restart: This Maple session accompanies the paper "Three quadrant walks by invariants" and considers the case of double Kreweras' steps.The proofs are in Section 6.

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCYiIiJGLyomSSJ0R0YoRi8sLkkieEdGKEYvSSJ5R0YoRi8qJkYzISIiRjRGNkYvKiRGM0Y2Ri8qJEY0RjZGLyomRjNGL0Y0Ri9GL0YvRjY3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcUNHRigsLComLCYiIiJGMSomSSJ0R0YoRjEsLkkieEdGKEYxSSJ5R0YoRjEqJkY1ISIiRjZGOEYxKiRGNUY4RjEqJEY2RjhGMSomRjVGMUY2RjFGMUYxRjhGMS1JIkNHRig2JEY1RjZGMUY4RjFGMSoqRjNGMSwmRjFGMUY5RjFGMUY2RjgtSSNDbUdGKDYjRjlGMUY4KipGM0YxRjVGOCwmRjFGMUY6RjFGMS1GQjYjRjpGMUY4KipGM0YxRjVGOEY2RjhJJEMwMEdGKEYxRjg3I0Yu

Cmser:=proc(n) local i: local res:res:=coeff(convert(Cser(n),polynom),y,0): add(coeff(res,x,-i)/x^i,i=1..n+2): end: C00ser:=proc(n) coeff(coeff(convert(Cser(n),polynom),y,0),x,0): end: Cser:=proc(n) option remember: if n=0 then 1 else normal(series(subs(C(x,y)=Cser(n-1),C00=C00ser(n-1),Cm(1/x)=Cmser(n-1),Cm(1/y)=subs(x=y,Cmser(n-1)),eqC+C(x,y)),t,n+1)): fi: end: map(expand,Cser(4));

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

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

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

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

LCoiIiJGIyomLCoqJClJInlHNiIiIiRGI0YjKiYiIidGIylGKCIiI0YjRiMqJiIjOUYjRihGI0YjIiIpRiNGIylJInRHRilGKkYjRiMqJiwoKiRGLUYjRiMqJkYuRiNGKEYjRiMiIiZGI0YjKUYzRi5GI0YjKiZGKEYjRjNGI0Yj

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZDZXhwckdGKCwoLUkjRERHRig2IyomSSJ4R0YoIiIiSSJ5R0YoRjRGNComRjMhIiItSSJVR0YoNiQqJEYzRjdGMkY0RjQqJkY1RjctRjk2JCokRjVGN0YyRjRGNDcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZTdGVwc0dGKCwuSSJ4R0YoIiIiSSJ5R0YoRjAqJkYvISIiRjFGM0YwKiRGL0YzRjAqJEYxRjNGMComRi9GMEYxRjBGMDcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSd0U3RlcHNHRigsLkkieEdGKCIiIkkieUdGKEYwKiZGLyEiIkYxRjNGMCokRi9GM0YwKiRGMUYzRjAqJkYvRjBGMUYwRjA3I0Yu

The polynomials V for the companion model Vpy:=coeff(tSteps, x,1);Vmy:=coeff(tSteps, x,-1);V0y:=coeff(tSteps, x,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRWcHlHRigsJkkieUdGKCIiIkYwRjA3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRWbXlHRigsJiIiIkYvKiRJInlHRighIiJGLzcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRWMHlHRigsJkkieUdGKCIiIiokRi8hIiJGMDcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRIbXhHRigsJiIiIkYvKiRJInhHRighIiJGLzcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcURHRigsLComLCYqJiwmSSJ5R0YoIiIiKiRGMyEiIkY0RjRJInRHRihGNEY2RjRGNEY0LUkjRERHRig2I0YzRjRGNkY0RjQqKEY3RjRGM0Y2LUY5NiMiIiFGNEY2KioiIiNGNEY3RjQsJkY0RjRGNUY0RjQtSSJVR0YoNiRGPkYzRjRGNCoqRkBGNEY3RjRGM0Y2LUZDNiRGPkY+RjRGNjcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcVVHRigsLCooSSJ4R0YoIiIiLUkiVUdGKDYkRjBJInlHRihGMSwmRjFGMSomSSJ0R0YoRjEsLkYwRjFGNUYxKiZGMCEiIkY1RjtGMSokRjBGO0YxKiRGNUY7RjEqJkYwRjFGNUYxRjFGMUY7RjFGOyoqRjhGMUYwRjEsJkY1RjFGMUYxRjEtSSNEREdGKDYjRjVGMUYxKipGOEYxRjVGOywmRjBGMUYxRjFGMS1GMzYkRjAiIiFGMUY7KihGOEYxLCZGMUYxRj1GMUYxLUYzNiRGSEY1RjFGOyooRjhGMUY1RjstRjM2JEZIRkhGMUYxNyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSllcVVmaW5hbEdGKCwsKioiIiMiIiJJInRHRihGMSwmSSJ4R0YoRjFGMUYxRjEtSSJVR0YoNiRGNCIiIUYxISIiKiYsMCoqRjBGMUYyRjEpRjRGMEYxKUkieUdGKEYwRjFGMSoqRjBGMUYyRjFGPUYxRj9GMUYxKipGMEYxRjJGMUY0RjFGPkYxRjEqKEYwRjFGMkYxRjRGMUYxKihGMEYxRjJGMUY/RjFGMSooRjBGMUY0RjFGP0YxRjkqJkYwRjFGMkYxRjFGMS1GNjYkRjRGP0YxRjEqJkYyRjEtSSNEREdGKDYjRjhGMUY5KiYsLEZBRjEqKkYwRjFGMkYxRjRGMUY/RjFGMSomRjJGMUY+RjFGMUYyRjFGP0Y5RjEtRko2I0Y/RjFGMUY/RjE3I0Yu

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

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

K:=1-t*(x+y+1/(x*y)+1/x+1/y+x*y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCYiIiJGLyomSSJ0R0YoRi8sLkkieEdGKEYvSSJ5R0YoRi8qJkYzISIiRjRGNkYvKiRGM0Y2Ri8qJEY0RjZGLyomRjNGL0Y0Ri9GL0YvRjY3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcVFHRigsLComLCYiIiJGMSomSSJ0R0YoRjEsLkkieEdGKEYxSSJ5R0YoRjEqJkY1ISIiRjZGOEYxKiRGNUY4RjEqJEY2RjhGMSomRjVGMUY2RjFGMUYxRjhGMS1JIlFHRig2JEY1RjZGMUY4RjFGMSoqRjNGMUY2RjgsJkYxRjFGOUYxRjEtRj02JEY1IiIhRjFGOCoqRjNGMUY1RjgsJkYxRjFGOkYxRjEtRj02JEZDRjZGMUY4KipGM0YxRjVGOEY2RjgtRj02JEZDRkNGMUYxNyNGLg==

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

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

Two pairs of invariants (Table 3). The first one is rational I0x:=1/x-x-(1+2*t)/t/(1+x); J0y:=subs(x=y,I0x);

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LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRKMHlHRigsKCokSSJ5R0YoISIiIiIiRjBGMSooLCZGMkYyKiYiIiNGMkkidEdGKEYyRjJGMkY3RjEsJkYwRjJGMkYyRjFGMTcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdyYXRpbzBHRigqKiwmSSJ4R0YoIiIiSSJ5R0YoISIiRjFJInRHRihGMywmRjBGMUYxRjFGMywmRjJGMUYxRjFGMzcjRi4=

Decoupling of xy fx:=(x-t-t*x^2)/t/(1+x); gy:=-1/y; convert(fx,parfrac,x)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmeEdGKCooLCgqJkkidEdGKCIiIilJInhHRigiIiNGMiEiIkYxRjZGNEYyRjJGMUY2LCZGNEYyRjJGMkY2NyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNneUdGKCwkKiRJInlHRighIiJGMTcjRi4=

LChJInhHNiIhIiIqJiwmIiIiRihJInRHRiRGKEYoRilGJUYoKigsJiomIiIjRihGKUYoRiVGKEYlRihGKUYlLCZGI0YoRihGKEYlRig=

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

KiosMCooSSJ0RzYiIiIiKUkieEdGJiIiI0YnKUkieUdGJkYqRidGJyooRiVGJ0YoRidGLEYnRicqKEYlRidGKUYnRitGJ0YnKiZGJUYnRilGJ0YnKiZGJUYnRixGJ0YnKiZGKUYnRixGJyEiIkYlRidGJ0YlRjIsJkYpRidGJ0YnRjJGLEYy

LCQqKEkieEc2IiIiIkkidEdGJSEiIiwmRiRGJkYmRiZGKEYo

Invariants related to quadrant walks I1x:=t*(1+x)*Q(x,0)-fx; J1y:=-t*(1+y)*Q(0,y)+t*Q(0,0)+gy;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRJMXhHRigsJiooSSJ0R0YoIiIiLCZJInhHRihGMUYxRjFGMS1JIlFHRig2JEYzIiIhRjFGMSooLCgqJkYwRjEpRjMiIiNGMSEiIkYwRj1GM0YxRjFGMEY9RjJGPUY9NyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRKMXlHRigsKCooSSJ0R0YoIiIiLCZJInlHRihGMUYxRjFGMS1JIlFHRig2JCIiIUYzRjEhIiIqJkYwRjEtRjU2JEY3RjdGMUYxKiRGM0Y4Rjg3I0Yu

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

KiwsMCooSSJ0RzYiIiIiKUkieEdGJiIiI0YnKUkieUdGJkYqRidGJyooRiVGJ0YoRidGLEYnRicqKEYlRidGKUYnRitGJ0YnKiZGJUYnRilGJ0YnKiZGJUYnRixGJ0YnKiZGKUYnRixGJyEiIkYlRidGJywoKiotSSJRR0YmNiRGKUYsRidGJUYnRilGJ0YsRidGJyooRjVGJ0YlRidGLEYnRidGJ0YnRidGJUYyLCZGKUYnRidGJ0YyRixGMg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdyYXRpbzFHRigsJCoqSSJ4R0YoIiIiLCgqKi1JIlFHRig2JEYwSSJ5R0YoRjFJInRHRihGMUYwRjFGN0YxRjEqKEY0RjFGOEYxRjdGMUYxRjFGMUYxRjghIiIsJkYwRjFGMUYxRjpGOjcjRi4=

LCYqKEkieEc2IiIiIkkieUdGJUYmLUkiUUdGJTYkRiRGJ0YmISIiKihGJEYmSSJ0R0YlRissJkYkRiZGJkYmRitGKw==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZTdGVwc0dGKCwuSSJ4R0YoIiIiSSJ5R0YoRjAqJkYvISIiRjFGM0YwKiRGL0YzRjAqJEYxRjNGMComRi9GMEYxRjBGMDcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSd0U3RlcHNHRigsLkkieEdGKCIiIkkieUdGKEYwKiZGLyEiIkYxRjNGMCokRi9GM0YwKiRGMUYzRjAqJkYvRjBGMUYwRjA3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRWcHlHRigsJkkieUdGKCIiIkYwRjA3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRWbXlHRigsJiIiIkYvKiRJInlHRighIiJGLzcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRWMHlHRigsJkkieUdGKCIiIiokRi8hIiJGMDcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjb0dGKCwoKioiIiMiIiJJInRHRihGMUkieEdGKEYxLCZJInlHRihGMUYxRjFGMUYxKiYsJkY1RjEqJEY1ISIiRjFGMUYyRjFGMUYxRjk3I0Yu

The functions F and G and H of Table 5 Fx:=-1-x-1/x+(1+2*t)/t/(1+x); Gy:=1/(1+y)/t;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwqIiIiISIiSSJ4R0YoRjAqJEYxRjBGMCooLCZGL0YvKiYiIiNGL0kidEdGKEYvRi9GL0Y3RjAsJkYxRi9GL0YvRjBGLzcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKComLCZJInlHRigiIiJGMUYxISIiSSJ0R0YoRjI3I0Yu

factor(y-Gy*co-Fx);

LCQqMCwmSSJ4RzYiIiIiSSJ5R0YmISIiRicsMCooSSJ0R0YmRicpRiUiIiNGJylGKEYuRidGJyooRixGJ0YtRidGKEYnRicqKEYsRidGJUYnRi9GJ0YnKiZGLEYnRiVGJ0YnKiZGLEYnRihGJ0YnKiZGJUYnRihGJ0YpRixGJ0YnRiVGKUYsRiksJkYlRidGJ0YnRilGKEYpLCZGKEYnRidGJ0YpRik=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRIeHlHRigqKiwmSSJ4R0YoIiIiSSJ5R0YoISIiRjFJInRHRihGMywmRjBGMUYxRjFGMywmRjJGMUYxRjFGMzcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNTeUdGKCwmKiZJInlHRigiIiItSSNEREdGKDYjRjBGMUYxKiYsJkYwRjFGMUYxISIiSSJ0R0YoRjdGMTcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNSeEdGKCwuKioiIiMiIiJJInRHRihGMSwmSSJ4R0YoRjFGMUYxRjEtSSJVR0YoNiRGNCIiIUYxRjFGMUYxRjRGMSokRjQhIiJGMSooLCZGMUYxKiZGMEYxRjJGMUYxRjFGMkY6RjNGOkY6KiZGMkYxLUkjRERHRig2I0Y4RjFGMTcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZEZWx0YUdGKCwmKiQpLCYqJiwmSSJ5R0YoIiIiKiRGNCEiIkY1RjVJInRHRihGNUY3RjVGNSIiI0Y1RjUqKiIiJUY1KUY4RjlGNSwmRjRGNUY1RjVGNSwmRjVGNUY2RjVGNUY3NyNGLg==

We now check the identity of Proposition 3.2 factor(Rx^2-Delta*Sy^2): Divisibility by K=tK factor(subs(isolate(eqUfinal,U(x,0)),%)/K):denom(%);

KixJInhHNiIiIiJJInlHRiRGJSksJkYmRiVGJUYlIiIjRiUpSSJ0R0YkRilGJSksJkYjRiVGJUYlRilGJQ==

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

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

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

IiIh

Sy;

LCYqJkkieUc2IiIiIi1JI0RER0YlNiNGJEYmRiYqJiwmRiRGJkYmRiYhIiJJInRHRiVGLEYm

Rx;Delta;

LC4qKiIiIyIiIkkidEc2IkYlLCZJInhHRidGJUYlRiVGJS1JIlVHRic2JEYpIiIhRiVGJUYlRiVGKUYlKiRGKSEiIkYlKigsJkYlRiUqJkYkRiVGJkYlRiVGJUYmRi9GKEYvRi8qJkYmRiUtSSNEREdGJzYjRi1GJUYl

LCYqJCksJiomLCZJInlHNiIiIiIqJEYoISIiRipGKkkidEdGKUYqRixGKkYqIiIjRipGKioqIiIlRiopRi1GLkYqLCZGKEYqRipGKkYqLCZGKkYqRitGKkYqRiw=

Ix:=Rx^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeEdGKCokKSwuKioiIiMiIiJJInRHRihGMywmSSJ4R0YoRjNGM0YzRjMtSSJVR0YoNiRGNiIiIUYzRjNGM0YzRjZGMyokRjYhIiJGMyooLCZGM0YzKiZGMkYzRjRGM0YzRjNGNEY8RjVGPEY8KiZGNEYzLUkjRERHRig2I0Y6RjNGM0YyRjM3I0Yu

Jy:=Delta*Sy^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNKeUdGKComLCYqJCksJiomLCZJInlHRigiIiIqJEY1ISIiRjZGNkkidEdGKEY2RjhGNkY2IiIjRjZGNioqIiIlRjYpRjlGOkY2LCZGNUY2RjZGNkY2LCZGNkY2RjdGNkY2RjhGNiksJiomRjVGNi1JI0RER0YoNiNGNUY2RjYqJkY+RjhGOUY4RjZGOkY2NyNGLg==

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

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

series(ratio,x,1);

KydJInhHNiIsJCooSSJ5R0YkIiIiSSJ0R0YkISIiLCZGJ0YoRihGKEYqRipGKi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIh

series(ratio,y,1);

KydJInlHNiIqKEkieEdGJCIiIkkidEdGJCEiIiwmRiZGJ0YnRidGKUYpLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiciIiE=

series(Ix,x,1);

KylJInhHNiIiIiIhIiMsKiooIiIlRiVJInRHRiRGJS1JIlVHRiQ2JCIiIUYuRiVGJSIiI0YlKihGL0YlLCZGJUYlKiZGL0YlRipGJUYlRiVGKiEiIkYzKihGL0YlRipGJS1JI0RER0YkNiNGLkYlRiVGMy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlRi4=

I0 has poles at -1 and 0, while I1 has a pole at -1 only I0x; I1x;

LCgqJEkieEc2IiEiIiIiIkYkRiYqKCwmRidGJyomIiIjRidJInRHRiVGJ0YnRidGLEYmLCZGJEYnRidGJ0YmRiY=

LCYqKEkidEc2IiIiIiwmSSJ4R0YlRiZGJkYmRiYtSSJRR0YlNiRGKCIiIUYmRiYqKCwoKiZGJEYmKUYoIiIjRiYhIiJGJEYyRihGJkYmRiRGMkYnRjJGMg==

What do we need to subtract to Ix to hope to have a constant ? We first consider the pole at 0 series(Ix-I0x^2,x,1);

KydJInhHNiIsKCooIiIlIiIiSSJ0R0YkRigtSSJVR0YkNiQiIiFGLUYoRigiIiNGKCooRi5GKEYpRigtSSNEREdGJDYjRi1GKEYoISIiLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGLQ==

A2:=factor(coeff(%,x,-1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNBMkdGKCwoKigiIiUiIiJJInRHRihGMS1JIlVHRig2JCIiIUY2RjFGMSIiI0YxKihGN0YxRjJGMS1JI0RER0YoNiNGNkYxRjE3I0Yu

No more pole at 0 factor(series(Ix-I0x^2-A2*I0x,x,2));

KydJInhHNiIsMCooIiIlIiIiKS1JIlVHRiQ2JCIiIUYtIiIjRigpSSJ0R0YkRi5GKEYoKipGJ0YoRipGKC1JI0RER0YkNiNGLUYoRi9GKEYoKiYpRjJGLkYoRi9GKEYoKihGJ0YoRjBGKC0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRig2I0YrRixGKEYoKigiIilGKEYwRihGKkYoRigqKEYuRihGMEYoRjJGKEYoIiImRihGLS1JIk9HRjxGP0Yo

What about the pole at x=-1 ? factor(series(Ix-I0x^2-A2*I0x,x=-1,2));

KyksJkkieEc2IiIiIkYmRiYsJCoqIiIlRiYsJkYmRiYqJiIiI0YmSSJ0R0YlRiZGJkYmLCYqJkYtRiYtSSJVR0YlNiQiIiFGM0YmRiZGJkYmRiZGLSEiIkYmRjQsJComLC4qJiktSSNEREdGJTYjRjNGLEYmKUYtIiIkRiZGNCooIiIpRiYtRjE2JEY0RjNGJilGLUYsRiZGJiooRixGJkZDRiZGOkYmRiYqKEYpRiZGLUYmRkFGJkYmKiYiIihGJkYtRiZGJkYpRiZGJkYtRjRGNEYzLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiZGJg==

series(I1x,x=-1,1);

KyksJkkieEc2IiIiIkYmRiYsJComLCYqJiIiI0YmSSJ0R0YlRiYhIiJGJkYtRiZGLEYtRi1GLSwkKiYsJkYmRiZGKkYmRiZGLEYtRi0iIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJkYm

A1:=factor(op(1,%%)/op(1,%))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNBMUdGKCwmKigiIiUiIiJJInRHRihGMS1JIlVHRig2JCIiIUY2RjFGMUYwRjE3I0Yu

No more pole at -1 either factor(series(Ix-I0x^2-A2*I0x-A1*I1x,x=-1,2));

KycsJkkieEc2IiIiIkYmRiYsMComKS1JI0RER0YlNiMiIiEiIiNGJilJInRHRiVGLkYmRiYqKCIiKUYmRjBGJi1JIlVHRiU2JCEiIkYtRiZGNiooRjJGJkYwRiYtRjQ2JEYtRi1GJkYmKihGLkYmRjBGJkYqRiZGNiomIiIlRiZGM0YmRjYqJkY8RiZGOEYmRiZGJkYmRi0tSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJkYm

What happened to J(y) if we modify it as we did for I(x)? An encouraging sign: no pole at 0 map(factor,series(Jy-J0y^2-A2*J0y-A1*J1y,y,3));

KydJInlHNiIsMComKS1JI0RER0YkNiMiIiEiIiMiIiIpSSJ0R0YkRixGLUYtKigiIilGLUYvRi0tSSJVR0YkNiRGK0YrRi1GLSooIiInRi1GL0YtRihGLSEiIiooRixGLS0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKUYqRi1GL0YtRi0qJiIiJUYtRjJGLUYtKiZGLEYtRihGLUY3IiIkRi1GKy1JIk9HRjw2I0YtRi0=

At y=-1, we need an identity involving D(-1) to prove that there is no pole either map(factor,series(Jy-J0y^2-A2*J0y-A1*J1y,y=-1,2));

KycsJkkieUc2IiIiIkYmRiYsJCoqIiIjRiYsJkYmRiYqJkYpRiZJInRHRiVGJkYmRiYsLCooRilGJkYsRiYtSSJVR0YlNiQiIiFGMkYmRiYqJkYsRiYtSSNEREdGJTYjRjJGJkYmKihGKUYmLUY1NiMhIiJGJkYsRiZGOkY4RjpGJkYmRiZGLEY6RiZGOi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YmRjI=

subs(y=-1,eqD);

