqIiM7IiIiSSZhbHBoYUdGKEYxSSZnYW1tYUdGJkYxKUkidEdGKCIiI0YxISIiKigiIilGMUYyRjFGNUYxRjEqJiIjN0YxKUklYmV0YUdGKEY2RjFGMTcjRi4=

We have thus eliminated g2 and g3 edQ2:=op(4,factor(subs(g2=g2s,edQ1)));

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

Now the series Q_{0,i} will start to come in factor(series(edQ2,y,1));

KydJInlHNiIsLiIiIiEiIiooIiInRiYtSSJRR0YkNiMiIiFGJilJInRHRiQiIiRGJkYmKiZGMEYmSSViZXRhR0YkRiZGJiooRilGJilJJmdhbW1hRyUqcHJvdGVjdGVkRyIiI0YmKUYvRjdGJkYnKihGKUYmRjVGJkYvRiZGJiomIiIpRiZGOEYmRiZGLS1JIk9HNiRGNkkoX3N5c2xpYkdGJDYjRiZGJg==

Expression of beta betas:=solve(coeff(%,y,0),beta);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZiZXRhc0dGKCwsKigiIiMiIiItSSJRR0YoNiMiIiFGMSlJInRHRigiIiRGMSEiIiooRjBGMSlJJmdhbW1hR0YmRjBGMSlGN0YwRjFGMSooRjBGMUY8RjFGN0YxRjkqJiMiIilGOEYxRj1GMUY5I0YxRjhGMTcjRi4=

edQ3:=(factor(subs(beta=betas,edQ2))): nops(%);

IiMiKQ==

factor(series(edQ3,y,2));

KydJInlHNiIsNioqIiM3IiIiLUkiUUdGJDYjIiIhRihJJmdhbW1hRyUqcHJvdGVjdGVkR0YoKUkidEdGJCIiJEYoRihJJmFscGhhR0YkRigqJiIiI0YoRi1GKCEiIiooIiIlRigpRi1GMUYoKUYwRjRGKEY1KigiIidGKClGLUY0RihGMEYoRigqKCIjNUYoRilGKEY5RihGNSooIiIpRihGL0YoLS1JIkRHNiRGLkkoX3N5c2xpYkdGJDYjRipGK0YoRigqJkZARihGMEYoRjUqKCIjO0YoRi1GKEY5RihGKComRjdGKEY5RihGKEYoLUkiT0dGRDYjRihGNA==

Expression of alpha alphas:=solve(coeff(%,y,1),alpha);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbHBoYXNHRigsNCoqIiM3IiIiLUkiUUdGKDYjIiIhRjFJJmdhbW1hR0YmRjEpSSJ0R0YoIiIkRjEhIiIqKCIiJUYxKUY2RjlGMSlGOCIiI0YxRjEqKCIiKUYxRjdGMS0tSSJER0YlNiNGM0Y0RjFGOiooIiM1RjFGMkYxRj5GMUYxKigiIidGMSlGNkY/RjFGOEYxRjoqKCIjO0YxRjZGMUY+RjFGOiomRjxGMUY+RjFGOiomRj9GMUY2RjFGMSomRkFGMUY4RjFGMTcjRi4=

edQ4:=factor(subs(alpha=alphas,edQ3)):nops(%);

IiNw

factor(series(edQ4,y,3));

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

Now we do not have an expression of gamma, but a quartic equation for it eq1:=collect(coeff(%,y,2),gamma, factor);

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

We finally eliminate gamma factor(resultant(eq1,edQ4,gamma));nops(%);

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

IiIj

Here is the first order DE for Q(y) involving the series Q_{0,i} (here m=3) edQ:=collect(op(1,op(2,factor(resultant(eq1,edQ4,gamma)))),[diff(Q(y), y),Q(y)],distributed,factor);;

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

Checking this DE n:=20: normal(series(eval(subs(Q(y)=Qvser(n),Q(0)=subs(y=0,Qvser(n)), (D(Q))(0)=subs(y=0,diff(Qvser(n),y)),((D@@2)(Q))(0)=subs(y=0,diff(Qvser(n),y$2)),edQ)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0A=

Introducing the series Q_{0,i} (here denoted Q_i) edQ:=subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,edQ);

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

degree(edQ,Q0);degree(edQ,Q1);degree(edQ,Q2);

IiIj

IiIi

IiIi

dedQ:=factor(diff(edQ,y)): indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkjUTBHRihJI1ExR0YoSSNRMkdGKEkidEdGKEkieUdGKC1JIlFHRig2I0YxLUklZGlmZkdGJjYkRjJGMS1GNjYkRjItSSIkR0YmNiRGMSIiIzcjPCpGLUYuRi9GMEYxRjJGNS1GNjYkRjVGMQ==

We now want to obtain polynomial coefficients in t and y. We eliminate the Q_i one after the other, increasing the order of the DE res1:=factor(resultant(edQ,dedQ,Q2));nops(%);

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

IiIm

Choosing and checking the correct factor (the first one cannot cancel because of the factors y and the CT 1) edQbis:=collect(op(5,res1),[diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0,Q1],factor);

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KUkjUTBHRihJI1ExR0YoSSJ0R0YoSSJ5R0YoLUkiUUdGKDYjRjAtSSVkaWZmR0YmNiRGMUYwLUY1NiRGMS1JIiRHRiY2JEYwIiIjNyM8KUYtRi5GL0YwRjFGNC1GNTYkRjRGMA==

degree(edQbis,Q1);

IiIi

Eliminating Q1 dedQbis:=factor(diff(edQbis,y)): factor(resultant(edQbis,dedQbis,Q1)): nops(%);

IiIk

edQter:=collect(%%,[diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0],factor);;

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KUkjUTBHRihJInRHRihJInlHRigtSSJRR0YoNiNGLy1JJWRpZmZHRiY2JEYwRi8tRjQ2JEYwLUkiJEdGJjYkRi8iIiMtRjQ2JEYwLUY5NiRGLyIiJDcjPClGLUYuRi9GMEYzLUY0NiRGM0YvLUY0NiRGQ0Yv

Eliminating Q0 dedQter:=op(4,factor(diff(edQter,y))):nops(%);

IiNd

res2:=factor(resultant(edQter,dedQter,Q0)):nops(res2);

IiIl

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

IiND

KiQpSSJ0RzYiIiIlIiIi

SSJ5RzYi

nops(op(4,res2));

IiROIg==

edQpol:=collect(op(4,res2),[diff(Q(y), y, y, y, y),diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y)],factor);;

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

Let us check n:=30: normal(series(eval(subs(Q(y)=Qvser(n),edQpol)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0o=

We note that expanding the 1st order DE or the one we have just obtained gives the same information (relating Q3 to Q0, Q1 and Q2) convert(factor(subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,series(edQpol,y,1))),polynom);

LDQqKiIjTyIiIkkjUTBHNiJGJUkjUTFHRidGJSlJInRHRiciIiRGJUYlKigiIz1GJSlGJiIiI0YlKUYqRi9GJSEiIiooIiInRiUtLS1JI0BARzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YnNiRJIkRHRjhGKzYjSSJRR0YnNiMiIiFGJUYwRiVGMSooRiRGJUYmRiVGMEYlRiUqKCIjc0YlRihGJUYwRiVGJSooRi1GJUYmRiVGKkYlRjEqKCIjYUYlRipGJUkjUTJHRidGJUYlKiZGLUYlRihGJUYxKiZGJEYlRipGJUYl

convert(normal(subs(Q(0)=Q0, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,series(edQ,y,4))),polynom);

KiYsNCoqIiM3IiIiSSNRMEc2IkYmSSNRMUdGKEYmKUkidEdGKCIiJEYmRiYqKCIiJ0YmKUYnIiIjRiYpRitGMEYmISIiKihGMEYmLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKDYkSSJER0Y4Riw2I0kiUUdGKDYjIiIhRiZGMUYmRjIqKEYlRiZGJ0YmRjFGJkYmKigiI0NGJkYpRiZGMUYmRiYqKEYuRiZGJ0YmRitGJkYyKigiIz1GJkYrRiZJI1EyR0YoRiZGJiomRi5GJkYpRiZGMiomRiVGJkYrRiZGJkYmKUkieUdGKEYsRiY=

factor(%%/%);