LCoqJiwmIiIiRiUqJiIiI0YlSSJ0RzYiRiVGJUYlLUkjRERHRik2IyEiIkYlRi1GJUYlKiZGKEYlLUYrNiMiIiFGJUYlKihGJ0YlRihGJS1JIlVHRik2JEYxRjFGJUYl

factor(subs(isolate(%,DD(-1)),%%))

KyUsJkkieUc2IiIiIkYmRiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJiIiIQ==

Thus there is hope! So let us consider the ratio, and expand it, first in x. ser:=map(factor,series(ratio-ratio0*(I0x+J0y)-A2*ratio0-A1*ratio1,x,2));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzZXJHRigrJ0kieEdGKCoqLDIqKClJInlHRigiIiMiIiItSSNEREdGKDYjRjRGNkkidEdGKEY2ISIiKipGNUY2LUkiVUdGKDYkIiIhRjRGNkY6RjZGNEY2RjsqKEY1RjZGOkY2LUY+NiRGQEZARjZGNiomRjdGNkY6RjZGOyomRjRGNkY3RjZGNiomRjpGNi1GODYjRkBGNkY2KihGNUY2Rj1GNkY6RjZGO0Y0RjtGNkY6RjtGNEY2LCZGNEY2RjZGNkY7RkAtSSJPR0YlNiNGNkY2NyNGLg==

Let us check that the coefficient of x^0 vanishes factor(subs(isolate(eqD,U(0,y)),coeff(ser,x,0)));

IiIh

Now let us expand the ratio in y: it is a multiple of y ser:=map(factor,series(ratio-ratio0*(I0x+J0y)-A2*ratio0-A1*ratio1,y,2));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzZXJHRigrJUkieUdGKC1JIk9HRiU2IyIiIkYzNyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZleHByeEdGKCwqKiYsJiooIiIlIiIiSSJ0R0YoRjMtSSJVR0YoNiQiIiFGOEYzRjNGMkYzRjMsJiooRjRGMywmSSJ4R0YoRjNGM0YzRjMtSSJRR0YoNiRGPEY4RjNGMyooLCgqJkY0RjMpRjwiIiNGMyEiIkY0RkVGPEYzRjNGNEZFRjtGRUZFRjNGRSomLChGMUYzRkRGMyooRkRGM0Y0RjMtSSNEREdGKDYjRjhGM0YzRjMsKCokRjxGRUYzRjxGRSooLCZGM0YzKiZGREYzRjRGM0YzRjNGNEZFRjtGRUZFRjNGRSokKUZMRkRGM0ZFKiQpLC4qKkZERjNGNEYzRjtGMy1GNkY/RjNGM0YzRjNGPEYzRk1GM0ZORkUqJkY0RjNGSUYzRjNGREYzRjM3I0Yu

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

Let us check n:=10: normal(series(subs(U(x,0)=subs(y=0,User(n)),Q(x,0)=subs(y=0,Qser(n)),DD(0)=subs(y=0,Dser(n)),U(0,0)=subs(x=0,y=0,User(n)), exprx),t,n));

KzdJInRHNiIiIiIiIiEhIiNGJSIiJiIiIyIiJyIiJCIjbSIiJSIkUyNGKCIlYDhGKiIlSWoiIigiJlFJJCIiKS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIq

n:=10: normal(series(subs(DD(y)=Dser(n), Q(0,y)=subs(x=0,Qser(n)),Q(0,0)=subs(x=0,y=0,Qser(n)),DD(0)=subs(y=0,Dser(n)),U(0,0)=subs(x=0,y=0,User(n)), expry),t,n));

KzdJInRHNiIiIiIiIiEhIiNGJSIiJiIiIyIiJyIiJCIjbSIiJSIkUyNGKCIlYDhGKiIlSWoiIigiJlFJJCIiKS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIq

The solution of quadrant walks with double Kreweras steps, taken from earlier papers algN:=-N*(1-N)^2+t*(1-2*N+6*N^2-2*N^3+N^4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdOR0YoLCYqJkkiTkdGKCIiIiksJkYxRjFGMCEiIiIiI0YxRjQqJkkidEdGKEYxLCwqJClGMCIiJUYxRjEqJkY1RjEpRjAiIiRGMUY0KiYiIidGMSlGMEY1RjFGMSomRjVGMUYwRjFGNEYxRjFGMUYxNyNGLg==

tN:=solve(algN,t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN0TkdGKCooSSJOR0YoIiIiKSwmRjAhIiJGL0YwIiIjRjAsLCokKUYvIiIlRjBGMComRjRGMClGLyIiJEYwRjMqJiIiJ0YwKUYvRjRGMEYwKiZGNEYwRi9GMEYzRjBGMEYzNyNGLg==

with(gfun): Nser:=proc(n) op(2,algeqtoseries(algN,t,N,n,true)): end: Nser(4); Nser(5);

KytJInRHNiIiIiJGJSIiJiIiJCIiKSIiJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlRiY=

Ky1JInRHNiIiIiJGJSIiJiIiJCIiKSIiJSIjaUYmLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIic=

unassign('N'):Deltapx:=1-2*N*(1+N^2)/(1-N)^2*x+x^2*N^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShEZWx0YXB4R0YoLCgiIiJGLyosIiIjRi9JIk5HRihGLywmKiQpRjJGMUYvRi9GL0YvRi8pLCZGL0YvRjIhIiJGMUY4SSJ4R0YoRi9GOComKUY5RjFGL0Y1Ri9GLzcjRi4=

Known expression of Q(x,0) Qx0:=(N*(1-N)+2*x*N-(1-N)*x^2)/2/t/x/N/(1-N)/(1+x)^2*sqrt(Deltapx)- (N*(1-N)^2+N*(N^3+4*N^2-5*N+2)*x-(1-2*N+7*N^2-4*N^3)*x^2+x^3*N*(1-N)^2)/2/t/x/N/(1-N)^2/(1+x)^2;

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

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

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

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

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

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

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIio=

factor(subs(t=tN,series(Qx0,x,2)));

KydJInhHNiIsJCooLCwqJClJIk5HRiQiIiUiIiJGLComIiIjRiwpRioiIiRGLCEiIiomIiInRiwpRipGLkYsRiwqJkYuRixGKkYsRjFGLEYsRiwsKCokRjRGLEYsRjVGLEYsRjFGLCksJkYsRjFGKkYsRitGMUYxIiIhLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRixGLA==

Q00:=convert(%,polynom);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRRMDBHRigsJCooLCwqJClJIk5HRigiIiUiIiJGNSomIiIjRjUpRjMiIiRGNSEiIiomIiInRjUpRjNGN0Y1RjUqJkY3RjVGM0Y1RjpGNUY1RjUsKCokRj1GNUY1Rj5GNUY1RjpGNSksJkY1RjpGM0Y1RjRGOkY6NyNGLg==

A smaller expression Q00tN:=Q00*tN/t;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZRMDB0TkdGKCwkKiosKCokKUkiTkdGKCIiIyIiIkY1KiZGNEY1RjNGNUY1RjUhIiJGNSksJkY1RjdGM0Y1RjRGN0YzRjVJInRHRihGN0Y3NyNGLg==

Back to I1x I1x;

LCYqKEkidEc2IiIiIiwmSSJ4R0YlRiZGJkYmRiYtSSJRR0YlNiRGKCIiIUYmRiYqKCwoKiZGJEYmKUYoIiIjRiYhIiJGJEYyRihGJkYmRiRGMkYnRjJGMg==

I1xexpl:=collect(subs(Q(x,0)=Qx0,sqrt(1-2*N*(N^2+1)*x/(1-N)^2+x^2*N^2)=sqrt(Dpx),I1x),Dpx,factor);

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

We can simplify this a bit. We first ignore the term with the square root part2:=convert(subs(Dpx=0,t=tN,%),parfrac,x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZwYXJ0MkdGKCwqKiYjIiIiIiIjRjFJInhHRihGMUYxKipGMEYxLCwqJkYyRjEpSSJOR0YoIiIlRjFGMSokKUY4IiIkRjFGMSomRjxGMSlGOEYyRjFGMUY4ISIiRjFGMUYxKSwmRjFGP0Y4RjFGMkY/RjhGP0Y/KixGMEYxLCgqJEY3RjFGMSomRjJGMUY+RjFGMUYxRjFGMSwmRjNGMUYxRjFGP0ZARj9GOEY/RjEqJkYwRjFGM0Y/Rj83I0Yu

We can simplify the term with a pole at x=-1 (take the correct term!) factor(resultant(numer(c-op(3,part2)),algN,N)); {solve(%,c)}

LCQqJiIjOyIiIiksKioqIiIjRiVJImNHNiJGJUkidEdGK0YlSSJ4R0YrRiVGJSooRilGJUYqRiVGLEYlRiUqJkYpRiVGLEYlISIiRiVGMCIiJUYlRiU=

PCMsJCoqIyIiIiIiIyIiIiwmIiIiIiIiKiYiIiMiIiJJInRHNiIiIiIiIiIiIiJJInRHNiIhIiIsJkkieEc2IiIiIiIiIiIiIiEiIiIiIg==

I1expl:=op(1,I1xexpl)+ subsop(3=op(%),part2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdJMWV4cGxHRigsLCowIyIiIiIiI0YxLCwqJkkiTkdGKEYxKUkieEdGKEYyRjEhIiIqJClGNUYyRjFGMSooRjJGMUY1RjFGN0YxRjgqJEY2RjFGMUY1RjhGMSwmRjdGMUYxRjFGOEY3RjhGNUY4LCZGMUY4RjVGMUY4KUkkRHB4R0YoRjBGMUYxKiZGMEYxRjdGMUYxKipGMEYxLCwqJkYyRjEpRjUiIiVGMUYxKiQpRjUiIiRGMUYxKiZGSUYxRjpGMUYxRjVGOEYxRjFGMSlGPkYyRjhGNUY4RjgqKkYwRjEsJkYxRjEqJkYyRjFJInRHRihGMUYxRjFGT0Y4Rj1GOEYxKiZGMEYxRjdGOEY4NyNGLg==

We recognize I0/2 in the above expression I0x

LCgqJEkieEc2IiEiIiIiIkYkRiYqKCwmRidGJyomIiIjRidJInRHRiVGJ0YnRidGLEYmLCZGJEYnRidGJ0YmRiY=

I1expl+I0x/2

LCYqMCMiIiIiIiNGJSwsKiZJIk5HNiJGJSlJInhHRipGJkYlISIiKiQpRilGJkYlRiUqKEYmRiVGKUYlRixGJUYtKiRGK0YlRiVGKUYtRiUsJkYsRiVGJUYlRi1GLEYtRilGLSwmRiVGLUYpRiVGLSlJJERweEdGKkYkRiVGJSoqRiRGJSwsKiZGJkYlKUYpIiIlRiVGJSokKUYpIiIkRiVGJSomRj1GJUYvRiVGJUYpRi1GJUYlRiUpRjNGJkYtRilGLUYt

I2 should contain only one term! (otherwise we have taken the wrong c above) I2x:=I1expl+I0x/2+(2*N^4+N^3+3*N^2-N+1)/(2*N*(-1+N)^2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRJMnhHRigsJCowIyIiIiIiI0YxLCwqJkkiTkdGKEYxKUkieEdGKEYyRjEhIiIqJClGNUYyRjFGMSooRjJGMUY1RjFGN0YxRjgqJEY2RjFGMUY1RjhGMSwmRjdGMUYxRjFGOEY3RjhGNUY4LCZGMUY4RjVGMUY4KUkkRHB4R0YoRjBGMUYxNyNGLg==

I2pap:=(N+2*x*N/(1-N)-x^2)/2/x/N/(1+x)*sqrt(Dpx)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZJMnBhcEdGKCwkKi4jIiIiIiIjRjEsKEkiTkdGKEYxKipGMkYxSSJ4R0YoRjFGNEYxLCZGMUYxRjQhIiJGOEYxKiQpRjZGMkYxRjhGMUY2RjhGNEY4LCZGNkYxRjFGMUY4KUkkRHB4R0YoRjBGMUYxNyNGLg==

factor(I2pap/I2x)

IiIi

Ixform:=I0^2+AA2*I0+AA1*I1+AA0; Jyform:=J0^2+AA2*J0+AA1*J1+AA0;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdJeGZvcm1HRigsKiomSSRBQTFHRigiIiJJI0kxR0YoRjFGMSomSSRBQTJHRihGMUkjSTBHRihGMUYxKiQpRjUiIiNGMUYxSSRBQTBHRihGMTcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdKeWZvcm1HRigsKiomSSRBQTFHRigiIiJJI0oxR0YoRjFGMSomSSRBQTJHRihGMUkjSjBHRihGMUYxKiQpRjUiIiNGMUYxSSRBQTBHRihGMTcjRi4=

Ixformnew:=collect(subs(I1=-I0/2-(2*N^4+N^3+3*N^2-N+1)/(2*N*(-1+N)^2)+I2, Ixform),[I0,I2,AA0],factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpJeGZvcm1uZXdHRigsLCokKUkjSTBHRigiIiMiIiJGMyomLCYqJiNGM0YyRjNJJEFBMUdGKEYzISIiSSRBQTJHRihGM0YzRjFGM0YzKiZGOEYzSSNJMkdGKEYzRjMqLEY3RjNGOEYzLCwqJkYyRjMpSSJOR0YoIiIlRjNGMyokKUZBIiIkRjNGMyomRkVGMylGQUYyRjNGM0ZBRjlGM0YzRjMpLCZGM0Y5RkFGM0YyRjlGQUY5RjlJJEFBMEdGKEYzNyNGLg==

A2t:=factor(A2-A1/2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBMnRHRigsJCooIiIjIiIiSSJ0R0YoRjEsJi1JIlVHRig2JCIiIUY3RjEtSSNEREdGKDYjRjdGMUYxRjE3I0Yu

A0t:=A0-A1*(2*N^4+N^3+3*N^2-N+1)/(2*N*(-1+N)^2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBMHRHRigsJkkjQTBHRigiIiIqLCNGMCIiI0YwLCYqKCIiJUYwSSJ0R0YoRjAtSSJVR0YoNiQiIiFGO0YwRjBGNkYwRjAsLComRjNGMClJIk5HRihGNkYwRjAqJClGPyIiJEYwRjAqJkZCRjApRj9GM0YwRjBGPyEiIkYwRjBGMCksJkYwRkVGP0YwRjNGRUY/RkVGRTcjRi4=

Ixform:=I0x^2+tA2*I0x+AA1*I2x+tA0;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdJeGZvcm1HRigsKioyIyIiIiIiI0YxSSRBQTFHRihGMSwsKiZJIk5HRihGMSlJInhHRihGMkYxISIiKiQpRjZGMkYxRjEqKEYyRjFGNkYxRjhGMUY5KiRGN0YxRjFGNkY5RjEsJkY4RjFGMUYxRjlGOEY5RjZGOSwmRjFGOUY2RjFGOSlJJERweEdGKEYwRjFGMSokKSwoKiRGOEY5RjFGOEY5KigsJkYxRjEqJkYyRjFJInRHRihGMUYxRjFGSUY5Rj5GOUY5RjJGMUYxKiZGREYxSSR0QTJHRihGMUYxSSR0QTBHRihGMTcjRi4=

Let us check by comparing the expansions of Ix and Ixform n:=6: normal(series(subs(tA2=A2t, AA1=A1,Dpx=Deltapx,N=Nser(n+2), U(0,0)=subs(x=0,y=0,User(n+2)), DD(0)=subs(y=0,Dser((n+2))), Ixform),t,n));

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

normal(series(subs(U(x,0)=subs(y=0,User(n+2)),DD(0)=subs(y=0,Dser((n+2))),Ix),t,n));

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

factor(series(%%-%,t,n))

Ky9JInRHNiIiIiMhIiIsJkkkdEEwR0YkIiIiRilGKSIiIUYlRilGJkYlIiIpIiIkLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRikiIiU=

tA0ser:=expand(solve(convert(%,polynom),tA0));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSd0QTBzZXJHRigsLComIiIjIiIiSSJ0R0YoISIiRjNGMUYzKiZGMEYxRjJGMUYzKiQpRjJGMEYxRjEqJiIiKUYxKUYyIiIkRjFGMzcjRi4=

For J1(y) expand(J1y);

LCoqKEkidEc2IiIiIi1JIlFHRiU2JCIiIUkieUdGJUYmRitGJiEiIiomRiRGJkYnRiZGLComRiRGJi1GKDYkRipGKkYmRiYqJEYrRixGLA==

J1y1:=collect(subs(Q(0,y)=subs(x=y,Qx0),Q(0,0)=Q00tN,sqrt(1-2*N*(N^2+1)*y/(1-N)^2+y^2*N^2)=sqrt(Dpy),J1y),Dpy,factor);

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

Let us ignore the part with the square root convert(subs(Dpy=0,%),parfrac,y);

LCoqJiMiIiIiIiNGJUkieUc2IkYlRiUqKkYkRiUsLComRiZGJSlJIk5HRigiIiVGJUYlKiQpRi0iIiRGJUYlKiZGMUYlKUYtRiZGJUYlRi0hIiJGJUYlRiUpLCZGJUY0Ri1GJUYmRjRGLUY0RjQqLEYkRiUsKCokRixGJUYlKiZGJkYlRjNGJUYlRiVGJUYlLCZGJ0YlRiVGJUY0RjVGNEYtRjRGJSomRiRGJUYnRjRGNA==

We recognize what we had for I1(x). Only the sign before the square root has changed part2

LCoqJiMiIiIiIiNGJUkieEc2IkYlRiUqKkYkRiUsLComRiZGJSlJIk5HRigiIiVGJUYlKiQpRi0iIiRGJUYlKiZGMUYlKUYtRiZGJUYlRi0hIiJGJUYlRiUpLCZGJUY0Ri1GJUYmRjRGLUY0RjQqLEYkRiUsKCokRixGJUYlKiZGJkYlRjNGJUYlRiVGJUYlLCZGJ0YlRiVGJUY0RjVGNEYtRjRGJSomRiRGJUYnRjRGNA==

J0y;

LCgqJEkieUc2IiEiIiIiIkYkRiYqKCwmRidGJyomIiIjRidJInRHRiVGJ0YnRidGLEYmLCZGJEYnRidGJ0YmRiY=

map(factor,subs(t=tN,collect(J1y1+J0y/2,Dpy,factor)));

LCYqMCMiIiIiIiNGJSwsKiZJIk5HNiJGJSlJInlHRipGJkYlISIiKiQpRilGJkYlRiUqKEYmRiVGKUYlRixGJUYtKiRGK0YlRiVGKUYtRiUsJkYsRiVGJUYlRi1GLEYtRilGLSwmRiVGLUYpRiVGLSlJJERweUdGKkYkRiVGLSoqRiRGJSwsKiZGJkYlKUYpIiIlRiVGJSokKUYpIiIkRiVGJSomRj1GJUYvRiVGJUYpRi1GJUYlRiUpRjNGJkYtRilGLUYt

Jyform:=J0y^2+tA2*J0y-AA1*subs(x=y,Dpx=Dpy,I2x)+tA0;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdKeWZvcm1HRigsKiokKSwoKiRJInlHRighIiIiIiJGM0Y0KigsJkY1RjUqJiIiI0Y1SSJ0R0YoRjVGNUY1RjpGNCwmRjNGNUY1RjVGNEY0RjlGNUY1KiZJJHRBMkdGKEY1RjFGNUY1KjIjRjVGOUY1SSRBQTFHRihGNSwsKiZJIk5HRihGNSlGM0Y5RjVGNCokKUZDRjlGNUY1KihGOUY1RkNGNUYzRjVGNCokRkRGNUY1RkNGNEY1RjtGNEYzRjRGQ0Y0LCZGNUY0RkNGNUY0KUkkRHB5R0YoRj9GNUY0SSR0QTBHRihGNTcjRi4=

Let us check n:=6: normal(series(subs(tA2=A2t, AA1=A1,Dpy=subs(x=y,Deltapx),N=Nser(n+2), U(0,0)=subs(x=0,y=0,User(n+2)), DD(0)=subs(y=0,Dser((n+2))), Jyform),t,n));

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

normal(series(subs(DD(y)=Dser(n+2),DD(0)=subs(y=0,Dser((n+2))),Jy),t,n));

KzNJInRHNiIqJCksJkkieUdGJCIiIkYpRikiIiMhIiIhIiMsJCoqRipGKSwmKiQpRigiIiRGKUYpRilGK0YpRiZGK0YoRitGKUYrKigsLCokKUYoIiInRilGKSomIiIpRilGMUYpRisqJiIjNUYpKUYoRipGKUYrKiYiIiVGKUYoRilGK0YpRilGKUY8RitGJkYrIiIhLCQqKkYqRilGKEYrLCoqJkYyRilGMUYpRikqJkY+RilGPEYpRilGPUYpRilGK0YpRidGK0YrRiksJComLCoqJkY+RilGMUYpRikqJiIjN0YpRjxGKUYpKiZGMkYpRihGKUYpIiM8RilGKUYnRitGK0YqLCQqKkYqRilGKEYrLC4qJkYqRikpRigiIiZGKUYpKiZGN0YpKUYoRj5GKUYpKiYiI0BGKUYxRilGKSomIiM5RilGPEYpRikqJiIjTUYpRihGKUYpRlJGK0YpRidGK0YrRjIsJCoqRipGKUYoRissMComRipGKUY2RilGKSomRjtGKUZRRilGKSomIiNLRilGVEYpRikqJiIjISlGKUYxRilGKSomIiNCRilGPEYpRikqJiIkTCJGKUYoRilGKUY5RitGKUYnRitGK0Y+LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRilGUg==

factor(series(%%-%,t,n))