LCQqJiIiJCIiIilJInlHNiJGJCEiIkYl

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

stepset:=[[0,1],[1,0],[0,-1],[-1,1],[1,-1]];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShzdGVwc2V0R0YoNyc3JCIiISIiIjckRjFGMDckRjAhIiI3JEY0RjE3JEYxRjQ3I0Yu

liste:=convert(map(s->x^s[1]*y^s[2],stepset),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwsSSJ5R0YoIiIiSSJ4R0YoRjAqJEYvISIiRjAqJkYxRjNGL0YwRjAqJkYxRjBGL0YzRjA3I0Yu

Number of walks of length n ending at (i,j) co:=proc(i,j,n) option remember: if i<0 or j<0 or n<0 then 0 elif n=0 and i=0 and j=0 then 1 elif n=0 then 0 else convert(map (s-> co(i-s[1],j-s[2],n-1),stepset),`+`): fi: end: co(0,0,2);

IiIi

The series Q(x,y;t) Qser:=proc(N) add(add(add(co(i,j,n)*x^i*y^j,i=0..n),j=0..n)*t^n,n=0..N): end: Qser(4);

LCwiIiJGIyomLCZJInhHNiJGI0kieUdGJ0YjRiNJInRHRidGI0YjKiYsLiokKUYmIiIjRiNGIyooRi5GI0YmRiNGKEYjRiMqJClGKEYuRiNGI0YmRiNGKEYjRiNGI0YjKUYpRi5GI0YjKiYsNiokKUYmIiIkRiNGIyooRjdGI0YtRiNGKEYjRiMqKEY3RiNGJkYjRjFGI0YjKiQpRihGN0YjRiMqJkY3RiNGLUYjRiMqKCIiJUYjRiZGI0YoRiNGIyomRjdGI0YxRiNGIyomRj5GI0YmRiNGIyomRjdGI0YoRiNGI0YjRiNGIylGKUY3RiNGIyomLEAqJClGJkY+RiNGIyooRj5GI0Y2RiNGKEYjRiMqKCIiJ0YjRi1GI0YxRiNGIyooRj5GI0YmRiNGO0YjRiMqJClGKEY+RiNGIyomRklGI0Y2RiNGIyooIiM2RiNGLUYjRihGI0YjKihGT0YjRiZGI0YxRiNGIyomRklGI0Y7RiNGIyomRk9GI0YtRiNGIyooIiM7RiNGJkYjRihGI0YjKiYiIilGI0YxRiNGIyomRlZGI0YmRiNGIyomRlZGI0YoRiNGI0Y3RiNGIylGKUY+RiNGIw==

The series Q(0,y;t), for walks ending on the vertical axis Qvser:=proc(N) add(add(co(0,j,n)*y^j,j=0..n)*t^n,n=0..N): end: Qvser(5);

LC4iIiJGIyomSSJ0RzYiRiNJInlHRiZGI0YjKiYsKCokKUYnIiIjRiNGI0YnRiNGI0YjRiMpRiVGLEYjRiMqJiwqKiQpRiciIiRGI0YjKiZGMkYjRitGI0YjKiZGMkYjRidGI0YjRiNGI0YjKUYlRjJGI0YjKiYsLCokKUYnIiIlRiNGIyomIiInRiNGMUYjRiMqJiIiKUYjRitGI0YjKiZGPkYjRidGI0YjRjJGI0YjKUYlRjpGI0YjKiYsLiokKUYnIiImRiNGIyomIiM1RiNGOUYjRiMqJiIjP0YjRjFGI0YjKiYiI0lGI0YrRiNGIyomIiM+RiNGJ0YjRiNGPkYjRiMpRiVGRUYjRiM=

The kernel K:=expand(x*y*(t*liste-1));subs(x=0,K);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMUkieUdGKEYxRjEqKEYwRjFGM0YxKUY1RjRGMUYxKiZGMEYxRjJGMUYxKiZGMEYxRjdGMUYxKiZGMEYxRjNGMUYxKiZGM0YxRjVGMSEiIjcjRi4=

KiZJInRHNiIiIiIpSSJ5R0YkIiIjRiU=

The discriminant tilde d(y) dy:=factor(discrim(K,x));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkeUdGKCwwKiYpSSJ0R0YoIiIjIiIiKUkieUdGKCIiJUYzRjMqKEY2RjNGMEYzKUY1IiIkRjMhIiIqKEYyRjNGMEYzKUY1RjJGM0Y6KihGMkYzRjFGM0Y4RjNGOiokRjBGM0YzKihGMkYzRjFGM0Y1RjNGOiokRjxGM0YzNyNGLg==

Decoupling function G:=-1/y^2+1/t/y+y*(1+t)/t-y^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJHR0YoLCoqJClJInlHRigiIiMhIiJGMyomSSJ0R0YoRjNGMUYzIiIiKihGMUY2LCZGNkY2RjVGNkY2RjVGM0Y2KiRGMEY2RjM3I0Yu

Connection between S(y) (or Q(0,y)) and the weak invariant w eqSw:=subs(x=0,K)*Q(y)-G=alpha/(w(y)-beta)+gamma;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcVN3R0YoLywsKigtSSJRR0YoNiNJInlHRigiIiJJInRHRihGNSlGNCIiI0Y1RjUqJEY3ISIiRjUqJkY2RjpGNEY6RjoqKEY0RjUsJkY1RjVGNkY1RjVGNkY6RjoqJEY3RjVGNSwmKiZJJmFscGhhR0YoRjUsJi1JIndHRihGM0Y1SSViZXRhR0YoRjpGOkY1SSZnYW1tYUdGJkY1NyNGLg==

wyQ:=op(2,isolate(eqSw,w(y)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR3eVFHRigsJiomLC4qKC1JIlFHRig2I0kieUdGKCIiIkkidEdGKEY2KUY1IiIjRjYhIiIqJEY4RjpGOiomRjdGOkY1RjpGNiooRjVGNiwmRjZGNkY3RjZGNkY3RjpGNiokRjhGNkY6SSZnYW1tYUdGJkY2RjpJJmFscGhhR0YoRjZGOkklYmV0YUdGKEY2NyNGLg==

series(wyQ,y,3);

KylJInlHNiJJJWJldGFHRiQiIiFJJmFscGhhR0YkIiIjLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiIk

DE for w(y) eqw:=4*dy*(diff(w(y),y))^2-(4*w(y)^3-g2*w(y)-g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcXdHRigsKiooIiIlIiIiLDAqJilJInRHRigiIiNGMSlJInlHRihGMEYxRjEqKEYwRjFGNEYxKUY4IiIkRjEhIiIqKEY2RjFGNEYxKUY4RjZGMUY8KihGNkYxRjVGMUY6RjFGPCokRjRGMUYxKihGNkYxRjVGMUY4RjFGPCokRj5GMUYxRjEpLUklZGlmZkdGJjYkLUkid0dGKDYjRjhGOEY2RjFGMSomRjBGMSlGR0Y7RjFGPComSSNnMkdGKEYxRkdGMUYxSSNnM0dGKEYxNyNGLg==

subs(w(y)=wyQ,eqw);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsKiooIiIlIiIiLDAqJilJInRHRigiIiNGLylJInlHRihGLkYvRi8qKEYuRi9GMkYvKUY2IiIkRi8hIiIqKEY0Ri9GMkYvKUY2RjRGL0Y6KihGNEYvRjNGL0Y4Ri9GOiokRjJGL0YvKihGNEYvRjNGL0Y2Ri9GOiokRjxGL0YvRi8pLUklZGlmZkdGJjYkLCYqJiwuKigtSSJRR0YoNiNGNkYvRjNGL0Y8Ri9GOiokRjxGOkY6KiZGM0Y6RjZGOkYvKihGNkYvLCZGL0YvRjNGL0YvRjNGOkYvRkBGOkkmZ2FtbWFHRiZGL0Y6SSZhbHBoYUdGKEYvRjpJJWJldGFHRihGL0Y2RjRGL0YvKiZGLkYvKUZFRjlGL0Y6KiZJI2cyR0YoRi9GRUYvRi9JI2czR0YoRi9GKw==

res:=numer(factor(eval(%))): nops(res);

IiRlKQ==

DE for Q(0,y) edQ:=collect(res,[diff(Q(y), y),Q],factor): indets(edQ);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KkkmYWxwaGFHRihJJWJldGFHRihJI2cyR0YoSSNnM0dGKEkidEdGKEkieUdGKC1JIlFHRig2I0YyLUklZGlmZkdGJjYkRjNGMkYr

Now we want to eliminate the unknown series, at the cost of introducing series Q_{0,i} factor(series(edQ,y,1));