Ky9JInRHNiIiIiMhIiIsJkkkdEEwR0YkIiIiRilGKSIiIUYlRilGJkYlIiIpIiIkLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRikiIiU=

normal(subs(tA0=tA0ser, convert(%,polynom)));

IiIh

Jyform

LCoqJCksKCokSSJ5RzYiISIiIiIiRidGKSooLCZGKkYqKiYiIiNGKkkidEdGKEYqRipGKkYvRiksJkYnRipGKkYqRilGKUYuRipGKiomSSR0QTJHRihGKkYlRipGKiomSSRBQTFHRihGKiwoKjAjRipGLkYqLCwqJkkiTkdGKEYqKUYnRi5GKkYpKiQpRjpGLkYqRioqKEYuRipGOkYqRidGKkYpKiRGO0YqRipGOkYpRipGMEYpRidGKUY6RiksJkYqRilGOkYqRikpSSREcHlHRihGN0YqRioqLEY3RiosKCokKUY6IiIlRipGKiomRi5GKkY9RipGKkYqRipGKkYwRikpRkBGLkYpRjpGKUYqKipGN0YqRixGKkYvRilGMEYpRilGKkYpSSR0QTBHRihGKg==

Ixform

LCoqJkkkQUExRzYiIiIiLCgqMCNGJiIiI0YmLCwqJkkiTkdGJUYmKUkieEdGJUYqRiYhIiIqJClGLUYqRiZGJiooRipGJkYtRiZGL0YmRjAqJEYuRiZGJkYtRjBGJiwmRi9GJkYmRiZGMEYvRjBGLUYwLCZGJkYwRi1GJkYwKUkkRHB4R0YlRilGJkYmKixGKUYmLCgqJClGLSIiJUYmRiYqJkYqRiZGMkYmRiZGJkYmRiZGNUYwKUY2RipGMEYtRjBGJioqRilGJiwmRiZGJiomRipGJkkidEdGJUYmRiZGJkZDRjBGNUYwRjBGJkYmKiQpLCgqJEYvRjBGJkYvRjAqKEZBRiZGQ0YwRjVGMEYwRipGJkYmKiZGRkYmSSR0QTJHRiVGJkYmSSR0QTBHRiVGJg==

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

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

series(ratio,x,1);

KydJInhHNiIsJCooSSJ5R0YkIiIiSSJ0R0YkISIiLCZGJ0YoRihGKEYqRipGKi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIh

series(ratio,y,1);

KydJInlHNiIqKEkieEdGJCIiIkkidEdGJCEiIiwmRiZGJ0YnRidGKUYpLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiciIiE=

series(Ix,x,1);

KylJInhHNiIiIiIhIiMsKiooIiIlRiVJInRHRiRGJS1JIlVHRiQ2JCIiIUYuRiVGJSIiI0YlKihGL0YlLCZGJUYlKiZGL0YlRipGJUYlRiVGKiEiIkYzKihGL0YlRipGJS1JI0RER0YkNiNGLkYlRiVGMy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlRi4=

I1 has a pole at -1 only I1x;series(%,x,1)

LCYqKEkidEc2IiIiIiwmSSJ4R0YlRiZGJkYmRiYtSSJRR0YlNiRGKCIiIUYmRiYqKCwoKiZGJEYmKUYoIiIjRiYhIiJGJEYyRihGJkYmRiRGMkYnRjJGMg==

KydJInhHNiIsJiomSSJ0R0YkIiIiLUkiUUdGJDYkIiIhRixGKEYoRihGKEYsLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGKA==

What do we need to subtract to Ix to hope to have a constant ? We first get rid of the pole at 0 series(Ix*(I1x-subs(x=0,I1x))^2,x,1);

KyVJInhHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiE=

What about the pole at x=-1 ? Now we have a pole of order 4. factor(series(Ix*(I1x-subs(x=0,I1x))^2,x=-1,2));

KyksJkkieEc2IiIiIkYmRiYqJiksJkYmRiYqJiIiI0YmSSJ0R0YlRiZGJiIiJUYmKUYsRi0hIiIhIiUsJCoqRitGJilGKSIiJEYmLCoqJilGLEYrRiYtSSJRR0YlNiQiIiFGO0YmRiYqJkY3RiYtSSNEREdGJTYjRjtGJkYmRipGJkYmRiZGJkYuRi9GLyEiJC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YmISIj

series(I1x,x=-1,1);

KyksJkkieEc2IiIiIkYmRiYsJComLCYqJiIiI0YmSSJ0R0YlRiYhIiJGJkYtRiZGLEYtRi1GLSwkKiYsJkYmRiZGKkYmRiZGLEYtRi0iIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJkYm

Anew4:=factor(op(1,%%)/op(1,%)^4)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZBbmV3NEdGKCIiIjcjRi4=

factor(series(Ix*(I1x-subs(x=0,I1x))^2-I1x^4,x=-1,2));

KycsJkkieEc2IiIiIkYmRiYsJCoqIiIjRiYpLCZGJkYmKiZGKUYmSSJ0R0YlRiZGJiIiJEYmLCoqJilGLUYpRiYtSSJRR0YlNiQiIiFGNUYmRiYqJkYxRiYtSSNEREdGJTYjRjVGJkYmRiwhIiJGJkY6RiYpRi0iIiVGOkY6ISIkLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiYhIiM=

series(I1x,x=-1,1); Anew3:=factor(op(1,%%)/op(1,%)^3)

KyksJkkieEc2IiIiIkYmRiYsJComLCYqJiIiI0YmSSJ0R0YlRiYhIiJGJkYtRiZGLEYtRi1GLSwkKiYsJkYmRiZGKkYmRiZGLEYtRi0iIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJkYm

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZBbmV3M0dGKCwkKigiIiMiIiIsKiomKUkidEdGKEYwRjEtSSJRR0YoNiQiIiFGOUYxRjEqJkY0RjEtSSNEREdGKDYjRjlGMUYxKiZGMEYxRjVGMSEiIkYxRj9GMUY1Rj9GPzcjRi4=

factor(series(Ix*(I1x-subs(x=0,I1x))^2-I1x^4-Anew3*I1x^3,x=-1,3));

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

series(I1x,x=-1,1); Anew2:=factor(op(1,%%)/op(1,%)^2)

KyksJkkieEc2IiIiIkYmRiYsJComLCYqJiIiI0YmSSJ0R0YlRiYhIiJGJkYtRiZGLEYtRi1GLSwkKiYsJkYmRiZGKkYmRiZGLEYtRi0iIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJkYm

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZBbmV3MkdGKCwkKiYsPComKS1JIlFHRig2JCIiIUY2IiIjIiIiKUkidEdGKCIiJUY4ISIiKipGO0Y4RjNGOC1JI0RER0YoNiNGNkY4RjlGOEY8KiYpRj5GN0Y4RjlGOEY8KigiIilGOC1JIlVHRig2JEY8RjZGOClGOiIiJEY4RjgqKCIjNUY4RjNGOEZIRjhGOCooRjdGOEY+RjhGSEY4RjgqKEY7RjgtRjRGR0Y4RkhGOEY4KihGO0Y4RkVGOClGOkY3RjhGOCooRjtGOEZQRjhGM0Y4RjgqKEY3RjhGUEY4Rj5GOEY4KihGN0Y4Rk5GOEZQRjhGOComIiInRjhGUEY4RjhGOEY8RjhGUEY8Rjw3I0Yu

factor(series(Ix*(I1x-subs(x=0,I1x))^2-I1x^4-Anew3*I1x^3-Anew2*I1x^2,x=-1,3));

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

series(I1x,x=-1,1); Anew1:=factor(op(1,%%)/op(1,%))

KyksJkkieEc2IiIiIkYmRiYsJComLCYqJiIiI0YmSSJ0R0YlRiYhIiJGJkYtRiZGLEYtRi1GLSwkKiYsJkYmRiZGKkYmRiZGLEYtRi0iIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJkYm

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

factor(series(Ix*(I1x-subs(x=0,I1x))^2-I1x^4-Anew3*I1x^3-Anew2*I1x^2-Anew1*I1x,x=-1,3));

KyUsJkkieEc2IiIiIkYmRiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJiIiIQ==

Could the above series be a constant? exprx:=Ix*(I1x-subs(x=0,I1x))^2-I1x^4-Anew3*I1x^3-Anew2*I1x^2-Anew1*I1x: n:=20: serx:=factor(series(subs(U(x,0)=subs(y=0,User(n)),Q(x,0)=subs(y=0,Qser(n)), Q(0,0)=subs(x=0,y=0,Qser(n)),U(-1,0)=subs(x=-1,y=0,User(n)),Q(-1,0)=subs(x=-1,y=0,Qser(n)), (D[1](U))(-1, 0)= subs(x=-1,diff(subs(y=0,User(n)),x)), (D[1](Q))(-1, 0)= subs(x=-1,diff(subs(y=0,Qser(n)),x)), DD(0)=subs(y=0,Dser(n)), exprx),t,n))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVzZXJ4R0YoK0VJInRHRigiIiIiIiEhIiNGMCEiJiIiIyEjNSIiJCEjViIiJSEkYyIiIiYhJC8oIiInISUjPiQiIighJm9gIiIiKSEmL2UoIiIqIScpMyRRIiM1ISg7Myg+IiM2ISl3R0c1IiM3ISk/ISlIYSIjOCEqTy9dKkciIzkhK1NOZWM6IiM6LUkiT0dGJTYjRjAiIzs3I0Yu

What happened to J(y) if we modify it as we did for I(x)? Let us simply check the expansion subs(x=0,I1x);

LCYqJkkidEc2IiIiIi1JIlFHRiU2JCIiIUYqRiZGJkYmRiY=

expry:=Jy*(J1y-subs(x=0,I1x))^2-J1y^4-Anew3*J1y^3-Anew2*J1y^2-Anew1*J1y: n:=20: sery:=factor(series( subs(Q(0,y)=subs(x=0,Qser(n)), Q(0,0)=subs(x=0,y=0,Qser(n)),U(0,0)=subs(x=0,y=0,User(n)), U(-1,0)=subs(x=-1,y=0,User(n)),Q(-1,0)=subs(x=-1,y=0,Qser(n)), (D[1](U))(-1, 0)= subs(x=-1,diff(subs(y=0,User(n)),x)), (D[1](Q))(-1, 0)= subs(x=-1,diff(subs(y=0,Qser(n)),x)), DD(0)=subs(y=0,Dser(n)), DD(y)=Dser(n), expry),t,n))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVzZXJ5R0YoK0lJInRHRigiIiIiIiEhIiNGMCEiJiIiIyEjNSIiJCEjViIiJSEkYyIiIiYhJC8oIiInISUjPiQiIighJm9gIiIiKSEmL2UoIiIqIScpMyRRIiM1ISg7Myg+IiM2ISl3R0c1IiM3ISk/ISlIYSIjOCEqTy9dKkciIzkhK1NOZWM6IiM6ISs/RjtIJSkiIzshLHc4QEtmJSIjPC1JIk9HRiU2I0YwIiM9NyNGLg==

series(serx-sery,t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzs=

We first relate I2(x) and I0(x) Alg:=I2x;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRBbGdHRigsJCowIyIiIiIiI0YxLCwqJkkiTkdGKEYxKUkieEdGKEYyRjEhIiIqJClGNUYyRjFGMSooRjJGMUY1RjFGN0YxRjgqJEY2RjFGMUY1RjhGMSwmRjdGMUYxRjFGOEY3RjhGNUY4LCZGMUY4RjVGMUY4KUkkRHB4R0YoRjBGMUYxNyNGLg==

Deltapx;

LCgiIiJGIyosIiIjRiNJIk5HNiJGIywmKiQpRiZGJUYjRiNGI0YjRiMpLCZGI0YjRiYhIiJGJUYtSSJ4R0YnRiNGLSomKUYuRiVGI0YqRiNGIw==

Algsq:=factor(subs(Dpx=Deltapx,Alg^2));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZBbGdzcUdGKCwkKjAjIiIiIiIlRjEpLCwqJkkiTkdGKEYxKUkieEdGKCIiI0YxISIiKiQpRjZGOUYxRjEqKEY5RjFGNkYxRjhGMUY6KiRGN0YxRjFGNkY6RjlGMSwyKiYpRjZGMkYxRjdGMUYxKihGOUYxKUY2IiIkRjFGN0YxRjoqKEY5RjFGQ0YxRjhGMUY6KiZGPEYxRjdGMUYxRjtGMUY9RjoqJkY5RjFGNkYxRjpGMUYxRjEpLCZGOEYxRjFGMUY5RjpGN0Y6RjxGOiksJkYxRjpGNkYxRjJGOkYxNyNGLg==

res:=convert(%,parfrac,x);

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

We look at the pole at x=0 I0x

LCgqJEkieEc2IiEiIiIiIkYkRiYqKCwmRidGJyomIiIjRidJInRHRiVGJ0YnRidGLEYmLCZGJEYnRidGJ0YmRiY=

factor(subs(t=tN,Algsq-I0x^2/4));

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

factor(series(%,x,1));

KydJInhHNiIsJCoqIyIiIiIiI0YoLCoqJClJIk5HRiQiIiRGKEYoKiQpRi1GKUYoISIiRi1GMUYoRjFGKEYtRjEpLCZGKEYxRi1GKEYpRjFGMUYxLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIiE=

c1N:=coeff(%,x,-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRjMU5HRigsJCoqIyIiIiIiI0YxLCoqJClJIk5HRigiIiRGMUYxKiQpRjZGMkYxISIiRjZGOkYxRjpGMUY2RjopLCZGMUY6RjZGMUYyRjpGOjcjRi4=

The minimal polynomial of c1 has degree 4 factor(resultant(numer(c1N-c),algN,N));

LDwqKCIkYyMiIiIpSSJjRzYiIiIlRiUpSSJ0R0YoIiIkRiVGJSooIiQ3JkYlKUYnRixGJUYqRiUhIiIqKEYkRiVGL0YlKUYrIiIjRiVGMCooIiQjPkYlKUYnRjNGJUYqRiVGMCooIiRbJUYlRidGJUYqRiVGJSooIiMhKUYlRjZGJUYrRiVGJSooIiQhW0YlRidGJUYyRiVGJSomIiQvJEYlRipGJUYlKigiJDciRiVGJ0YlRitGJUYlKiYiJEMjRiVGMkYlRiUqJiIiKUYlRidGJUYwKiYiI0tGJUYrRiVGJUYpRjA=

factor(subs(t=tN,Algsq-I0x^2/4-I0x*c1N));

LCQqLCMiIiIiIiVGJSwmKiQpSSJORzYiIiIjRiVGJUYlRiVGJSwsKiZGJkYlKUYqRiZGJUYlKiYiIipGJSlGKiIiJEYlISIiKiYiIzhGJUYpRiVGJUYqRjRGJUYlRiUpLCZGJUY0RipGJUYzRjRGKUY0RjQ=

c0N:=%;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRjME5HRigsJCosIyIiIiIiJUYxLCYqJClJIk5HRigiIiNGMUYxRjFGMUYxLCwqJkYyRjEpRjZGMkYxRjEqJiIiKkYxKUY2IiIkRjEhIiIqJiIjOEYxRjVGMUYxRjZGP0YxRjFGMSksJkYxRj9GNkYxRj5GP0Y1Rj9GPzcjRi4=

Minimal polynomial of c0 factor(resultant(numer(%-c),algN,N));

LEwqKCImJVE7IiIiKUkiY0c2IiIiJUYlKUkidEdGKCIiJ0YlRiUqKEYkRiUpRiciIiRGJUYqRiUhIiIqKEYkRiVGLkYlKUYrIiImRiVGMCooIiZ3WCNGJSlGJyIiI0YlRipGJUYlKigiJSc0JUYlRi5GJSlGK0YpRiVGMCooIiZfIlxGJUY2RiVGMkYlRjAqKCImY10lRiVGJ0YlRipGJUYlKigiJjtbJEYlRjZGJUY6RiVGMCooIiZHUChGJUYnRiVGMkYlRiUqJiInL1I3RiVGKkYlRiUqKCIlWz9GJUY2RiUpRitGL0YlRjAqKCImKzcmRiVGJ0YlRjpGJUYlKiYiJydwZCJGJUYyRiVGJSooIiZHRCNGJUYnRiVGR0YlRiUqJiInR0A2RiVGOkYlRiUqKCIlN3BGJUYnRiUpRitGN0YlRiUqJiImXz4nRiVGR0YlRiUqKCIlQzVGJUYnRiVGK0YlRiUqJiImITNBRiVGUkYlRiUqJiIlS2NGJUYrRiVGJUZWRiU=

The following two expressions just differ by the sign before AA1. Moreover, recall that J0=I0 Ixform;Jyform

LCoqMiMiIiIiIiNGJUkkQUExRzYiRiUsLComSSJOR0YoRiUpSSJ4R0YoRiZGJSEiIiokKUYrRiZGJUYlKihGJkYlRitGJUYtRiVGLiokRixGJUYlRitGLkYlLCZGLUYlRiVGJUYuRi1GLkYrRi4sJkYlRi5GK0YlRi4pSSREcHhHRihGJEYlRiUqJCksKCokRi1GLkYlRi1GLiooLCZGJUYlKiZGJkYlSSJ0R0YoRiVGJUYlRj5GLkYzRi5GLkYmRiVGJSomRjlGJUkkdEEyR0YoRiVGJUkkdEEwR0YoRiU=

LCoqJCksKCokSSJ5RzYiISIiIiIiRidGKSooLCZGKkYqKiYiIiNGKkkidEdGKEYqRipGKkYvRiksJkYnRipGKkYqRilGKUYuRipGKiomSSR0QTJHRihGKkYlRipGKioyI0YqRi5GKkkkQUExR0YoRiosLComSSJOR0YoRiopRidGLkYqRikqJClGOEYuRipGKiooRi5GKkY4RipGJ0YqRikqJEY5RipGKkY4RilGKkYwRilGJ0YpRjhGKSwmRipGKUY4RipGKSlJJERweUdGKEY0RipGKUkkdEEwR0YoRio=

Ixff:=I0^2+tA2*I0+AA1*I2+tA0;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVJeGZmR0YoLCoqJkkkQUExR0YoIiIiSSNJMkdGKEYxRjEqJClJI0kwR0YoIiIjRjFGMSomRjVGMUkkdEEyR0YoRjFGMUkkdEEwR0YoRjE3I0Yu

Jxff:=I0^2+tA2*I0-AA1*I2+tA0;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVKeGZmR0YoLCoqJkkkQUExR0YoIiIiSSNJMkdGKEYxISIiKiQpSSNJMEdGKCIiI0YxRjEqJkY2RjFJJHRBMkdGKEYxRjFJJHRBMEdGKEYxNyNGLg==

expand(Ixff*Jxff);

LDAqJilJJEFBMUc2IiIiIyIiIilJI0kyR0YmRidGKCEiIiokKUkjSTBHRiYiIiVGKEYoKihGJ0YoKUYuIiIkRihJJHRBMkdGJkYoRigqJilGLkYnRigpRjNGJ0YoRigqKEYnRihGNUYoSSR0QTBHRiZGKEYoKipGJ0YoRi5GKEY4RihGM0YoRigqJClGOEYnRihGKA==

P4:=subs(I2^2=I0^2/4+c1*I0+c0,I0=u,%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNQNEdGKCwwKiQpSSJ1R0YoIiIlIiIiRjMqKCIiI0YzKUYxIiIkRjNJJHRBMkdGKEYzRjMqJilGMUY1RjMpRjhGNUYzRjMqKEY1RjNGOkYzSSR0QTBHRihGM0YzKipGNUYzRjFGM0Y9RjNGOEYzRjMqJClGPUY1RjNGMyomKUkkQUExR0YoRjVGMywoKiYjRjNGMkYzRjpGM0YzKiZJI2MxR0YoRjNGMUYzRjNJI2MwR0YoRjNGMyEiIjcjRi4=

This polynomial of degree 4 in u vanishes for u=I0(X), where X is one of the two roots of Delta(x), denoted X+ and X_, that are finite at x=0 map(allvalues,algeqtoseries(numer(Delta),t,y,10, true));

NyQsOEkidEc2IiIiIiomIiIjRiYpRiQjIiIkRihGJkYmKiZGKEYmKUYkRihGJkYmKiZGK0YmKUYkIyIiJkYoRiZGJiomIiIqRiYpRiRGK0YmRiYqJiMiI3oiIiVGJilGJCMiIihGKEYmRiYqJiIjTUYmKUYkRjhGJkYmKiYjIiRaJiIiKUYmKUYkI0YzRihGJkYmKiYiJGkiRiYpRiRGMUYmRiYqJiMiJlZHIyIja0YmKUYkIyIjNkYoRiZGJi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyokKUYkIiInRiZGJiw4RiRGJkYnISIiRixGJkYuRllGMkYmRjVGWUY8RiZGP0ZZRkVGJkZIRllGT0Ym

minimal polynomial of X+ and X- (related to Delta+) numer(factor(subs(t=tN,y=x,Delta)));

KiYsMiomKUkiTkc2IiIiJSIiIilJInhHRiciIiNGKUYpKihGLEYpKUYmIiIkRilGKkYpISIiKihGLEYpRi5GKUYrRilGMComKUYmRixGKUYqRilGKSokRjNGKUYpKihGLEYpRiZGKUYrRilGMComRixGKUYmRilGMEYpRilGKSwyKiRGJUYpRilGMUYwRjJGKSomRixGKUYuRilGMCooRixGKUYmRilGKkYpRjBGNEYpRjVGMCokRipGKUYpRik=

Choose the correct factor subs(N=0,op(1,%));

IiIi

algXpm:=collect(op(2,%%),x,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdYcG1HRigsKComKUkieEdGKCIiIyIiIiksJkYzISIiSSJOR0YoRjNGMkYzRjMqKkYyRjNGN0YzLCYqJClGN0YyRjNGM0YzRjNGM0YxRjNGNiomRjtGM0Y0RjNGMzcjRi4=

What are the corresponding values of I0 ? factor(resultant(algXpm, numer(subs(t=tN,I0-I0x )),x));

KiwpLCwqJClJIk5HNiIiIiUiIiJGKiomIiIjRiopRiciIiRGKiEiIiomIiInRiopRidGLEYqRioqJkYsRipGJ0YqRi9GKkYqRixGKkYtRiopLCYqJEYyRipGKkYqRipGLEYqKSwmRipGL0YnRioiIiZGKiw+KiYpSSNJMEdGKEYsRipGJkYqRioqKEYsRipGPUYqKUYnRjlGKkYqKiQpRidGMUYqRioqKEYuRipGPEYqRi1GKkYvKiRGP0YqRi8qKEYuRipGPEYqRjJGKkYqKiYiIzlGKkYmRipGKiomRjxGKkYnRipGLyooRilGKkY9RipGMkYqRi8qJiIjNUYqRi1GKkYvKihGLEYqRj1GKkYnRipGKiomIiM8RipGMkYqRioqJiIiKkYqRidGKkYvRilGKkYq

algI0Xpm:=collect(-op(5,%),I0,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlhbGdJMFhwbUdGKCwoKihJIk5HRigiIiIpLCZGMSEiIkYwRjEiIiRGMSlJI0kwR0YoIiIjRjFGNCosRjhGMUYwRjFGM0YxLCoqJClGMEY1RjFGMSokKUYwRjhGMUYxRjBGMUYxRjRGMUY3RjFGNComLCZGPUYxRjFGMUYxLCwqJClGMCIiJUYxRjFGO0Y0KiYiIzhGMUY+RjFGMSomIiIqRjFGMEYxRjRGREYxRjFGNDcjRi4=

P2:=subs(I0=u,algI0Xpm)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNQMkdGKCwoKihJIk5HRigiIiIpLCZGMSEiIkYwRjEiIiRGMSlJInVHRigiIiNGMUY0KixGOEYxRjBGMUYzRjEsKiokKUYwRjVGMUYxKiQpRjBGOEYxRjFGMEYxRjFGNEYxRjdGMUY0KiYsJkY9RjFGMUYxRjEsLCokKUYwIiIlRjFGMUY7RjQqJiIjOEYxRj5GMUYxKiYiIipGMUYwRjFGNEZERjFGMUY0NyNGLg==

Series expansions of the roots I0(X+) and I0(X-) n:=6: convert(subs(N=Nser(n), algI0Xpm),polynom): map(allvalues,algeqtoseries(%,t,I0,3));

NyQsKiomIiIjIiIiKUkidEc2IiNGJkYlISIiRiZGJkYmKiRGJ0YmRiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YpNiNGKEYmLCpGJEYrRiZGJkYsRitGLUYm

A root of I(x) algX:=(op(1,Ix));algX:=numer(%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdYR0YoLC4qKiIiIyIiIkkidEdGKEYxLCZJInhHRihGMUYxRjFGMS1JIlVHRig2JEY0IiIhRjFGMUYxRjFGNEYxKiRGNCEiIkYxKigsJkYxRjEqJkYwRjFGMkYxRjFGMUYyRjpGM0Y6RjoqJkYyRjEtSSNEREdGKDYjRjhGMUYxNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdYR0YoLDQqKiIiIyIiIi1JIlVHRig2JEkieEdGKCIiIUYxKUkidEdGKEYwRjEpRjUiIiRGMUYxKioiIiVGMUYyRjFGN0YxKUY1RjBGMUYxKihGN0YxLUkjRERHRig2I0Y2RjFGPUYxRjEqKkYwRjFGN0YxRjJGMUY1RjFGMSooRjdGMUY/RjFGNUYxRjEqJkY4RjFGOUYxRjEqKEYwRjFGOEYxRj1GMUYxRjhGMUY1ISIiNyNGLg==

eqpap:=-X+t*(1+X)/(1+2*t)*(2*t*(1+X)*X*U(X,0)+1+X+X^2+t*X*DD(0))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZlcXBhcEdGKCwmSSJYR0YoISIiKipJInRHRigiIiIsJkYzRjNGL0YzRjMsJkYzRjMqJiIiI0YzRjJGM0YzRjAsLCosRjdGM0YyRjNGNEYzRi9GMy1JIlVHRig2JEYvIiIhRjNGM0YzRjNGL0YzKiQpRi9GN0YzRjMqKEYyRjNGL0YzLUkjRERHRig2I0Y9RjNGM0YzRjM3I0Yu

factor(subs(X=x,eqpap)/algX)

KiQsJiIiIkYkKiYiIiNGJEkidEc2IkYkRiQhIiI=

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

KzdJInRHNiIiIiJGJSIiJEYmIiIlRiciI0UiIiYiIyMpIiInIiQqUiIiKCIlJXAiIiIpIiU9IikiIioiJnElUSIjNS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiM2

The value of I0(X) series(subs(x=Xser(n),I0x),t,5)

KytJInRHNiIhIiIiIiEhIiQiIiIhIiMiIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIiJA==

Double root? Yes, since this derivative is non-zero numer(diff(I0x,x));

LCwqJkkidEc2IiIiIilJInhHRiUiIiVGJiEiIiooIiIjRiZGJEYmKUYoIiIkRiZGKiooRixGJkYkRiZGKEYmRioqJClGKEYsRiZGJkYkRio=

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

KzVJInRHNiIhIiIiIiJGJSIiIyEiIyIiJUYlIiImISIqIiInISM3IiIoISNhIiIpISMlKiIiKi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YmIiM1

We have thus found an identity between two polynomials in u diff1:=collect(P4-(u-I0X)^2*P2/(1-N)^3/N,u,normal);

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

coeff(diff1,u,4);

IiIh

I0Xsol:=collect(solve(coeff(diff1,u,3),I0X),[tA2],factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdJMFhzb2xHRigsJkkkdEEyR0YoISIiKiYsKiokKUkiTkdGKCIiJCIiIkY3KiQpRjUiIiNGN0Y3RjVGN0Y3RjBGNyksJkY3RjBGNUY3RjpGMEY3NyNGLg==

Let us check this n:=7: normal(series(subs(x=Xser(n+2),I0x),t,n));

Ky9JInRHNiIhIiIiIiEhIiQiIiIhIiMiIiMhIzoiIiQhI1MiIiUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIiJg==

A2t;

LCQqKCIiIyIiIkkidEc2IkYlLCYtSSNEREdGJzYjIiIhRiUtSSJVR0YnNiRGLEYsRiVGJUYl

n:=8: normal(series(subs(tA2=subs(x=0,y=0,User(n)+Dser(n))*2*t,N=Nser(n), I0Xsol),t,n));

KzVJInRHNiIhIiIiIiEhIiQiIiIhIiMiIiMhIzoiIiQhI1MiIiUhJC8jIiImISRHKSIiJyElUlMiIigtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIiKQ==

We go on, after replacing I0X by its expression diff2:=map(factor,subs(I0X=I0Xsol,diff1));

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

tA0sol:=collect(factor(solve(coeff(diff2,u,2),tA0)),[tA2,AA1],factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSd0QTBzb2xHRigsKCooLCoqJClJIk5HRigiIiQiIiJGNSokKUYzIiIjRjVGNUYzRjVGNSEiIkY1KSwmRjVGOUYzRjVGOEY5SSR0QTJHRihGNUY1KiYjRjUiIilGNSlJJEFBMUdGKEY4RjVGNSooLDIqJClGMyIiKEY1RjUqJiIiJUY1KUYzIiInRjVGNSomRjRGNSlGMyIiJkY1RjkqJiIjN0Y1KUYzRkhGNUY1KiYiIzpGNUYyRjVGOSomIiM1RjVGN0Y1RjUqJkZNRjVGM0Y1RjlGOEY1RjVGM0Y5KUY7RkhGOUY5NyNGLg==

diff3:=map(factor,subs(tA0=tA0sol,c1=c1N,diff2));

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

Depends on A1^2, not A1 tA2sol:=collect(factor(solve(coeff(diff3,u,1),tA2)),AA1,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSd0QTJzb2xHRigsJCoqIiIjIiIiLCYqKCwqKiQpSSJOR0YoIiIkRjFGMSokKUY3RjBGMSEiIkY3RjtGMUY7RjEpLCZGMUY7RjdGMSIiJUYxKUkkQUExR0YoRjBGMUYxKigiIztGMSwqRjVGMUY5RjFGN0YxRjFGO0YxKSwoRjlGMUY3RjtGMUYxRjhGMUY7RjEpRj1GMEY7LCYqKEY/RjFGN0YxRjxGMUYxKiZGQkYxRkRGMUYxRjtGOzcjRi4=

We should check that the denominator does not vanishes A1;

LCYqKCIiJSIiIkkidEc2IkYlLUkiVUdGJzYkIiIhRitGJUYlRiRGJQ==

n:=6: normal(series(subs(AA1=4+4*t*subs(x=0,y=0,User(n)),N=Nser(n),denom(tA2sol)),t,n));

KzFJInRHNiIiIzsiIiEhI2siIiIiJDciIiIjISQrJSIiJCIkVyYiIiUhJSU9JCIiJi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIn

Constant coefficient diff4:=numer(map(factor,subs(tA2=tA2sol,c0=c0N,diff3)));nops(%);

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

IiIj

subs(N=0,op(1,diff4));subs(N=0,op(2,diff4));

LCYqJiIjayIiIkkkQUExRzYiRiVGJSIkYyNGJQ==

LCYqJiIjayIiIkkkQUExRzYiRiUhIiIiJGMjRiU=

A1;

LCYqKCIiJSIiIkkidEc2IkYlLUkiVUdGJzYkIiIhRitGJUYlRiRGJQ==

CHOOSE THE CORRECT FACTOR algA1N:=collect(numer(op(2,diff4)),AA1,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdBMU5HRigsLCooKUkiTkdGKCIiIyIiIiksJkYzISIiRjFGMyIiKUYzKUkkQUExR0YoIiIlRjNGMyosRjpGM0YxRjMpLCZGMUYzRjNGMyIiJEYzKUY1IiIoRjMpRjlGPkYzRjYqLCIjS0YzRjFGMyksKCokRjBGM0YzRjFGNkYzRjNGPkYzKUY1RjpGMylGOUYyRjNGMyosIiNrRjMpRjVGPkYzRjxGM0ZERjNGOUYzRjMqJiIkYyNGMylGRSIiJ0YzRjM3I0Yu

Let us check A1;

LCYqKCIiJSIiIkkidEc2IkYlLUkiVUdGJzYkIiIhRitGJUYlRiRGJQ==

n:=10: series(4*(1+t*subs(x=0,y=0,User(n))),t,n)

KzdJInRHNiIiIiUiIiFGJSIiI0YlIiIkIiNTRiUiJDsiIiImIiRrJyIiJyIlN0ciIigiJilbOSIiKSImPywoIiIqLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiM1

series(subs(AA1=4*(1+t*subs(x=0,y=0,User(n))),N=Nser(n),algA1N),t,n+1);

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

with(algcurves): genus(algA1N,N,AA1);

IiIm

A1 has degree 16 over Q(t), but in fact A1^2 has degree 8. factor(resultant(algA1N,algN,N)): nops(%);degree(%%,AA1); algA1:=collect(%%%,AA1,factor);

IiN1

IiM7

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

subs(A1=-A1,algA1)/algA1

IiIi

Let us check that the description of A1 given in the paper, at the beginning of Section 6.1, is correct. We first introduce M algM:=-M+t*(1+2*M+4*M^2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdNR0YoLCZJIk1HRighIiIqJkkidEdGKCIiIiwoKiYiIiVGMylGLyIiI0YzRjMqJkY4RjNGL0YzRjNGM0YzRjNGMzcjRi4=

Mt:=op(2,[solve(algM,M)])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNNdEdGKCwkKigjIiIiIiIpRjEsKComIiIjRjFJInRHRihGMUYxRjEhIiIqJCksKComIiM3RjEpRjZGNUYxRjcqJiIiJUYxRjZGMUY3RjFGMSNGMUY1RjFGMUYxRjZGN0Y3NyNGLg==

tM:=solve(algM,t)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN0TUdGKComSSJNR0YoIiIiLCgqJiIiJUYwKUYvIiIjRjBGMComRjVGMEYvRjBGMEYwRjAhIiI3I0Yu

Mser:=proc(n) series(Mt,t,n+1): end: Mser(5)

Ky1JInRHNiIiIiJGJSIiI0YmIiIpIiIkIiNLIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIiY=

algP1M:=-P1*(1 - P1)^2*(4*M + 1)^3+M*(M+1)^3*(1+P1)^4;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdQMU1HRigsJiooSSNQMUdGKCIiIiksJkYxRjFGMCEiIiIiI0YxKSwmKiYiIiVGMUkiTUdGKEYxRjFGMUYxIiIkRjFGNCooRjpGMSksJkY6RjFGMUYxRjtGMSksJkYxRjFGMEYxRjlGMUYxNyNGLg==

map(allvalues,algeqtoseries(convert(subs(M=Mser(n),algP1M),polynom),t,P1,3))

NyYrK0kidEc2IiIiIiEiIkYmIiIhISIkRiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJiIiIywqRiZGJiomIiIlRiYpRiQjRiZGMEYmRiYqJiIiKUYmRiRGJkYmLUYrNiMqJClGJCMiIiRGMEYmRiYsKkYmRiZGMkYnRjZGJkY4RiYrK0YkRiZGJkYnRjBGM0Y9RipGMw==

P1ser:=proc(n) op(4, map(allvalues,algeqtoseries(convert(subs(M=Mser(n+6),algP1M),polynom),t,P1,n))): end: P1ser(5)

Ky9JInRHNiIiIiJGJSEiIiIiIyIiJSIiJCEiI0YoIiNJIiImLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIic=

The definition of A1 in the paper A1P1:=4*(M + 1)*sqrt(P1*(M + 1)/M)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVBMVAxR0YoLCQqKCIiJSIiIiwmSSJNR0YoRjFGMUYxRjEpKihJI1AxR0YoRjFGMkYxRjMhIiIjRjEiIiNGMUYxNyNGLg==

First terms A1ser:=proc(n) series(subs(M=Mser(n),P1=P1ser(n),A1P1),t,n) : end: Let us compare A1ser(10); series(subs(U(0,0)=subs(x=0,y=0,User(n)),A1),t,n)

KzdJInRHNiIiIiUiIiFGJSIiI0YlIiIkIiNTRiUiJDsiIiImIiRrJyIiJyIlN0ciIigiJilbOSIiKSImPywoIiIqLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiM1

KzdJInRHNiIiIiUiIiFGJSIiI0YlIiIkIiNTRiUiJDsiIiImIiRrJyIiJyIlN0ciIigiJilbOSIiKSImPywoIiIqLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiM1

Equation of A1 (as defined in the paper) over Q(M) algA1Mpap:=collect(op(3,factor(resultant(numer(A1P1^2-AA1^2),algP1M,P1))),AA1,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSphbGdBMU1wYXBHRigsLComKUkkQUExR0YoIiIpIiIiKUkiTUdGKCIiJUYzRjMqKiIjO0YzKUY1IiIjRjMsLComRjZGM0Y0RjNGMyomIiNfRjMpRjUiIiRGMyEiIiomIiNPRjNGOUYzRkEqJkYyRjNGNUYzRkFGM0ZBRjMpRjEiIidGM0YzKiwiJDcmRjNGNUYzLCwqJkZARjNGNEYzRjMqJiIjdEYzRj9GM0YzKiYiI2RGM0Y5RjNGMyomIiM6RjNGNUYzRjNGM0YzRjMpLCZGNUYzRjNGM0ZARjMpRjFGNkYzRjMqKiIlJzQlRjNGO0YzKUZSRkZGMylGMUY6RjNGMyomIiZPYidGMylGUiIjN0YzRjM3I0Yu

Over Q(N): do we recover the equation derived via invariants? algNM:=M*(1-N)^2-N

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdOTUdGKCwmKiZJIk1HRigiIiIpLCZGMUYxSSJOR0YoISIiIiIjRjFGMUY0RjU3I0Yu

MN:=solve(algNM,M)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNNTkdGKComSSJOR0YoIiIiKSwmRjAhIiJGL0YwIiIjRjM3I0Yu

factor([solve(algNM,N)])

NyQsJCooIyIiIiIiI0YmSSJNRzYiISIiLCgqJkYnRiZGKEYmRiZGJkYmKiQpLCYqJiIiJUYmRihGJkYmRiZGJkYlRiZGJkYmRiYsJCooRiVGJkYoRiosKEYsRiZGJkYmRi1GKkYmRiY=

NM:=op(2,%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNOTUdGKCwkKigjIiIiIiIjRjFJIk1HRighIiIsKComRjJGMUYzRjFGMUYxRjEqJCksJiomIiIlRjFGM0YxRjFGMUYxRjBGMUY0RjFGMTcjRi4=

res:=factor(resultant(algA1Mpap,algNM,M))

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

This is divisible by the equation we have found for A1 factor(res/algA1N)

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

and the remaining factor does not vanish for the correct value of A1 n:=4: series(subs(N=Nser(n),AA1=A1,U(0,0)=subs(x=0,y=0,User(n)),%%),t,n)

Ky1JInRHNiIiJDcmIiIhISVbPyIiIiIlJzQlIiIjISY3TCIiIiQtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIiJQ==

Hence the description of A1 given in the paper is correct/ From this point on, we can check the expressions given in the paper for tA2, tA0 and C00. We will explain later (next section in the Maple file) how these expressions where obtained. Back to tA2 tA2sol;

LCQqKiIiIyIiIiwmKigsKiokKUkiTkc2IiIiJEYlRiUqJClGK0YkRiUhIiJGK0YwRiVGMEYlKSwmRiVGMEYrRiUiIiVGJSlJJEFBMUdGLEYkRiVGJSooIiM7RiUsKkYpRiVGLkYlRitGJUYlRjBGJSksKEYuRiVGK0YwRiVGJUYtRiVGMEYlKUYyRiRGMCwmKihGNEYlRitGJUYxRiVGJSomRjdGJUY5RiVGJUYwRjA=

The (first) value given in the theorem tA2pap:=(1+2*M)^2/M-AA1/4-4*(1+M)^3/M/AA1

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSd0QTJwYXBHRigsKComKSwmKiYiIiMiIiJJIk1HRihGNEY0RjRGNEYzRjRGNSEiIkY0KiYjRjQiIiVGNEkkQUExR0YoRjRGNioqRjlGNCksJkY1RjRGNEY0IiIkRjRGNUY2RjpGNkY2NyNGLg==

OK! numer(factor(subs(M=MN,tA2pap-tA2sol))): factor(%/algA1N)

ISIi

Another check, this time with expansions A2t

LCQqKCIiIyIiIkkidEc2IkYlLCYtSSNEREdGJzYjIiIhRiUtSSJVR0YnNiRGLEYsRiVGJUYl

n:=10:series(subs(DD(0)=subs(y=0,Dser(n)), U(0,0)=subs(x=0,y=0,User(n)),A2t),t,n)

KzdJInRHNiIiIiMiIiJGJUYlIiM3IiIkIiNPIiIlIiR5IiIiJiIkWygiIiciJV9PIiIoIiYhUTwiIikiJmtxKSIiKi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YmIiM1

series(subs(M=Mser(n),AA1=A1ser(n),tA2pap),t,10)

KzNJInRHNiIiIiMiIiJGJUYlIiM3IiIkIiNPIiIlIiR5IiIiJiIkWygiIiciJV9PIiIoLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiYiIik=

Degree of tA2: only 8, when one could expect 16 Equation for tA2 over Q(N) algtA2N:=collect(op(5,factor(resultant(numer(tA2sol-tA2),algA1N,AA1))),tA2,factor);

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

But over Q(t), degree 8 factor(resultant(algtA2N,algN,N));

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

algtA2:=collect(op(1,op(2,%)),tA2,factor);

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

algtA2M:=collect(op(1,factor(resultant(algtA2N,algNM,N))),tA2,factor)

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

We introduce the series P2 unassign('P2') algP2M:=P2^4-(1+4*M)^3*P2^2+4*M*(1+M)^3*(1+4*M)^3

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdQMk1HRigsKCokKUkjUDJHRigiIiUiIiJGMyomKSwmKiZGMkYzSSJNR0YoRjNGM0YzRjMiIiRGMylGMSIiI0YzISIiKipGMkYzRjhGMyksJkY4RjNGM0YzRjlGM0Y1RjNGMzcjRi4=

map(allvalues,algeqtoseries(subs(M=convert(Mser(10),polynom),algP2M),t,P2,3))