KydJInlHNiIsJComKUkidEdGJCIiJSIiIiwoKiZGKUYqKUklYmV0YUdGJCIiJEYqRioqJkYuRipJI2cyR0YkRiohIiJJI2czR0YkRjJGKkYyIiIhLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRipGKg==

Expression of g3 g3s:=solve(coeff(%,y,0),g3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnM3NHRigqJkklYmV0YUdGKCIiIiwmKiYiIiVGMClGLyIiI0YwRjBJI2cyR0YoISIiRjA3I0Yu

edQ1:=op(5,factor(subs(g3=g3s,edQ))):nops(%);

IiQiSA==

factor(series(edQ1,y,1));

KydJInlHNiIsJComKUkidEdGJCIiJCIiIiwoKigiIztGKkkmYWxwaGFHRiRGKilGKCIiI0YqRioqJiIjN0YqKUklYmV0YUdGJEYwRiohIiJJI2cyR0YkRipGKkY1IiIhLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRipGKg==

Expression of g2 g2s:=solve(coeff(%,y,0),g2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnMnNHRigsJiooIiM7IiIiSSZhbHBoYUdGKEYxKUkidEdGKCIiI0YxISIiKiYiIzdGMSlJJWJldGFHRihGNUYxRjE3I0Yu

We have thus eliminated g2 and g3 edQ2:=op(5,factor(subs(g2=g2s,edQ1)));

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

The series Q_{0,i} do not come in yet factor(series(edQ2,y,1));

KydJInlHNiIsJComKUkidEdGJCIiIyIiIiwqKigiIzdGKkkmZ2FtbWFHJSpwcm90ZWN0ZWRHRipGJ0YqRioqJiIiKUYqRidGKiEiIiomIiIkRipJJWJldGFHRiRGKkYyRipGKkYqRjIiIiEtSSJPRzYkRi9JKF9zeXNsaWJHRiQ2I0YqRio=

Expression of beta betas:=solve(coeff(%,y,0),beta);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZiZXRhc0dGKCwoKigiIiUiIiJJJmdhbW1hR0YmRjEpSSJ0R0YoIiIjRjFGMSomIyIiKSIiJEYxRjNGMSEiIiNGMUY5RjE3I0Yu

edQ3:=op(2,factor(subs(beta=betas,edQ2))): nops(%);

IiQ1Ig==

factor(series(edQ3,y,1));

KydJInlHNiIqJilJInRHRiQiIiMiIiIsMiooIiM/RiktSSJRR0YkNiMiIiFGKSlGJyIiJEYpRikqKCIjN0YpKUkmZ2FtbWFHJSpwcm90ZWN0ZWRHRihGKUYmRikhIiIqKCIjO0YpRjZGKUYmRilGKSomRjpGKUYmRilGKSomRihGKUY2RilGOEkmYWxwaGFHRiRGKSomIiM9RilGJ0YpRilGLEYpRilGMC1JIk9HNiRGN0koX3N5c2xpYkdGJDYjRilGKQ==

Expression of alpha alphas:=solve(coeff(%,y,0),alpha);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbHBoYXNHRigsMCooIiM/IiIiLUkiUUdGKDYjIiIhRjEpSSJ0R0YoIiIkRjEhIiIqKCIjN0YxKUkmZ2FtbWFHRiYiIiNGMSlGN0Y+RjFGMSooIiM7RjFGPUYxRj9GMUY5KiZGQUYxRj9GMUY5KiZGPkYxRj1GMUYxKiYiIz1GMUY3RjFGOUYwRjk3I0Yu

edQ4:=factor(subs(alpha=alphas,edQ3)):nops(%);

IiMnKQ==

factor(series(edQ4,y,2));

KydJInlHNiIsJCooIiNDIiIiKUkidEdGJCIiJUYoLCgqJkYqRigtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSJRR0YkNiMiIiFGKCEiIi1GNUY2RihGKEY4RihGOEYoLUkiT0dGMTYjRigiIiM=

Here we find the connection between Q0 and Q1, which is combinatorially clear Q0s:=1+t*D(Q)(0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRRMHNHRigsJiIiIkYvKiZJInRHRihGLy0tSSJER0YlNiNJIlFHRig2IyIiIUYvRi83I0Yu

factor(subs(Q(0)=Q0s,series(edQ4,y,3)));