NyYrK0kidEc2IiIiIiIiISIiJUYmIiM1IiIjLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiYiIiQrK0YkISIiRichIiVGJiEjNUYqRitGMSwoKiZGKkYmKUYkI0YmRipGJkYmKiYiIipGJilGJCNGMUYqRiZGJi1GLDYjKiQpRiRGKkYmRiYsKEY3RjNGOkYzRj5GJg==

P2ser:=proc(n) op(1, map(allvalues,algeqtoseries(subs(M=convert(Mser(n+3),polynom),algP2M),t,P2,n))): end: P2ser(5)

Ky9JInRHNiIiIiIiIiEiIiVGJSIjNSIiIyIjVSIiJCIkdyJGJy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIm

Let us check the two expressions of P2 given in the paper P2bis:=(1+4*M)^(3/2)*(1-P1)/(1+P1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZQMmJpc0dGKCooKSwmKiYiIiUiIiJJIk1HRihGM0YzRjNGMyMiIiQiIiNGMywmRjNGM0kjUDFHRighIiJGMywmRjNGM0Y5RjNGOjcjRi4=

factor(resultant(numer(P2bis^2-P2^2),algP1M,P1)/algP2M)

LCQqKCIjOyIiIiksJiomIiIlRiVJIk1HNiJGJUYlRiVGJSIiJ0YlLDoqJiIkYyNGJSlGKiIiKEYlRiUqJiIkZypGJSlGKkYsRiVGJSomIiUjUiJGJSlGKiIiJkYlRiUqKCIja0YlKUYqIiIkRiUpSSNQMkdGKyIiI0YlISIiKiYiJCEpKkYlKUYqRilGJUYlKigiI1tGJSlGKkY/RiVGPUYlRkAqJClGPkYpRiVGJSomIiRbJEYlRjtGJUYlKigiIzdGJUYqRiVGPUYlRkAqJiIjZ0YlRkZGJUYlKiRGPUYlRkBGKEYlRiVGJQ==

P2ter:=M*AA1/4+4*(1+M)^3/AA1

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZQMnRlckdGKCwmKigjIiIiIiIlRjFJJEFBMUdGKEYxSSJNR0YoRjFGMSooRjJGMSksJkY0RjFGMUYxIiIkRjFGMyEiIkYxNyNGLg==

factor(resultant(numer(P2-P2ter),algA1Mpap,AA1)/algP2M)

LCQqKiIrJ0huXEglIiIiKSwmSSJNRzYiRiVGJUYlIiM3RiUsOiomIiRjI0YlKUYoIiIoRiVGJSomIiRnKkYlKUYoIiInRiVGJSomIiUjUiJGJSlGKCIiJkYlRiUqKCIja0YlKUYoIiIkRiUpSSNQMkdGKSIiI0YlISIiKiYiJCEpKkYlKUYoIiIlRiVGJSooIiNbRiUpRihGPkYlRjxGJUY/KiQpRj1GQ0YlRiUqJiIkWyRGJUY6RiVGJSooRipGJUYoRiVGPEYlRj8qJiIjZ0YlRkZGJUYlKiRGPEYlRj8qJkZDRiVGKEYlRiVGJUZCRiVGJQ==

factor(tA2pap+P2ter/M)

KiYpLCYqJiIiIyIiIkkiTUc2IkYnRidGJ0YnRiZGJ0YoISIi

Back to tA0 tA0solA1:=collect(factor(subs(tA2=tA2sol,tA0sol)),AA1,factor);

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

The expression in the paper tA0pap:=AA1^2/8+(2*M^3-14*M^2-12*M-3)/M+(AA1/8/M+2*(M+1)^3/AA1/M^2)*((1+4*M)^(3/2)-(1+2*M)^2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSd0QTBwYXBHRigsKComIyIiIiIiKUYxKUkkQUExR0YoIiIjRjFGMSomLCoqJkY1RjEpSSJNR0YoIiIkRjFGMSomIiM5RjEpRjpGNUYxISIiKiYiIzdGMUY6RjFGP0Y7Rj9GMUY6Rj9GMSomLCYqKEYwRjFGNEYxRjpGP0YxKipGNUYxKSwmRjpGMUYxRjFGO0YxRjRGP0Y+Rj9GMUYxLCYqJCksJiomIiIlRjFGOkYxRjFGMUYxI0Y7RjVGMUYxKiQpLCYqJkY1RjFGOkYxRjFGMUYxRjVGMUY/RjFGMTcjRi4=

Replace the square root in the above expression using: NM*2*M-(1+2*M)

LCQqJCksJiomIiIlIiIiSSJNRzYiRihGKEYoRigjRigiIiNGKCEiIg==

tA0papalt:=AA1^2/8+(2*M^3-14*M^2-12*M-3)/M+(AA1/8/M+2*(M+1)^3/AA1/M^2)*(-(N*2*M-(1+2*M))^3-(1+2*M)^2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSp0QTBwYXBhbHRHRigsKComIyIiIiIiKUYxKUkkQUExR0YoIiIjRjFGMSomLCoqJkY1RjEpSSJNR0YoIiIkRjFGMSomIiM5RjEpRjpGNUYxISIiKiYiIzdGMUY6RjFGP0Y7Rj9GMUY6Rj9GMSomLCYqKEYwRjFGNEYxRjpGP0YxKipGNUYxKSwmRjpGMUYxRjFGO0YxRjRGP0Y+Rj9GMUYxLCYqJCksKCooRjVGMUY6RjFJIk5HRihGMUYxKiZGNUYxRjpGMUY/RjFGP0Y7RjFGPyokKSwmRk5GMUYxRjFGNUYxRj9GMUYxNyNGLg==

numer(factor(subs(M=MN,tA0solA1-tA0papalt))):factor(%/algA1N)

LCoqJClJIk5HNiIiIiQiIiJGKCokKUYlIiIjRihGKEYlRihGKCEiIg==

polynomial equations for tA0 algtA0N:=collect(op(6,factor(resultant(numer(tA0-tA0solA1),algA1N,AA1))),tA0,factor);

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

factor(resultant(algtA0N,algN,N)):nops(%); degree(%%,tA0);

IiRjIw==

IiM7

algtA0:=collect(factor(resultant(algtA0N,algN,N)),tA0,factor): Checking the series expansions series(subs(M=Mser(n),AA1=A1ser(n),tA0pap),t,n)

KzdJInRHNiIhIiMhIiJGJiIiIUYlIiIiRigiIiMhIikiIiRGJSIiJSEjcyIiJiEkYCIiIichJSU0IiIiKC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIp

tA0ser

LCwqJiIiIyIiIkkidEc2IiEiIkYoRiVGKComRiRGJUYmRiVGKCokKUYmRiRGJUYlKiYiIilGJSlGJiIiJEYlRig=

Finally, the expression of C00=D(0) A1;A2t

LCYqKCIiJSIiIkkidEc2IkYlLUkiVUdGJzYkIiIhRitGJUYlRiRGJQ==

LCQqKCIiIyIiIkkidEc2IkYlLCYtSSNEREdGJzYjIiIhRiUtSSJVR0YnNiRGLEYsRiVGJUYl

expand((A2t-A1/2)/2)

LCYiIiIhIiIqJkkidEc2IkYjLUkjRERHRic2IyIiIUYjRiM=

This is the expression derived from the invariant approach C00timest:=(tA2-AA1/2)/2+1

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpDMDB0aW1lc3RHRigsKComIyIiIiIiI0YxSSR0QTJHRihGMUYxKiYjRjEiIiVGMUkkQUExR0YoRjEhIiJGMUYxNyNGLg==

tC00pap:=1+(1+2*M)^2/2/M-3*AA1/8-2*(1+M)^3/M/AA1

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSh0QzAwcGFwR0YoLCoiIiJGLyooI0YvIiIjRi8pLCYqJkYyRi9JIk1HRihGL0YvRi9GL0YyRi9GNiEiIkYvKiYjIiIkIiIpRi9JJEFBMUdGKEYvRjcqKkYyRi8pLCZGNkYvRi9GL0Y6Ri9GNkY3RjxGN0Y3NyNGLg==

C00timestsol:=subs(tA2=tA2sol,C00timest)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS1DMDB0aW1lc3Rzb2xHRigsKCooLCYqKCwqKiQpSSJOR0YoIiIkIiIiRjcqJClGNSIiI0Y3ISIiRjVGO0Y3RjtGNyksJkY3RjtGNUY3IiIlRjcpSSRBQTFHRihGOkY3RjcqKCIjO0Y3LCpGM0Y3RjhGN0Y1RjdGN0Y7RjcpLChGOEY3RjVGO0Y3RjdGNkY3RjtGNylGPUY6RjssJiooRj9GN0Y1RjdGPEY3RjcqJkZCRjdGREY3RjdGO0Y7KiYjRjdGPkY3RkBGN0Y7RjdGNzcjRi4=

numer(factor(subs(M=MN,C00timestsol-tC00pap))): factor(%/algA1N)

IiIi

This completes the proof of the expressions given in theorem. For the degrees, see further down in the Maple session

D0 has degree 16 C00timest

LCgqJiMiIiIiIiNGJUkkdEEyRzYiRiVGJSomI0YlIiIlRiVJJEFBMUdGKEYlISIiRiVGJQ==

res:=factor(resultant(numer(-tD0+subs(tA2=tA2sol,C00timest)),algA1N,AA1)); nops(%)

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

IiIn

algtD0N:=collect(op(6,res),tD0,factor)

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

algtD0:=factor(resultant(algtD0N,algN,N)):nops(%); degree(%%,tD0);

IiReIg==

IiM7

Singularity analysis of A1? We proceed step by step, using the algebraic description of the paper factor(Mt)

LCQqKCMiIiIiIilGJSwoKiYiIiNGJUkidEc2IkYlRiVGJSEiIiokKSwkKiYsJkYlRiVGKEYlRiUsJkYlRiwqJiIiJ0YlRipGJUYlRiVGLCNGJUYpRiVGJUYlRipGLEYs

Dominant singularity at 1/6 (the other one at t=-1/2) Msing:=series(subs(t=1/6*(1-epsilon),Mt),epsilon,4)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZNc2luZ0dGKCw0IyIiIiIiI0YwKihGL0YwKSIiJEYvRjApSShlcHNpbG9uR0YoRi9GMCEiIiomI0Y0IiIlRjBGNkYwRjAqKCMiIigiIztGMEYzRjApRjYjRjRGMUYwRjcqJkY5RjApRjZGMUYwRjAqKCMiJDYiIiRjI0YwRjNGMClGNiMiIiZGMUYwRjcqJkY5RjApRjZGNEYwRjAqKCMiJCgpKSIlWz9GMEYzRjApRjYjRj1GMUYwRjctSSJPR0YlNiMqJClGNkY6RjBGMDcjRi4=

Xritical value of M: 1/2 subs(t=1/6,Mt)

IyIiIiIiIw==

plot(Mt,t=-1/6..1/6)

6*-%'CURVESG6$7fw7$$!2mmmmmmmm"!#<$!2Nhb@'fuR8!#<7$$!2W"He;89\;!#<$!1;=-jp`F8!#;7$$!2'zD=.E*Qj"!#<$!27g"[r&**oJ"!#<7$$!2Ov],$Qu;;!#<$!2mN-rA?\I"!#<7$$!1)f>RQ"[*f"!#;$!2O0lzMVGH"!#<7$$!1x?3$)4I#e"!#;$!2`3?rG03G"!#<7$$!2lIhAbsjc"!#<$!1vINYuip7!#;7$$!1>rv%ez)\:!#;$!2;%==Dd.e7!#<7$$!2V&3<MD#G`"!#<$!2%p`K)4GgC"!#<7$$!1W@w&=?e^"!#;$!2"oc\"yQSB"!#<7$$!2V^p0=J$)\"!#<$!2nL*)\q%o@7!#<7$$!1#Qw_&o#H["!#;$!1z$47k%y57!#;7$$!2u$3]L_el9!#<$!2X8W&3C\)>"!#<7$$!1M,O0C<[9!#;$!2QF$z3k7'="!#<7$$!1tz#*=>RJ9!#;$!2j'Qd^o=u6!#<7$$!1T#['HO:;9!#;$!2u]kq!\Kj6!#<7$$!2_u:)HQ.)R"!#<$!2/D%oeSQ]6!#<7$$!2XczD)Ro#Q"!#<$!2u>*e"z*RR6!#<7$$!2&G!R6YG[O"!#<$!2J!*\R*ofE6!#<7$$!1zC;*HB!\8!#;$!2ZiusqS_6"!#<7$$!1(y!\JDoJ8!#;$!2CdL$p_v-6!#<7$$!2$)*H$*>+<:8!#<$!2(pP*3XS34"!#<7$$!0m)RY4%zH"!#:$!2D'4G\7Qy5!#<7$$!1k%f&y#>@G"!#;$!26hh*3Z"p1"!#<7$$!14=OsM0l7!#;$!2NlH+**=X0"!#<7$$!2%**)zf*oKZ7!#<$!12!GcY7;/"!#;7$$!2#y*G"fe*=B"!#<$!1)GRd<^.."!#;7$$!2X"He;*H_@"!#<$!2T_6U"3;=5!#<7$$!2#z"prO7!)>"!#<$!2#4oH%*f`05!#<7$$!2E&Q5a$o6="!#<$!1V>iq)R:$**!#<7$$!2R8g`!4([;"!#<$!1x;We"R9")*!#<7$$!2"z"prOvn9"!#<$!1%3e^0Pxn*!#<7$$!2=.t7s:08"!#<$!0m=tyysb*!#;7$$!2"opsy]:86!#<$!0nSV#=KG%*!#;7$$!2rS"GcPU(4"!#<$!0O*)z#o:6$*!#;7$$!2:khcED-3"!#<$!1DE:]*>F=*!#<7$$!2u:)HEG/k5!#<$!1t8c0h`h!*!#<7$$!2-O0W^Fr/"!#<$!0<"*Qf5X$*)!#;7$$!2)HE>0(*eI5!#<$!1`/L'=k*4))!#<7$$!2)*G"f^hF85!#<$!1#eiGo">z')!#<7$$!1Fa3<9,m**!#<$!17%)49S'Gb)!#<7$$!0#Ryc`[&z*!#;$!1)*Q'fu%GB%)!#<7$$!1e%e$Q8PE'*!#<$!1xVF%y!Q%H)!#<7$$!1sz#*=J(4Z*!#<$!1`=Q2gdv")!#<7$$!0d!yA#pGH*!#;$!1,gZrg)*Q!)!#<7$$!1sV([(4dL"*!#<$!1nL&y%eU;z!#<7$$!1O/U<esj*)!#<$!1["oQMN`y(!#<7$$!1\K)*HB;,))!#<$!1iZJjgXfw!#<7$$!1C<,pkl?')!#<$!1#3"[vg?>v!#<7$$!1;khc;Nk%)!#<$!1cD@wpM(R(!#<7$$!17C['HWpG)!#<$!1hvdvHces!#<7$$!1x'oquc_7)!#<$!1<V)=&3jJr!#<7$$!1bWAyfR[z!#<$!1**4$Rd"H#*p!#<7$$!1JhAX]z&z(!#<$!0'z'p3U;(o!#;7$$!1$))4`R+?i(!#<$!1E7M"Q]Pt'!#<7$$!1%e$QVL7au!#<$!0P:O!Q0+m!#;7$$!1OSZhfN'G(!#<$!1[#en"\%fY'!#<7$$!1PT\ld?>r!#<$!1=U\%)Q#=L'!#<7$$!1')QWbniep!#<$!1Yu8%3#\-i!#<7$$!1C['HfO]y'!#<$!1H2c'oQ@1'!#<7$$!0`R7e#f>m!#;$!1W(H!f"Qy#f!#<7$$!1OPT\0ZXk!#<$!1.<:@1#fy&!#<7$$!1BX!4=[yG'!#<$!1#o`2)e$pl&!#<7$$!16*[k&fk8h!#<$!1dA#*fR!Q^&!#<7$$!20`R7e!zYf!#=$!1\")Q!*y7w`!#<7$$!1<MoO8M!y&!#<$!1$)*fHy5#Q_!#<7$$!2%*Q6cX%[1c!#=$!0,]7WIN4&!#;7$$!1LIF@fNYa!#<$!2;)Q4)H*pf\!#=7$$!1m*f')RiBG&!#<$!2l4:F6c?#[!#=7$$!2;f^O1%H,^!#=$!1'fouh&QpY!#<7$$!2:%ze<:NP\!#=$!1V&\*4N^IX!#<7$$!1.R6c&*opZ!#<$!2D`uOx[yQ%!#=7$$!1*\m*f;=*f%!#<$!1L7`CZ3UU!#<7$$!204=Os^CW%!#=$!2E*pk)z&[2T!#=7$$!1prw'o#QvU!#<$!1hEl1$\L'R!#<7$$!1fZG!R&e4T!#<$!2'R.=cli>Q!#=7$$!2kZ&4>QBKR!#=$!1Fg_b/7lO!#<7$$!29^o.uGdx$!#=$!1T;m&z/"GN!#<7$$!1:#4&oV:&f$!#<$!1y))*)H8ApL!#<7$$!1#*zf>**>KM!#<$!2&>(G%*3'4DK!#=7$$!0D]+,l4F$!#;$!0*p4hJy"3$!#;7$$!2u7#4&o&e(4$!#=$!2-w4q1vo#H!#=7$$!1E^-0]]BH!#<$!1dL&R0!\qF!#<7$$!1Z!4=O<^w#!#<$!2$Gr;a)[ui#!#=7$$!2b+o-s;pf#!#=$!2[+*3*RWZZ#!#=7$$!2WLLLLpFV#!#=$!2V4#f&z4\K#!#=7$$!2i&3<M=$[D#!#=$!21"4e8mch@!#=7$$!2ujgaUm35#!#=$!0B"QE>W>?!#;7$$!2K!R6c:.C>!#=$!2&>kC#)zGb=!#=7$$!1j?T#[(zd<!#<$!2irK$RB1+<!#=7$$!1:id[!pJf"!#<$!0u^*3zWX:!#;7$$!1#H"f^'4%G9!#<$!2ik1*\*3)*Q"!#=7$$!2hR7e\l.E"!#=$!2w>HJM?,B"!#=7$$!2%\E'e]TA3"!#=$!2v#*Gow)yf5!#=7$$!1/C"e\#>*=*!#=$!1aSfJ%)HE!*!#=7$$!1%z_06#*pd(!#=$!1B)4K][bY(!#=7$$!1M3;Kkd`e!#=$!1%4)=kch'y&!#=7$$!1u>1zC"R8%!#=$!0*=k-+H+T!#<7$$!10!*zf>T!f#!#=$!2k&[WD"Hrd#!#>7$$!1[?2")GC?w!#>$!1:3u_Xm3w!#>7$$"1t>V'Gd)yw!#>$"12Jp!f'o!p(!#>7$$"1&R$oOtGdD!#=$"2uHe(\>]qD!#>7$$"1_bWAy4lU!#=$"1o&Q6V6@I%!#=7$$"1O!zC;p**y&!#=$"1x%odQ1'ee!#=7$$"0jhcYY[](!#<$"20e$Q=&y4i(!#>7$$"10sx)3"4J#*!#=$"1x$3q\_!3%*!#=7$$"2(HHD<J"\4"!#=$"2k%Hp'G))*>6!#=7$$"2ZjgaU(>a7!#=$"2(>(y$p)>tG"!#=7$$"2D^-05F">9!#=$"2>5W*41$=Y"!#=7$$"2j:ljg(p*e"!#=$"2L())eXWnV;!#=7$$"2)fAX!4@(f<!#=$"2EZq[8Yj#=!#=7$$"1f&yB96Y$>!#<$"1rqd#G\d,#!#<7$$"2-))4`Ra')3#!#=$"2Lq4QvlQ=#!#=7$$"2cKlIhq?E#!#=$"1Xf3-JguB!#<7$$"21OsW*))>OC!#=$"2'Rc15TtnD!#=7$$"2&oRzeP+/E!#=$"03B**39av#!#;7$$"2))G"f^mQcF!#=$"20PX*RS?FH!#=7$$"2sC;*\YePH!#=$"1f13V2?LJ!#<7$$"2W0yA7$3"4$!#=$"2/"[]bN?4L!#=7$$"1<MoO$Q'pK!#<$"2&\h"3P0d^$!#=7$$"2%4*[k&**oFM!#=$"19(4#e$=,q$!#<7$$"2%Ge;Lw4,O!#=$"1DJ$4&eC/R!#<7$$"2nrV([FAmP!#=$"1zQaOLT+T!#<7$$"/r3%[8&QR!#:$"2&3i!)>8+2V!#=7$$"1i!zC;In4%!#<$"1jP!zir%)\%!#<7$$"17cXC#)QnU!#<$"1'GuHLKpq%!#<7$$"10ZF))RlWW!#<$"1%*p([3Wc#\!#<7$$"1>RycV'*)f%!#<$"0#pD&*y)y6&!#;7$$"2YbWAyBcw%!#=$"1FPVn_]F`!#<7$$"12>Qw#*zP\!#<$"1DX(pO&GYb!#<7$$"1w^.2%Ri5&!#<$"2M%fx)plDw&!#=7$$"2/OsW*Q@p_!#=$"2a;9#*3+S(f!#=7$$"13>Qw#p,X&!#<$"2cuoI@M8@'!#=7$$"1"QV`tlFh&!#<$"1_eFUC'pU'!#<7$$"1=S!3;sjy&!#<$"1L8O!QU(fm!#<7$$"1G'f_Q&oVf!#<$"1dq^=R,to!#<7$$"1%Gd9Hqc6'!#<$"1</nyHz3r!#<7$$"1G@4&o%\xi!#<$"1Z&f!>(4KL(!#<7$$"/,-/ykYk!#:$"1T%Q@$*G0d(!#<7$$"/u9'*e-7m!#:$"0lC6]Q`!y!#;7$$"103;K9;&y'!#<$"19aF>]?a!)!#<7$$"1s#))4`6>&p!#<$"1E#*)\aEpH)!#<7$$"1z(*G"fPC7(!#<$"1X?(ed?$[&)!#<7$$"1T_r4;b"H(!#<$"1t4WLl!4!))!#<7$$"1Cd9H)\pW(!#<$"1^J>iA(f.*!#<7$$"08$HDP0Dw!#;$"1RD@i;**3$*!#<7$$"0L*>t>N%y(!#;$"11/J_Xbc&*!#<7$$"1iKlIr>az!#<$"1)z?Y_MT#)*!#<7$$"1_/4=1w;")!#<$"2LG!*=i'Q35!#<7$$"1u>1zkE(H)!#<$"2l2oKicw."!#<7$$"1#Gd9HrNX)!#<$"2sg%*o%4Qj5!#<7$$"0H"f^'y4j)!#;$"2#yc%Q(*>I4"!#<7$$"0-05?mEz)!#;$"2DguEtc/7"!#<7$$"1U#\)pp_p*)!#<$"2[M^5F\4:"!#<7$$"1ov%G!z7A"*!#<$"2xp$G0$yw<"!#<7$$"1;Qw_D#fH*!#<$"2r26*)36'37!#<7$$"1=,p/'*zj%*!#<$"2Wh_xG2!R7!#<7$$"1j'*f')pcJ'*!#<$"2UBKm$>"*p7!#<7$$"1h&zD=<()z*!#<$"2$H+]],D,8!#<7$$"1;)HE>'Hf**!#<$"2CG)[bD*=L"!#<7$$"1))3^N')G85!#;$"2.PrHpMcO"!#<7$$"2rT$oOI$)H5!#<$"2B;FO^;%)R"!#<7$$"2o*f')R_CZ5!#<$"2B5Z#HbgL9!#<7$$"2w"prwu+j5!#<$"2ueZ$3m4m9!#<7$$"2(yC;*pF/3"!#<$"2U#G9!GVF]"!#<7$$"2rT$oOK6(4"!#<$"2$y!HVq1'Q:!#<7$$"2#G!R6;eP6"!#<$"1.Lq'Gk^d"!#;7$$"23BY#\Q9J6!#<$"2L&oLmjA9;!#<7$$"2p1!o-n:Z6!#<$"2V6t(p)R5l"!#<7$$"2MPT\0cN;"!#<$"1QKz>&='*o"!#;7$$"21@U%))Gm"="!#<$"2`'yPSQIL<!#<7$$"2id[I9d!)>"!#<$"2buNxE.Rx"!#<7$$"2)zf>RL#[@"!#<$"2L@G]@;l"=!#<7$$"2-s5)GT(=B"!#<$"2(*e%>[t0h=!#<7$$"27cXC7ZvC"!#<$"2N4'R#QSJ!>!#<7$$"1`18ESDk7!#;$"2(3ZoHQG\>!#<7$$"2V*)ydvL3G"!#<$"2(f4c,nZ'*>!#<7$$"1Ay*G"*o&)H"!#;$"2_D,'o)='[?!#<7$$"1>0x?%>UJ"!#;$"2a%4s*ymh4#!#<7$$"1\k&z&oFK8!#;$"1%Q>5qiH:#!#;7$$"22dZGIs&[8!#<$"2[PH)Gj91A!#<7$$"1XB!Qz&pk8!#;$"2ca">I5tgA!#<7$$"2s:)HEP.#Q"!#<$"28zCHq1=K#!#<7$$"2t&[I%zT%*R"!#<$"2u'[[%*)eeQ#!#<7$$"2eYE'e0G:9!#<$"2\ax))pjnW#!#<7$$"1p0yA15K9!#;$"2$***p\k'[9D!#<7$$"2i.u9O:&[9!#<$"2mlikL`Re#!#<7$$"2YG!R6"4jY"!#<$"1bb0uQ^jE!#;7$$"2iIhAl0<["!#<$"2O4x3$*=kt#!#<7$$"0)f>R"*Q*\"!#:$"1fO$[H;b#G!#;7$$"2U;mla7g^"!#<$"1QenB3O:H!#;7$$"2([(\**QvC`"!#<$"1k`eLaB6I!#;7$$"22C['H8&*[:!#<$"1_u-OLY:J!#;7$$"2.8E_ubdc"!#<$"1)R%Qr+IKK!#;7$$"106AW"oNe"!#;$"1K(Q4z`5P$!#;7$$"2iC\)p.!**f"!#<$"20^[OYrj^$!#<7$$"2B%=qtB-;;!#<$"2DgCO#G-%o$!#<7$$"2>w&[IlDL;!#<$"2F6X"=HK0R!#<7$$"1c7D][&=k"!#;$"/wb1%4F/%!#97$$"0v;+<`/l"!#:$"1$***Ha`;7U!#;7$$"2#3<M=*f&e;!#<$"2an*Q@e3IW!#<7$$"2mmmmmmmm"!#<$"1,++++++]!#;-%&COLORG6'%$RGBG$")C)eq%!")$""!!""$"('>!\&!")-%+_ATTRIBUTEG6#/%'sourceG%,mathdefaultG-%%VIEWG6$;$!+nmmm;!#5$"+nmmm;!#5%(DEFAULTG-&%&_AXISG6#"""6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-&%&_AXISG6#""#6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-%+AXESLABELSG6$-I#miG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"65Q"t6"/%'familyGQ!6"/%%sizeGQ#106"/%%boldGQ&false6"/%'italicGQ%true6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"Q!6"-%)_VISIBLEG6#"""-%%ROOTG6'-%)BOUNDS_XG6#$"#q!""-%)BOUNDS_YG6#$"$S"!""-%-BOUNDS_WIDTHG6#$"%!)[!""-%.BOUNDS_HEIGHTG6#$"%+[!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+]!""-%.BOUNDS_HEIGHTG6#$"%+]!""-%)CHILDRENG6"NiI=

Now the series N NM

LCQqKCMiIiIiIiNGJUkiTUc2IiEiIiwoKiZGJkYlRidGJUYlRiVGJSokKSwmKiYiIiVGJUYnRiVGJUYlRiVGJEYlRilGJUYl

Does the square root vanish before t=1/6? No. subs(M=-1/4,tM)

IyEiIiIiJA==

Thus N has radius 1/6 as well, with a unique dominant singularity at 1/6. Here is its singular expansion at 1/6 Nsing:=normal(series(subs(M=Msing,NM),epsilon,4))

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

Now the series P1 algP1M;

LCYqKEkjUDFHNiIiIiIpLCYqJiIiJUYmSSJNR0YlRiZGJkYmRiYiIiRGJiksJkYmISIiRiRGJiIiI0YmRiYqKEYrRiYpLCZGK0YmRiZGJkYsRiYpLCZGJkYmRiRGJkYqRiZGLw==

factor(discrim(%,P1)); factor(lcoeff(algP1M,P1))

LCQqLiIjayIiIiksJiomIiIjRiVJIk1HNiJGJUYlRiVGJUYpRiUpLCZGKkYlRiVGJSIiJEYlKSwmRihGJUYlISIiIiInRiUpLCYqJiIiJUYlRipGJUYlRiVGJSIiKkYlRipGJUYl

LCQqJkkiTUc2IiIiIiksJkYkRiZGJkYmIiIkRiYhIiI=

solve(subs(M=-1/4,algM),t)

IyEiIiIiJA==

We know that the radius must be a singularity (Pringsheim' theorem). Again, radius 1/6 with(plots): implicitplot(algP1M,M=0..1/2,P1=0..1,numpoints=10000)

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Around M=1/2 unassign('delta') map(allvalues,algeqtoseries(subs(M=1/2*(1-delta),algP1M),delta,P1,3))

NyYsKiIiJCIiIiomIiIjRiUpRicjRiVGJ0YlISIiKiYsJiomI0YnRiRGJSlGJEYpRiVGJSomIyIiJSIiKkYlKSIiJ0YpRiVGKkYlKUkmZGVsdGFHNiIjRiRGJ0YlRiUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0Y4NiMqJClGNyMiIiZGJ0YlRiUsKkYkRiVGJkYqKiYsJkYtRipGMEYlRiVGNkYlRiVGOkYlLCpGJEYlRiZGJSomLCZGLUYlRjBGJUYlRjZGJUYlRjpGJSwqRiRGJUYmRiUqJiwmRi1GKkYwRipGJUY2RiVGJUY6RiU=

Choose the rigth expansion evalf(%)

NyYsKCQiKndHZHIiISIqIiIiKiYkIilKJVFnJ0YmRicpSSZkZWx0YUc2IiMiIiQiIiNGJ0YnLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGLTYjKiQpRiwjIiImRjBGJ0YnLChGJEYnRighIiJGMUYnLCgkIitDclVHZUYmRicqJiQiK1pFT1ZBRiZGJ0YrRidGJ0YxRicsKEY+RidGQEY8RjFGJw==

P1Msing:=op(2,%%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShQMU1zaW5nR0YoLCoiIiQiIiIqJiIiI0YwKUYyI0YwRjJGMCEiIiomLCYqJiNGMkYvRjApRi9GNEYwRjUqJiMiIiUiIipGMCkiIidGNEYwRjBGMClJJmRlbHRhR0YoI0YvRjJGMEYwLUkiT0dGJTYjKiQpRkIjIiImRjJGMEYwNyNGLg==

Msing

LDQjIiIiIiIjRiQqKEYjRiQpIiIkRiNGJClJKGVwc2lsb25HNiJGI0YkISIiKiYjRigiIiVGJEYqRiRGJCooIyIiKCIjO0YkRidGJClGKiNGKEYlRiRGLComRi5GJClGKkYlRiRGJCooIyIkNiIiJGMjRiRGJ0YkKUYqIyIiJkYlRiRGLComRi5GJClGKkYoRiRGJCooIyIkKCkpIiVbP0YkRidGJClGKiNGMkYlRiRGLC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRis2IyokKUYqRi9GJEYk

Finally, singular behaviour of P1 in terms of (1-6t) P1sing:=map(factor,series(subs(delta=1-2*Msing,P1Msing),epsilon,4))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdQMXNpbmdHRigsKiIiJCIiIiomIiIjRjApRjIjRjBGMkYwISIiKiojRjIiIipGMCwmKiZGL0YwKUYvRjRGMEYwKiZGMkYwKSIiJ0Y0RjBGNUYwKUYvI0YvIiIlRjApSShlcHNpbG9uR0YoRkBGMEY1LUkiT0dGJTYjKiQpRkMjIiImRkFGMEYwNyNGLg==

Now A1 A1sing:=map(normal,series(subs(P1=P1sing,M=Msing,A1P1),epsilon,4))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdBMXNpbmdHRigsLComIiInIiIiKUYwI0YxIiIjRjFGMSomRjBGMSkiIiRGM0YxISIiKipGNEYxKUkoZXBzaWxvbkdGKCNGNyIiJUYxKUY3I0YxRj1GMSwmKiRGMkYxRjgqJEY2RjFGMUYxRjEqKEY3RjFGO0YxRkBGMUY4LUkiT0dGJTYjKiQpRjsjIiImRj1GMUYxNyNGLg==

There we are, at last tC00pap

LCoiIiJGIyooI0YjIiIjRiMpLCYqJkYmRiNJIk1HNiJGI0YjRiNGI0YmRiNGKiEiIkYjKiYjIiIkIiIpRiNJJEFBMUdGK0YjRiwqKkYmRiMpLCZGKkYjRiNGI0YvRiNGKkYsRjFGLEYs

tC00sing:=collect(map(factor,map(expand,map(rationalize,series(subs(M=Msing, AA1=A1sing,tC00pap),epsilon,4)))),epsilon)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSl0QzAwc2luZ0dGKCwuKiYsKComIyIiJCIiIyIiIikiIicjRjVGNEY1ISIiKiYjRjMiIiVGNSlGM0Y4RjVGNUYzRjVGNUkoZXBzaWxvbkdGKEY1RjUqJiwmKihGOEY1KUYzI0Y1RjxGNUY2RjVGNSokKUYzRjtGNUY5RjUpRj5GO0Y1RjUiIiZGNSomRjNGNUY2RjVGOSomRjJGNUY9RjVGNS1JIk9HRiU2IyokKUY+I0ZHRjxGNUY1NyNGLg==

C00sing:=map(factor,collect(map(expand,series(tC00sing/(1-epsilon)*6,epsilon,3)),epsilon))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShDMDBzaW5nR0YoLC4iI0kiIiIqJiIjPUYwKSIiJyNGMCIiI0YwISIiKiYiIipGMCkiIiRGNUYwRjAqKkY7RjApRjsjRjAiIiVGMCwmKiRGM0YwRjAqJkY2RjBGOkYwRjdGMClJKGVwc2lsb25HRigjRjtGP0YwRjAqKCNGO0Y2RjAsKCooRjJGMEY6RjApRjZGNUYwRjBGOEY3IiNLRjdGMEZERjBGNy1JIk9HRiU2IyokKUZEIyIiJkY/RjBGMDcjRi4=

Let us check the constant term implicitplot(algtD0,t=0..1/6,tD0=0..0.3,numpoints=10000);evalf(subs(epsilon=0,tC00sing))

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LCYkIiokKXBnXCMhIioiIiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiFGJg==

Now the asymptotic behaviour of coefficients const:=subs(epsilon=1,op(4,C00sing))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZjb25zdEdGKCwkKigiIiQiIiIpRjAjRjEiIiVGMSwmKiQpIiInI0YxIiIjRjFGMSomRjpGMSlGMEY5RjEhIiJGMUYxNyNGLg==

evalf(const/GAMMA(-3/4));

JCIrQzpxJ0cpISM1

n:=50:plot([seq([1/i,coeff(C00ser(n),t,i)/6^i*i^(7/4)],i=1..n)],a=0..0.1,b=0.5..0.9,style=point);

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Ig==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv

On the diagonal (no periodicities for this model) Jy; Jyform

KiYsJiokKSwmKiYsJkkieUc2IiIiIiokRikhIiJGK0YrSSJ0R0YqRitGLUYrRisiIiNGK0YrKioiIiVGKylGLkYvRissJkYpRitGK0YrRissJkYrRitGLEYrRitGLUYrKSwmKiZGKUYrLUkjRERHRio2I0YpRitGKyomRjNGLUYuRi1GK0YvRis=

LCoqJCksKCokSSJ5RzYiISIiIiIiRidGKSooLCZGKkYqKiYiIiNGKkkidEdGKEYqRipGKkYvRiksJkYnRipGKkYqRilGKUYuRipGKiomSSR0QTJHRihGKkYlRipGKioyI0YqRi5GKkkkQUExR0YoRiosLComSSJOR0YoRiopRidGLkYqRikqJClGOEYuRipGKiooRi5GKkY4RipGJ0YqRikqJEY5RipGKkY4RilGKkYwRilGJ0YpRjhGKSwmRipGKUY4RipGKSlJJERweUdGKEY0RipGKUkkdEEwR0YoRio=

Jynew:=Jyform/Delta

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

Deltapx

LCgiIiJGIyosIiIjRiNJIk5HNiJGIywmKiQpRiZGJUYjRiNGI0YjRiMpLCZGI0YjRiYhIiJGJUYtSSJ4R0YnRiNGLSomKUYuRiVGI0YqRiNGIw==

map(collect,subs(sqrt(-(-y+7+4*sqrt(3))*(y-1))=sqrt(delta),map(factor,map(rationalize,map(simplify,map(factor,series(subs(Dpy=subs(x=y,Deltapx), tA2=tA2pap,tA0=tA0pap, t=1/6*(1-epsilon),M=Msing, N=Nsing,AA1=A1sing, Jynew),epsilon,1)))))),delta,factor)

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

res:=series(%,epsilon)

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

Let us examine the first two coefficients d0:=op(1,%)+op(2,%); d1:=subs(epsilon=1,op(3,%%))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkMEdGKCwmKjYiJDMiIiIiKSIiJCNGMSIiI0YxLCYqJClGNUY0RjFGMUYxISIiRjEsKEkieUdGKEY5RjVGOSokRjJGMUYxRjFGO0YxLChGO0YxIiIoRjkqJiIiJUYxRjJGMUYxRjksJkY7RjFGMUYxRjksKEY7RjlGPkYxRj9GMUY5LCZGO0YxRjFGOUY5KUkmZGVsdGFHRihGNEYxRjEqLiIjT0YxRkNGMSwuKipGM0YxRjJGMUY4RjEpRjtGNUYxRjEqKkYzRjFGMkYxRjhGMUY7RjFGMSokKUY7RjNGMUYxKiZGM0YxRkpGMUY5KiZGM0YxRjtGMUYxRjFGOUYxRj1GOUZCRjkpRkFGNUY5Rjk3I0Yu

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

We need to take the square root, and divide by y. First, consider the reciprocal of d0 collect(subs(sqrt(-(-y+7+4*sqrt(3))*(y-1))=sqrt(delta),factor(rationalize(subs(delta=-(-y+7+4*sqrt(3))*(y-1),1/d0)))),delta,factor);

LCYqNiMiIiIiIzdGJSkiIiQjRiUiIiNGJSwmKiQpRipGKUYlRiVGJSEiIkYlSSJ5RzYiRiUpLCZGL0YlRiVGJUYoRiUsKEYvRi4iIihGJSomIiIlRiVGJ0YlRiVGJSwoRi9GLkYqRi4qJEYnRiVGJUYlKSwoRi9GJUY4RiVGKkYlRipGLiksMioqRihGJUYnRiVGLUYlKUYvRipGJUYlKiQpRi9GKEYlRiUqKEYoRiVGPkYlRidGJUYuKipGKEYlRidGJUYtRiVGL0YlRiUqJkYoRiVGPkYlRi4qKEYoRiVGL0YlRidGJUYuKiZGKEYlRi9GJUYlRiVGLkYqRi4pSSZkZWx0YUdGMEYpRiVGLiowI0YlIiNPRiUsLkY9RiVGQkYlRj9GJUZDRi5GRUYlRiVGLkYlKUYyRipGJSksJkYvRiVGJUYuRipGJUYzRiVGOUYuRjtGLkYu

sd0_inv:=collect(simplify(sqrt(%),symbolic),delta)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShzZDBfaW52R0YoLCQqLiMiIiIiIidGMSwmSSJ5R0YoRjFGMUYxRjEpLChGNCEiIiIiKEYxKiYiIiVGMSkiIiQjRjEiIiNGMUYxRj1GMSksJiouRjxGMSwmRjxGNyomLCZGNEYxRj5GMUYxRjtGMUYxRjFGNEYxRjNGMSwmKiQpRj5GPUYxRjFGMUY3RjEpSSZkZWx0YUdGKEY9RjFGMSomLCYqLEY8RjFGNEYxRkdGMUYzRjFGO0YxRjEqJCksJkY0RjFGMUY3RjxGMUYxRjEpRk9GPkYxRjdGPUYxLCYqLEY8RjFGNEYxRkVGMUYzRjFGO0YxRjFGTUYxRjcsKEY0RjEqJEY7RjFGMUY+RjFGN0YxNyNGLg==

simplify(series(%,y,1))

KydJInlHNiIjISIiIiInIiIhLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiRi8=

We want the positive root c1:=-d1*sd0_inv/2/y;

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

factor(series(subs(delta=-(-y+7+4*sqrt(3))*(y-1),c1),y,4))

KytJInlHNiIsJiooIiIkIiIiKUYnI0YnIiIlRigpIiIjI0YoRi1GKEYoKiYiIidGKEYpRighIiIiIiEsKCooIiNDRihGKUYoRixGKEYoKigjIiNGRi1GKClGJyNGKEYrRihGLEYoRigqJiIjW0YoRilGKEYxRigsKComIyIlaj1GK0YoRilGKEYxKigiJGkiRihGOUYoRixGKEYoKigjRkAiIilGKEYpRihGLEYoRihGLS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRic=

evalf(%)

KytJInlHNiIkIStiSCJmKyUhIioiIiEkISlDKDQjcCEiKCIiIiQhKSM+SVcqRisiIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGLCIiJA==

Hdexpl:=-subs(delta=-(-y+7+4*sqrt(3))*(y-1),c1);

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

simplify(series(Hdexpl,y,5))

Ky1JInlHNiIsJCooIiIkIiIiKUYnI0YnIiIlRigsJiokKSIiIyNGKEYvRihGKEYvISIiRihGMSIiISwkKipGJ0YoLCoqJiIjVEYoRi5GKEYoKiYiI0tGKClGJ0YwRihGMSooIiNBRihGOkYoRi5GKEYoIiNrRjFGKEYpRigsJiomRi9GKEY6RihGKEYrRihGMUYxRigsJCosIyIjIikiIilGKCwqKigiJUZKRihGOkYoRi5GKEYoKiYiJTthRihGLkYoRigqJiIlaVdGKEY6RihGMSIlR3hGMUYoKUYnI0YoRitGKCksJkYvRigqJEY6RihGKEYvRjEsJiIiKEYoKiZGK0YoRjpGKEYoRjFGMUYvLCQqLCNGJyIjO0YoLCoiKldYQ1cjRjEqKCIpPFdoKipGKEY6RihGLkYoRigqJiIqSXNgcyJGKEYuRihGKComIipjWSxUIkYoRjpGKEYxRihGTUYoKUZQRidGMSlGUkYvRjFGMUYnLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGKw==

subs(sqrt(-(-y+7+4*sqrt(3))*(y-1))=s,H^2-Hdexpl^2);indets(%)