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

Now we do not have an expression of gamma, but a cubic equation for it eq1:=collect(coeff(%,y,2),gamma, factor);

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

We finally eliminate gamma res:=factor(resultant(eq1,subs(Q(0)=Q0s,edQ4),gamma));nops(%);

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

IiIk

Here is the first order DE for Q(y) involving the series Q_{0,i} (here m=3) edQ:=collect(op(1,op(3,res)),[diff(Q(y), y),Q(y)],distributed,factor): Checking this DE n:=30: normal(series(eval(subs(Q(y)=Qvser(n),Q(0)=subs(y=0,Qvser(n)), (D(Q))(0)=subs(y=0,diff(Qvser(n),y)),((D@@2)(Q))(0)=subs(y=0,diff(Qvser(n),y$2)),edQ)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0o=

Introducing the series Q_{0,i} (here denoted Q_i) edQ:=subs((D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,edQ): degree(edQ,Q1);degree(edQ,Q2);

IiIi

IiIi

dedQ:=factor(diff(edQ,y)): indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KUkjUTFHRihJI1EyR0YoSSJ0R0YoSSJ5R0YoLUkiUUdGKDYjRjAtSSVkaWZmR0YmNiRGMUYwLUY1NiRGMS1JIiRHRiY2JEYwIiIjNyM8KUYtRi5GL0YwRjFGNC1GNTYkRjRGMA==

We now want to obtain polynomial coefficients in t and y. We eliminate the Q_i one after the other, increasing the order of the DE res1:=factor(resultant(edQ,dedQ,Q2));nops(%);

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

IiIm

Choosing and checking the correct factor (the first one cannot cancel because of the factors y and the CT 1) edQbis:=collect(op(5,res1),[diff(Q(y), y, y),diff(Q(y), y),Q(y),Q0,Q1],factor);

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

indets(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8KEkjUTFHRihJInRHRihJInlHRigtSSJRR0YoNiNGLy1JJWRpZmZHRiY2JEYwRi8tRjQ2JEYwLUkiJEdGJjYkRi8iIiM3IzwoRi1GLkYvRjBGMy1GNDYkRjNGLw==

degree(edQbis,Q1);

IiIi

Eliminating Q1 dedQbis:=factor(diff(edQbis,y)): factor(resultant(edQbis,dedQbis,Q1)): nops(%);

IiNK

edQpol:=collect(%%,[diff(Q(y), y, y, y),diff(Q(y), y, y),diff(Q(y), y),Q(y)],factor);;

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

Let us check n:=30: normal(series(eval(subs(Q(y)=Qvser(n),edQpol)),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0o=

We note that expanding the 1st order DE or the one we have just obtained gives the same information (relating Q3 to Q0, Q1 and Q2) convert(factor(subs(Q(0)=Q0s, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,series(edQpol,y,3))),polynom);

KiYsMCooIiNDIiIiSSNRMUc2IkYmKUkidEdGKCIiJEYmISIiKigiIiVGJi0tLUkjQEBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRig2JEkiREdGM0YrNiNJIlFHRig2IyIiIUYmKUYqIiIjRiZGJiooRiVGJkYnRiZGPEYmRiwqKCIjW0YmRipGJkkjUTJHRihGJkYsKiZGJUYmRjxGJkYsKiZGJUYmRidGJkYmKiZGJUYmRipGJkYsRiYpSSJ5R0YoRj1GJg==

convert(normal(subs(Q(0)=Q0s, (D(Q))(0)=Q1, ((D@@2)(Q))(0)=2*Q2,series(edQ,y,4))),polynom);

KiYsMCooIyIjOyIiJCIiIilJInRHNiJGJ0YoLS0tSSNAQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKzYkSSJER0YwRic2I0kiUUdGKzYjIiIhRighIiIqKCIja0YoKUYqIiIjRihJI1EyR0YrRihGKCooIiNLRihJI1ExR0YrRihGKUYoRigqKEZARihGKkYoRkFGKEY5KihGQEYoRkFGKClGKiIiJUYoRigqJkZARihGKUYoRigqJkZARihGPEYoRihGKClJInlHRitGJ0Yo

factor(%%/%);

LCQqKCMiIiQiIiUiIiJJInRHNiIhIiJJInlHRilGKkYq

JSFH

JSFH

JSFH