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

PCVJIkhHNiJJInNHRiRJInlHRiQ=

res0:=factor(resultant(numer(%%),s^2-(1-y)*(-y+7+4*sqrt(3)),s)); nops(res0)

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

IiIq

algHd:=collect(op(2,res0),H,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdIZEdGKCwoKiosKEkieUdGKCIiIiIiKCEiIiomIiIlRjIpIiIkI0YyIiIjRjJGMkYyKSwoRjFGNEYzRjJGNUYyRjpGMiksJkYxRjJGMkY0RjNGMilJIkhHRihGNkYyRjQqLiIkMyJGMkY3RjIsJiokKUY6RjlGMkYyRjpGNEYyRjxGMilGPiIiJkYyKUZARjpGMkYyKigiJVsoKUYyLCZGOEY0KiZGOkYyRkVGMkYyRjIpLChGMUYyKiRGN0YyRjJGOkYyRjpGMkYyNyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlIc3NvbGVuc0dGKDckLCQqKiIjYSIiIixAKigpIiIkI0YyIiIjRjIpRjhGN0YyKUkieUdGKEY2RjJGMioqIiIqRjJGNUYyRjlGMilGO0Y4RjIhIiIqKEY4RjJGNUYyRjpGMkY/KioiIzpGMkY1RjJGOUYyRjtGMkYyKigiIzdGMkY5RjJGPkYyRj8qKCIjPUYyRj5GMkY1RjJGMiooIiIoRjJGNUYyRjlGMkY/KigiI0NGMkY5RjJGO0YyRjIqKCIjSUYyRjtGMkY1RjJGPyomRkpGMkY+RjJGMiomRkRGMkY5RjJGPyomIiM5RjJGNUYyRjIqJCksViIkdClGMiooIiVPT0YyRjlGMkY7RjJGPyooIiU9PUYyRjlGMilGOyIiJUYyRj8qKCIlLyopRjJGOUYyRjpGMkYyKigiJDMiRjJGOUYyKUY7IiImRjJGMiooIiVxSEYyRjlGMkY+RjJGPyooIiRPJEYyRjVGMkY5RjJGPyoqIiUpMyZGMkY1RjJGOUYyRjpGMkYyKioiJS88RjJGNUYyRjlGMkY+RjJGPyoqIiUrQEYyRjVGMkY5RjJGO0YyRj8qKiIjJSlGMkY1RjJGOUYyRmluRjJGMioqIiVLNUYyRjVGMkY5RjJGWUYyRj8qKCIiJ0YyRjlGMilGO0Zqb0YyRj8qJkY9RjJGW3BGMkYyKiYiJGkiRjJGaW5GMkY/KiYiJCNlRjJGOUYyRj8qJiIkLyZGMkY1RjJGMiooIiVjREYyRj5GMkY1RjJGMiooIiVdSkYyRjtGMkY1RjJGMiooIiVLd0YyRjVGMkY6RjJGPyooIiRFIkYyRjVGMkZpbkYyRj8qKCIlWzpGMkY1RjJGWUYyRjIqJiIlYWFGMkY7RjJGMiomIiViV0YyRj5GMkYyKiYiJmNMIkYyRjpGMkY/KiYiJUZGRjJGWUYyRjJGN0YyRj8qJiIjW0YyRjtGMkY/RkpGMkYyLDQqKEZaRjJGNUYyRjpGMkYyKiRGWUYyRj8qKCIjZ0YyRj5GMkY1RjJGPyomIiNBRjJGOkYyRjIqKEZbckYyRjtGMkY1RjJGMiomIiQ/IkYyRj5GMkY/KiZGWkYyRjVGMkY/KiYiJDEiRjJGO0YyRjJGSEY/Rj8pLCZGO0YyRjJGP0Y2Rj9GMiwkKipGMUYyLEBGNEYyRjxGP0ZARj9GQUYyRkNGP0ZFRjJGR0Y/RklGMkZLRj9GTUYyRk5GP0ZPRjJGUUYyRmVxRj9GSkYyRjJGZ3FGP0ZkckY/RjI3I0Yu

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

NyQrK0kieUc2IiQiK2ZRdC87ISIpIiIhJCIrbkQnXGEmRigiIiIkIitzbGJONyEiKCIiIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YsIiIkKytGJCQiK3VWMmAkKkYoRikkIisjcCp6KHAiISInRiwkIisnNDRUWSMhIiZGMEYxRjc=

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

NyQrK0kieUc2IiQiK2BIImYrJSEiKiIiISQiK2hCKDQjcEYoIiIiJCIrQmUsViUqRigiIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGLCIiJCsrRiQkIiskUkg2bipGKEYpJCIrUC1ueCgpISIpRiwkIitoVThjKCkhIihGL0YwRjY=

Hssol:=(simplify(factor(op(1,Hssolens))) assuming y>-1/1000 assuming y<1/1000)

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Hdfinal:=sqrt(%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShIZGZpbmFsR0YoKiQpKiosJioqIiNhIiIiLCYqJCkiIiMjRjRGOEY0RjRGNCEiIkY0LCgqJiwoKiQpSSJ5R0YoRjhGNEY0KiYiIiZGNEZARjRGOkY4RjpGNCkiIiRGOUY0RjQqJiIjOkY0RkBGNEY6RkRGOkY0KSomLChGQEY0IiIoRjoqJiIiJUY0RkNGNEY6RjQsJkZARjRGNEY6RjRGOUY0RjoqKkYzRjQsJkY2RjRGOEY6RjQsJiIjN0Y6KiYsJkZARjRGSkY6RjRGQ0Y0RjRGNClGTUY4RjRGOkY0KUZNRkxGOiwoRj5GNComIiM5RjRGQEY0RjpGNEY0RjpGSUY6RjlGNDcjRi4=

Hsdelta:=subs(sqrt((y-7-4*sqrt(3))*(y-1))=sdelta,Hssol)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShIc2RlbHRhR0YoKiosJioqIiNhIiIiLCYqJCkiIiMjRjJGNkYyRjJGMiEiIkYyLCgqJiwoKiQpSSJ5R0YoRjZGMkYyKiYiIiZGMkY+RjJGOEY2RjhGMikiIiRGN0YyRjIqJiIjOkYyRj5GMkY4RkJGOEYySSdzZGVsdGFHRihGMkY4KipGMUYyLCZGNEYyRjZGOEYyLCYiIzdGOComLCZGPkYyIiIoRjhGMkZBRjJGMkYyKSwmRj5GMkYyRjhGNkYyRjhGMilGTiIiJUY4LChGPEYyKiYiIzlGMkY+RjJGOEYyRjJGOCwoRj5GMkZMRjgqJkZQRjJGQUYyRjhGODcjRi4=

factor(coeff(Hsdelta,sdelta,0))

LCQqLiIjYSIiIikiIiQjRiUiIiNGJSwmKiQpRilGKEYlRiVGKSEiIkYlLChJInlHNiJGLSIiKEYlKiYiIiVGJUYmRiVGJUYtLChGL0YlRjFGLUYyRiVGLSksJkYvRiVGJUYtRilGLUYl

factor(expand(denom(%)))

LCQqJiwoKiQpSSJ5RzYiIiIjIiIiRioqJiIjOUYqRidGKiEiIkYqRipGKiksJkYnRipGKkYtRilGKkYt

factor(coeff(Hsdelta,sdelta,1))

LCQqMCIjYSIiIikiIiQjRiUiIiNGJSwmKiQpRilGKEYlRiVGJSEiIkYlLChJInlHNiJGLUYpRi0qJEYmRiVGJUYlLChGL0YtIiIoRiUqJiIiJUYlRiZGJUYlRi0sKEYvRiVGM0YtRjRGJUYtKSwmRi9GJUYlRi1GNUYtRi0=

factor(expand(denom(%)))

LCQqJiwoKiQpSSJ5RzYiIiIjIiIiRioqJiIjOUYqRidGKiEiIkYqRipGKiksJkYnRipGKkYtIiIlRipGLQ==

rationalize((-y-2+sqrt(3))^2/(y-7+4*sqrt(3)))

LCQqKCksKEkieUc2IiEiIiIiI0YoKiQpIiIkIyIiIkYpRi5GLkYpRi4sKEYmRigiIihGLiomIiIlRi5GK0YuRi5GLiwoKiQpRiZGKUYuRi4qJiIjOUYuRiZGLkYoRi5GLkYoRig=

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

KyVJInlHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIio=

Hdspap:=54*sqrt(3)*(sqrt(2)-1)/(1-y)^2/(1-14*y+y^2)*(sqrt(2)-(2-sqrt (3) +y)/(1-y)*sqrt((7+4*sqrt(3)-y)/(1-y)))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdIZHNwYXBHRigsJCouIiNhIiIiKSIiJCNGMSIiI0YxLCYqJClGNUY0RjFGMUYxISIiRjEpLCZGMUYxSSJ5R0YoRjlGNUY5LCgqJClGPEY1RjFGMSomIiM5RjFGPEYxRjlGMUYxRjksJkY3RjEqKCwoRjxGMUY1RjEqJEYyRjFGOUYxRjtGOSkqJiwoRjxGOSIiKEYxKiYiIiVGMUYyRjFGMUYxRjtGOUY0RjFGOUYxRjE3I0Yu

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

KyVJInlHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

simplify(sqrt(sqrt(2)-1))

KiQpLCYqJCkiIiMjIiIiRidGKUYpRikhIiJGKEYp

Hdpap:=3^(7/4)*sqrt(2)*sqrt((sqrt(2)-1))/(1-y)/sqrt((1-14*y+y^2))*(sqrt(2)-(2-sqrt (3) +y)/(1-y)*sqrt((7+4*sqrt(3)-y)/(1-y)))^(1/2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIZHBhcEdGKCwkKjAiIiQiIiIpRjAjRjAiIiVGMSkiIiMjRjFGNkYxKSwmKiRGNUYxRjFGMSEiIkY3RjEsJkYxRjFJInlHRihGO0Y7KSwoKiQpRj1GNkYxRjEqJiIjOUYxRj1GMUY7RjFGMUY3RjspLCZGOkYxKigsKEY9RjFGNkYxKiQpRjBGN0YxRjtGMUY8RjspKiYsKEY9RjsiIihGMSomRjRGMUZJRjFGMUYxRjxGO0Y3RjFGO0Y3RjFGMTcjRi4=

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

KydJInlHNiIiIiIiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIiKg==

Hdpap

LCQqMCIiJCIiIilGJCNGJCIiJUYlKSIiIyNGJUYqRiUpLCYqJEYpRiVGJUYlISIiRitGJSwmRiVGJUkieUc2IkYvRi8pLCgqJClGMUYqRiVGJSomIiM5RiVGMUYlRi9GJUYlRitGLyksJkYuRiUqKCwoRjFGJUYqRiUqJClGJEYrRiVGL0YlRjBGLykqJiwoRjFGLyIiKEYlKiZGKEYlRj5GJUYlRiVGMEYvRitGJUYvRitGJUYl

Let us check this numerically First, at (0,0) LL:=70: i:=0: ser:=coeff(Dser(LL), y,i): plot([seq([1/n,-coeff(ser,t,n)/6^n*n^(7/4)*GAMMA(-3/4)],n=1..LL)],a=0..0.1,b=2..5,style=point);

NistJSdDVVJWRVNHNiQ3Ym83JCQiIzUhIiIkIiIhISIiNyQkIiImISIiJCIrLERNZUEhIio3JCQiK0xMTExMISM1JCIrJm8mUUM3ISIqNyQkIiNEISIjJCIrPEcvS0QhIio3JCQiIiMhIiIkIitgVSU9OyMhIio3JCQiK25tbW07ISM1JCIrR2BWd0UhIio3JCQiK0g5ZEc5ISM1JCIrYiJba2ojISIqNyQkIiREIiEiJCQiK1JhKnklRyEiKjckJCIrNjY2NjYhIzUkIis9X2olKkchIio3JCQiIiIhIiIkIisyVV8sSSEiKjckJCIrIjQ0NDQqISM2JCIqdiVSZUkhIik3JCQiK0xMTEwkKSEjNiQiKyg0dGE3JCEiKjckJCIrI3AyQnAoISM2JCIrKClSOXZKISIqNyQkIitWciZHOSghIzYkIitcUWZCSyEiKjckJCIrbm1tbW0hIzYkIip0dVZFJCEiKTckJCIkRCchIiUkIisjPnI/SSQhIio3JCQiMjErKzUlSE4jKWUhIz0kIisnKkhXTkwhIio3JCQiK2NiYmJiISM2JCIrKkhvZk8kISIqNyQkIismKnk6al8hIzYkIitAOmckUiQhIio3JCQiIiYhIiMkIitIeCQqPU0hIio3JCQiK2laIT53JSEjNiQiKyhbYEBXJCEiKjckJCIrWFhYWFghIzYkIitLeGJqTSEiKjckJCIrKDNFeU0lISM2JCIrJSozTCRbJCEiKjckJCIrbm1tbVQhIzYkIipNbjtdJCEiKTckJCIiJSEiIyQiKyk+OCg9TiEiKjckJCIrWVE6WVEhIzYkIitAWmdNTiEiKjckJCIrL1BxLlAhIzYkIiomW1hcTiEiKTckJCIrciZHOWQkISM2JCIrZVhPak4hIio3JCQiK2llRltNISM2JCIrQTRVd04hIio3JCQiK0xMTExMISM2JCIrUjZxKWUkISIqNyQkIitfayFlQSQhIzYkIis7SUYrTyEiKjckJCIlREohIiYkIissbz42TyEiKjckJCIqLi4uLiQhIzUkIiskKmZfQE8hIio3JCQiK3JrPFRIISM2JCIraCQzOGokISIqNyQkIitkRzlkRyEjNiQiK2JtZVNPISIqNyQkIit5eHh4RiEjNiQiK24jKlJcTyEiKjckJCIrLkZxLUYhIzYkIithMnlkTyEiKjckJCIrWip5OmojISM2JCIqTWlkbSQhIik3JCQiK2tENWtEISM2JCIrc0FQdE8hIio3JCQiI0QhIiQkIiskPk8xbyQhIio3JCQiKlJDIVJDISM1JCIrQnVkKG8kISIqNyQkIisiUV80USMhIzYkIitNc0AlcCQhIio3JCQiKyZSImVEQiEjNiQiK1ZdZCtQISIqNyQkIit0c3NzQSEjNiQiK1Enb21xJCEiKjckJCIrQUFBQUEhIzYkIitIVl43UCEiKjckJCIrVkkiUjwjISM2JCIrLnI3PVAhIio3JCQiK3UmZnc3IyEjNiQiK2kyX0JQISIqNyQkIitMTEwkMyMhIzYkIitGITMoR1AhIio3JCQiK0ZqIjMvIyEjNiQiK18xcUxQISIqNyQkIiIjISIjJCIrMiY0JlFQISIqNyQkIis5VnlnPiEjNiQiK3BZOVZQISIqNyQkIitCcDJCPiEjNiQiK3phaFpQISIqNyQkIitgQ3onKT0hIzYkIitIMSQ+diQhIio3JCQiK189Jj0mPSEjNiQiKyI+KTRjUCEiKjckJCIrPT09PT0hIzYkIispcEQsdyQhIio3JCQiKydHOWR5IiEjNiQiK3AsLWtQISIqNyQkIitsZlFhPCEjNiQiK2kiKXluUCEiKjckJCIrSno4QzwhIzYkIisxZVZyUCEiKjckJCIrYV8iXHAiISM2JCIrVylvXHgkISIqNyQkIitubW1tOyEjNiQiK1xFUnlQISIqNyQkIitpVU1SOyEjNiQiK3BBciJ5JCEiKjckJCIrRUshSGgiISM2JCIrVUMkXHkkISIqNyQkIisoZSx0ZSIhIzYkIippZCEpeSQhIik3JCQiJkRjIiEiJyQiKyMqPjQieiQhIio3JCQiK1E6WVE6ISM2JCIrLyZSU3okISIqNyQkIis6Ojo6OiEjNiQiK3dRIXB6JCEiKjckJCIrOHRgI1wiISM2JCIrOScpbyp6JCEiKjckJCIrTiMpZXE5ISM2JCIrTHFSLVEhIio3JCQiK2lgRlw5ISM2JCIraEEuMFEhIio3JCQiK0g5ZEc5ISM2JCIraXNmMlEhIiotJSZDT0xPUkc2JyUkUkdCRyQiKUMpZXElISIpJCIiISEiIiQiKCc+IVwmISIpLSUrX0FUVFJJQlVURUc2Iy8lJ3NvdXJjZUclLG1hdGhkZWZhdWx0Ry0lJVZJRVdHNiQ7JCIiISEiIiQiIiIhIiI7JCIjPyEiIiQiI10hIiItJiUmX0FYSVNHNiMiIiI2Ji0lJkNPTE9SRzYmJSRSR0JHJCIiISEiIiQiIiEhIiIkIiIhISIiLSUqTElORVNUWUxFRzYjIiIhLSUqVEhJQ0tORVNTRzYjIiIhLSUtVFJBTlNQQVJFTkNZRzYjJCIiISEiIi0mJSZfQVhJU0c2IyIiIzYmLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSpMSU5FU1RZTEVHNiMiIiEtJSpUSElDS05FU1NHNiMiIiEtJS1UUkFOU1BBUkVOQ1lHNiMkIiIhISIiLSUrQVhFU0xBQkVMU0c2JC1JI21pRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNjVRImE2Ii8lJ2ZhbWlseUdRITYiLyUlc2l6ZUdRIzEwNiIvJSVib2xkR1EmZmFsc2U2Ii8lJ2l0YWxpY0dRJXRydWU2Ii8lKnVuZGVybGluZUdRJmZhbHNlNiIvJSpzdWJzY3JpcHRHUSZmYWxzZTYiLyUsc3VwZXJzY3JpcHRHUSZmYWxzZTYiLyUrZm9yZWdyb3VuZEdRKFswLDAsMF02Ii8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdNiIvJSdvcGFxdWVHUSZmYWxzZTYiLyUrZXhlY3V0YWJsZUdRJmZhbHNlNiIvJSlyZWFkb25seUdRJmZhbHNlNiIvJSljb21wb3NlZEdRJmZhbHNlNiIvJSpjb252ZXJ0ZWRHUSZmYWxzZTYiLyUraW1zZWxlY3RlZEdRJmZhbHNlNiIvJSxwbGFjZWhvbGRlckdRJmZhbHNlNiIvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSZmYWxzZTYiLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWM2Ii1JI21pRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNjVRImI2Ii8lJ2ZhbWlseUdRITYiLyUlc2l6ZUdRIzEwNiIvJSVib2xkR1EmZmFsc2U2Ii8lJ2l0YWxpY0dRJXRydWU2Ii8lKnVuZGVybGluZUdRJmZhbHNlNiIvJSpzdWJzY3JpcHRHUSZmYWxzZTYiLyUsc3VwZXJzY3JpcHRHUSZmYWxzZTYiLyUrZm9yZWdyb3VuZEdRKFswLDAsMF02Ii8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdNiIvJSdvcGFxdWVHUSZmYWxzZTYiLyUrZXhlY3V0YWJsZUdRJmZhbHNlNiIvJSlyZWFkb25seUdRJmZhbHNlNiIvJSljb21wb3NlZEdRJmZhbHNlNiIvJSpjb252ZXJ0ZWRHUSZmYWxzZTYiLyUraW1zZWxlY3RlZEdRJmZhbHNlNiIvJSxwbGFjZWhvbGRlckdRJmZhbHNlNiIvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSZmYWxzZTYiLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWM2Ii0lJlNUWUxFRzYjJSZQT0lOVEctJSlfVklTSUJMRUc2IyIiIi0lJVJPT1RHNictJSlCT1VORFNfWEc2IyQiJF0lISIiLSUpQk9VTkRTX1lHNiMkIiRTIiEiIi0lLUJPVU5EU19XSURUSEc2IyQiJUlMISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJT9NISIiLSUpQ0hJTERSRU5HNiItJStBTk5PVEFUSU9ORzYnLSUpQk9VTkRTX1hHNiMkIiIhISIiLSUpQk9VTkRTX1lHNiMkIiIhISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlK1MhIiItJS5CT1VORFNfSEVJR0hURzYjJCIlK1MhIiItJSlDSElMRFJFTkc2Ig==Ig==

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

KydJInlHNiIsJCoqIiIkIiIiKUYnI0YnIiIlRigpIiIjI0YoRi1GKCwmKiRGLEYoRihGKCEiIkYoRigiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYo

KydJInlHNiIkIiteSCJmKyUhIioiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGLw==

At (1,1) LL:=70: i:=1: ser:=coeff(Dser(LL), y,i): plot([seq([1/n,-coeff(ser,t,n)/6^n*n^(7/4)*GAMMA(-3/4)],n=1..LL)],a=0..0.1,b=5..7,style=point);

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Ig==

simplify(series(Hdpap,y,2));evalf(%);

KylJInlHNiIsJCoqIiIkIiIiKUYnI0YnIiIlRigpIiIjI0YoRi1GKCwmKiRGLEYoRihGKCEiIkYoRigiIiEsJCosRidGKEYpRihGLEYoLCoqKCIja0YoKUYnRi5GKEYsRihGKCIkWyJGMSomIiQ3IkYoRixGKEYoKiYiIyYpRihGOEYoRjFGKCwmIiM5RigqJiIiKUYoRjhGKEYoRjFGKEYoLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGLQ==

KylJInlHNiIkIiteSCJmKyUhIioiIiEkIis9Qyg0I3BGJyIiIi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YrIiIj

At (2,2) LL:=70: i:=2: ser:=coeff(Dser(LL), y,i): plot([seq([1/n,-coeff(ser,t,n)/6^n*n^(7/4)*GAMMA(-3/4)],n=1..LL)],a=0..0.1,b=6..10,style=point);

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NiI=

simplify(series(Hdpap,y,3));evalf(%);

KytJInlHNiIsJCoqIiIkIiIiKUYnI0YnIiIlRigpIiIjI0YoRi1GKCwmKiRGLEYoRihGKCEiIkYoRigiIiEsJCosRidGKEYpRihGLEYoLCoqKCIja0YoKUYnRi5GKEYsRihGKCIkWyJGMSomIiQ3IkYoRixGKEYoKiYiIyYpRihGOEYoRjFGKCwmIiM5RigqJiIiKUYoRjhGKEYoRjFGKEYoLCQqLiMiI0ZGQUYoLCoqKCIlIUcqRihGOEYoRixGKEYoKiYiJnVnIkYoRixGKEYoKiYiJldKIkYoRjhGKEYxIiZuRiNGMUYoRixGKEYpRigpLCYiIihGKComRitGKEY4RihGKEYtRjFGL0YxRjFGLS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRic=

KytJInlHNiIkIiteSCJmKyUhIioiIiEkIis9Qyg0I3BGJyIiIiQiKzEpPUlXKkYnIiIjLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRisiIiQ=

On the negative x-axis (no periodicities for this model) Ix; subs(U(x,0)=1/x*Cm(x), Ix);Ixform

KiQpLC4qKiIiIyIiIkkidEc2IkYnLCZJInhHRilGJ0YnRidGJy1JIlVHRik2JEYrIiIhRidGJ0YnRidGK0YnKiRGKyEiIkYnKigsJkYnRicqJkYmRidGKEYnRidGJ0YoRjFGKkYxRjEqJkYoRictSSNEREdGKTYjRi9GJ0YnRiZGJw==

KiQpLC4qLCIiIyIiIkkidEc2IkYnLCZJInhHRilGJ0YnRidGJ0YrISIiLUkjQ21HRik2I0YrRidGJ0YnRidGK0YnKiRGK0YsRicqKCwmRidGJyomRiZGJ0YoRidGJ0YnRihGLEYqRixGLComRihGJy1JI0RER0YpNiMiIiFGJ0YnRiZGJw==

LCoqMiMiIiIiIiNGJUkkQUExRzYiRiUsLComSSJOR0YoRiUpSSJ4R0YoRiZGJSEiIiokKUYrRiZGJUYlKihGJkYlRitGJUYtRiVGLiokRixGJUYlRitGLkYlLCZGLUYlRiVGJUYuRi1GLkYrRi4sJkYlRi5GK0YlRi4pSSREcHhHRihGJEYlRiUqJCksKCokRi1GLkYlRi1GLiooLCZGJUYlKiZGJkYlSSJ0R0YoRiVGJUYlRj5GLkYzRi5GLkYmRiVGJSomRjlGJUkkdEEyR0YoRiVGJUkkdEEwR0YoRiU=

map(collect,subs(sqrt(-(-x+7+4*sqrt(3))*(x-1))=sqrt(delta),map(factor,map(rationalize,map(simplify,map(factor,series(subs(Dpx=Deltapx, tA2=tA2pap,tA0=tA0pap, t=1/6*(1-epsilon),M=Msing, N=Nsing,AA1=A1sing, Ixform/t^2),epsilon,1)))))),delta,factor)

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

res:=series(%,epsilon)

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

Let us examine the first two coefficients d0:=collect(subs(epsilon=0,convert(res,polynom)),delta,factor); d1:=collect(subs(epsilon=1,op(6,res)),delta,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkMEdGKCwmKjIiJDMiIiIiKSIiJCNGMSIiI0YxLCYqJClGNUY0RjFGMUYxISIiRjEsKEkieEdGKEY5RjVGOSokRjJGMUYxRjEsJkY7RjFGMUY5RjFGO0Y5LCZGO0YxRjFGMUY5KUkmZGVsdGFHRihGNEYxRjEqLCIjT0YxLC4qKEYzRjEpIiInRjRGMSlGO0Y1RjFGMSokKUY7RjNGMUYxKihGM0YxRkVGMUY7RjFGMSomRjNGMUZHRjFGOSomRjNGMUY7RjFGMUYxRjlGMSlGPUYzRjFGR0Y5KUY+RjVGOUYxNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNkMUdGKCwmKjIiI08iIiIpIiIkI0YzIiIlRjEsJiokKSIiIyNGMUY5RjFGMUYxISIiRjEsKEkieEdGKEY7RjlGOyokKUYzRjpGMUYxRjEsJkY9RjFGMUY7RjFGPUY7LCZGPUYxRjFGMUY7KUkmZGVsdGFHRihGOkYxRjsqLEYwRjFGMkYxLDJGMUY7KioiIidGMUY/RjFGOEYxKUY9RjlGMUYxKipGR0YxRj9GMUY4RjFGPUYxRjEqJkYzRjFGPUYxRjEqKEZHRjFGPUYxRj9GMUY7KiQpRj1GM0YxRjEqJkYzRjFGSEYxRjsqKEZHRjFGSEYxRj9GMUY7RjFGPUY7RkFGO0YxNyNGLg==

We need to take the square root, and divide by 2(1+x). First, consider the reciprocal of d0 d0_inv:=collect(subs(sqrt(-(-x+7+4*sqrt(3))*(x-1))=sqrt(delta),factor(rationalize(subs(delta=-(-x+7+4*sqrt(3))*(x-1),1/d0)))),delta,factor);

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

series(d0_inv,x,3); evalf(%)

KydJInhHNiIsJCooIyIiIiIjT0YoLCYiIighIiIqJiIiJUYoKSIiJCNGKCIiI0YoRihGLCksJkYyRigqJEYvRihGKEYyRixGLEYyLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGMA==

KydJInhHNiIkIitYeXh4RiEjNiIiIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIiJA==

sd0_inv:=(collect(simplify(sqrt(d0_inv),symbolic),delta))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShzZDBfaW52R0YoLCQqMiMiIiIiIidGMUkieEdGKEYxLCZGM0YxRjFGMUYxKSwoRjNGMSomIiIlRjEpIiIkI0YxIiIjRjFGMSIiKCEiIkY7Rj4pLCYqLkY6RjEsJkY6Rj4qJiwmRjNGMUY8RjFGMUY5RjFGMUYxRjNGMUY0RjEsJiokKUY8RjtGMUYxRjFGPkYxKUkmZGVsdGFHRihGO0YxRjEqJiwmKixGOkYxRjNGMUZHRjFGNEYxRjlGMUYxKiQpLCZGM0YxRjFGPkY6RjFGMUYxKUZPRjxGMUYxRjtGMSwmKixGOkYxRjNGMUZFRjFGNEYxRjlGMUYxRk1GMUY+LChGM0YxKiRGOUYxRjFGPEYxRj5GT0Y+RjE3I0Yu

simplify(series(subs(delta=-(-x+7+4*sqrt(3))*(x-1), sd0_inv),x,2)) assuming x<1/1000 assuming x>-1/1000

KydJInhHNiIjIiIiIiInRiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJiIiIw==

We want the positive root c1:=d1*sd0_inv*x/4/(1+x);

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

factor(series(subs(delta=-(-x+7+4*sqrt(3))*(x-1),c1),x,4)) assuming x<1/1000

KytJInhHNiIsJCooIyIiJCIiIyIiIilGKCNGKCIiJUYqKUYpI0YqRilGKiEiIkYqLCgqKCMiI0ZGLUYqKUYoI0YqRi1GKkYuRipGKiomI0Y0RilGKkY1RipGMEYmRipGKSwoKiYjIiMiKUYpRipGNUYqRjAqKCNGPEYtRipGNUYqRi5GKkYqKigjIiNkIiM7RipGK0YqRi5GKkYqRigtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKkYt

evalf(%)

KytJInhHNiIkISsjcGtiJFshIioiIiIkISopKSlvI28kRiciIiMkISpnTXE3JSEiKSIiJC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIl

This is not yet the GF of the harmonic function, because of the term D0 in Ix Gmexpl:=-subs(delta=-(-x+7+4*sqrt(3))*(x-1),c1);

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

simplify(series(Gmexpl,x,5)) assuming x<1/11000

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

subs(sqrt(-(-x+7+4*sqrt(3))*(x-1))=s,G^2-Gmexpl^2);indets(%)

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

PCVJIkdHNiJJInNHRiRJInhHRiQ=

res0:=factor(resultant(numer(%%),s^2-(1-x)*(-x+7+4*sqrt(3)),s)); nops(res0)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVyZXMwR0YoLCQqMCIiJSIiIixcbyooIiZtJFJGMSkiIiQjRjEiIiNGMSlJInhHRigiIiZGMUYxKioiJDsjRjEpSSJHR0YoRjhGMUY1RjEpRjpGOEYxRjEqKiIkSyVGMUY+RjFGNUYxRjlGMUYxKipGQkYxRj5GMUY1RjEpRjpGMEYxISIiKipGPUYxRj5GMUY1RjEpRjpGNkYxRkUqKCImQUoiRjEpRjhGN0YxKUY6IiInRjFGMSooRklGMUZKRjFGOUYxRkUqKEY0RjFGNUYxRktGMUYxKigiJXVWRjFGSkYxKUY6IiIoRjFGRSooRjBGMSlGP0YwRjFGUUYxRjEqKEYwRjFGVEYxRktGMUYxKigiIzdGMUZURjFGOUYxRkUqKEZXRjFGVEYxRkRGMUZFKihGV0YxRlRGMUZHRjFGMSooRldGMUZURjFGQEYxRjEqKEYwRjFGVEYxRjpGMUZFKixGPUYxRj5GMUZKRjFGNUYxRjlGMUZFKixGPUYxRj5GMUZKRjFGNUYxRkRGMUYxKiwiJDMiRjFGPkYxRkpGMUY1RjFGR0YxRjEqLEZpbkYxRj5GMUZKRjFGNUYxRkBGMUZFKioiJldpI0YxRkpGMUY1RjFGOUYxRkUqLEZpbkYxRj5GMUZKRjFGNUYxRlFGMUYxKixGaW5GMUY+RjFGSkYxRjVGMUZLRjFGRSomRjBGMUZURjFGRSomIiVobEYxRlFGMUYxKiYiJiRvPkYxRktGMUZFKiZGY29GMUY5RjFGMSoqRlxvRjFGNUYxRkpGMUZLRjFGRSoqRj1GMUY+RjFGNUYxRlFGMUZFKipGPUYxRj5GMUY1RjFGS0YxRjEqKEZQRjFGSkYxRkRGMUYxKiZGYW9GMUZERjFGRUYxLCZGOkYxRjFGRUYxKSwoRjpGMSomRjBGMUY1RjFGMUZSRkVGOEYxKSwyKipGNkYxRjVGMUZKRjFGQEYxRjEqKkY2RjFGNUYxRkpGMUY6RjFGMSooRjZGMUZARjFGNUYxRkUqJEZHRjFGMSooRjZGMUY6RjFGNUYxRkUqJkY2RjFGQEYxRkUqJkY2RjFGOkYxRjFGMUZFRjBGMSksKEY6RjEqJEY1RjFGMUY4RjFGMEYxRkRGMUYxNyNGLg==

IiIo

algGm:=collect(op(2,res0),G,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdHbUdGKCwoKioiIiUiIiIpLCZJInhHRihGMUYxISIiIiIkRjEpLCZGNEYxRjFGMUYwRjEpSSJHR0YoRjBGMUYxKjAiJDMiRjEpRjYjRjEiIiNGMSwmKiQpRj9GPkYxRjFGP0Y1RjEpRjRGP0YxKUY4Rj9GMUYyRjEpRjpGP0YxRjEqLCIlKD0jRjEsJkY2RjUqJkY/RjFGQkYxRjFGMSwoRjRGMSomRjBGMUY9RjFGMSIiKEY1RjEpLChGNEYxKiRGPUYxRjFGP0YxRj9GMSlGNEYwRjFGNTcjRi4=

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

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

map(simplify,map(series,Gssolens,x,3))

NyQrJ0kieEc2IiwkKiYjIiNGIiIjIiIiKSIiJCNGK0YqRitGK0YqLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRitGLSsnRiQsJiooRilGK0YsRispRipGLkYrISIiKiYjIiMiKUYqRitGLEYrRitGKkYvRi0=

evalf(%)

NyQrJ0kieEc2IiQiKyJmbyNRQiEiKSIiIy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyIiIiIiJCsnRiQkIiptTT0sJUYoRilGKkYx

map(Z->series(sqrt(Z),x),%) assuming x>0

NyQrJ0kieEc2IiQiKyVwa2IkWyEiKiIiIi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YpIiIjKydGJCQiKyFbY0grI0YoRilGKkYw

Gssol:=(simplify(factor(op(1,Gssolens))) assuming x>-1/1000 assuming x<1/1000)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZHc3NvbEdGKCwkKi4jIiNGIiIjIiIiKUkieEdGKEYyRjMpIiIkI0YzRjJGMywmKigsJiokKUYyRjhGM0YzRjMhIiJGMywoRjVGM0YyRjMqJEY2RjNGPkYzKSomLChGNUYzIiIoRj4qJiIiJUYzRjZGM0Y+RjMsJkY1RjNGM0Y+RjNGOEYzRj4qJilGR0YyRjMsJkY8RjNGMkY+RjNGM0YzRklGPiksJkY1RjNGM0YzRjJGPkY+NyNGLg==

Gmfinal:=simplify(sqrt(Gssol)) assuming x>-1/1000 assuming x<1/1000

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShHbWZpbmFsR0YoLCQqMCIiJCIiIikiIiMjRjFGM0YxKUYwI0YwIiIlRjEpLCYqKCwmKiRGMkYxRjFGMSEiIkYxLChJInhHRihGMUYzRjEqJClGMEY0RjFGPUYxKSomLChGP0YxIiIoRj0qJkY3RjFGQUYxRj1GMSwmRj9GMUYxRj1GMUY0RjFGMSomKUZHRjNGMSwmRjxGMUYzRj1GMUY9RjRGMS1JJGFic0dGJjYjRj9GMUZHRj0sJiomRjNGMUY/RjFGMUYzRjFGPUY9NyNGLg==

Gsdelta:=subs(sqrt((x-7-4*sqrt(3))*(x-1))=sdelta,Gssol)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShHc2RlbHRhR0YoLCQqLiMiI0YiIiMiIiIpSSJ4R0YoRjJGMykiIiQjRjNGMkYzLCYqKCwmKiQpRjJGOEYzRjNGMyEiIkYzLChGNUYzRjJGMyokRjZGM0Y+RjNJJ3NkZWx0YUdGKEYzRj4qJiksJkY1RjNGM0Y+RjJGMywmRjxGM0YyRj5GM0YzRjNGQ0Y+KSwmRjVGM0YzRjNGMkY+Rj43I0Yu

factor(coeff(Gsdelta,sdelta,0))

LCQqLCMiI0YiIiMiIiIpSSJ4RzYiRiZGJykiIiQjRidGJkYnLCYqJClGJkYtRidGJ0YmISIiRicpLCZGKUYnRidGJ0YmRjFGMQ==

factor(coeff(Gsdelta,sdelta,1))

LCQqMCMiI0YiIiMiIiIpSSJ4RzYiRiZGJykiIiQjRidGJkYnLCYqJClGJkYtRidGJ0YnISIiRicsKEYpRjFGJkYxKiRGK0YnRidGJyksJkYpRidGJ0YxRiZGMSksJkYpRidGJ0YnRiZGMUYx

simplify(series(Gmexpl-Gmfinal,x,10)) assuming x>0

KyVJInhHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

Gmpap:=3^(7/4)*x*sqrt(sqrt(2)-1)/sqrt(2)/(1+x)*sqrt((sqrt(2)+sqrt((1-x)*(7+4*sqrt(3)-x))*(x+2-sqrt(3))/(1-x)^2))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZHbXBhcEdGKCwkKjAjIiIkIiIjIiIiKUYxI0YxIiIlRjNJInhHRihGMyksJiokKUYyI0YzRjJGM0YzRjMhIiJGPEYzRjtGMywmRjdGM0YzRjNGPSksJkY6RjMqKCkqJiwmRjdGPUYzRjNGMywoRjdGPSIiKEYzKiZGNkYzKUYxRjxGM0YzRjNGPEYzLChGN0YzRjJGMyokRkhGM0Y9RjMpRkRGMkY9RjNGPEYzRjM3I0Yu

simplify(series(Gmexpl/Gmpap,x,10)) assuming x>0

KydJInhHNiIiIiIiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNQ==

Gmpap

LCQqMCMiIiQiIiMiIiIpRiUjRiUiIiVGJ0kieEc2IkYnKSwmKiQpRiYjRidGJkYnRidGJyEiIkYxRidGMEYnLCZGK0YnRidGJ0YyKSwmRi9GJyooKSomLCZGK0YyRidGJ0YnLChGK0YyIiIoRicqJkYqRicpRiVGMUYnRidGJ0YxRicsKEYrRidGJkYnKiRGPUYnRjJGJylGOUYmRjJGJ0YxRidGJw==

simplify(series(Gmpap,x,1))

KyVJInhHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGKw==

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

LCgtSSNDbUc2IjYjSSJ4R0YlIiIiKiojRigiIiNGKEYnRigsJkYnRihGKEYoISIiLUkjRERHRiU2IyIiIUYoRigqKkYqRigsKiomSSJ0R0YlRigpRiciIiRGKEYoKihGK0YoRjVGKClGJ0YrRihGKEY1RihGJ0YtRigpRjVGK0YtKUYsRitGLUYo

C00sing

LC4iI0kiIiIqJiIjPUYkKSIiJyNGJCIiI0YkISIiKiYiIipGJCkiIiRGKUYkRiQqKkYvRiQpRi8jRiQiIiVGJCwmKiRGJ0YkRiQqJkYqRiRGLkYkRitGJClJKGVwc2lsb25HNiIjRi9GM0YkRiQqKCNGL0YqRiQsKCooRiZGJEYuRiQpRipGKUYkRiRGLEYrIiNLRitGJEY4RiRGKy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRjk2IyokKUY4IyIiJkYzRiRGJA==

Hmexpl:=Gmpap+x*3*3^(1/4)*(sqrt(6)-2*sqrt(3))/2/(1+x)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdIbWV4cGxHRigsJiowIyIiJCIiIyIiIilGMSNGMSIiJUYzSSJ4R0YoRjMpLCYqJClGMiNGM0YyRjNGM0YzISIiRjxGM0Y7RjMsJkY3RjNGM0YzRj0pLCZGOkYzKigpKiYsJkY3Rj1GM0YzRjMsKEY3Rj0iIihGMyomRjZGMylGMUY8RjNGM0YzRjxGMywoRjdGM0YyRjMqJEZIRjNGPUYzKUZERjJGPUYzRjxGM0YzKixGMEYzRjdGMylGMSNGM0Y2RjMsJiokKSIiJ0Y8RjNGMyomRjJGM0ZIRjNGPUYzRj5GPUYzNyNGLg==

Hmpap:=3^(7/4)*x*sqrt(sqrt(2)-1)/sqrt(2)/(1+x)*(sqrt(sqrt(2)+(2-sqrt(3)+x)/(1-x)*sqrt((7+4*sqrt(3)-x)/(1-x)))-sqrt(sqrt(2)-1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIbXBhcEdGKCwkKjAjIiIkIiIjIiIiKUYxI0YxIiIlRjNJInhHRihGMyksJiokKUYyI0YzRjJGM0YzRjMhIiJGPEYzRjtGMywmRjdGM0YzRjNGPSwmKiQpLCZGOkYzKigsKEY3RjNGMkYzKiQpRjFGPEYzRj1GMywmRjdGPUYzRjNGPSkqJiwoRjdGPSIiKEYzKiZGNkYzRkZGM0YzRjNGR0Y9RjxGM0YzRjxGM0YzKiRGOEYzRj1GM0YzNyNGLg==

simplify(series(Hmexpl/Hmpap,x,10))

KydJInhHNiIiIiIiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIiKg==

Let us check for some points (-i,0) at (-1,0) LL:=50: i:=1: ser:=coeff(subs(x=1/x,Cmser(LL)), x,i): plot([seq([1/n,-coeff(ser,t,n)/6^n*n^(7/4)*GAMMA(-3/4)],n=1..LL)],a=0..0.1,b=2..3,style=point);

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NiI=

simplify(series(Hmexpl,x,2)) assuming x<11000;evalf(%);

KydJInhHNiIsJCooIiIkIiIiKUYnI0YnIiIlRigsJiokKSIiIyNGKEYvRihGKEYoISIiRihGKEYoLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGLw==

KydJInhHNiIkIis4IzNFJEchIioiIiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIiIw==

at (-2,0) i:=2: ser:=coeff(subs(x=1/x,Cmser(LL)), x,i): plot([seq([1/n,-coeff(ser,t,n)/6^n*n^(7/4)*GAMMA(-3/4)],n=1..LL)],a=0..0.1,b=1..3,style=point);

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Ig==

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

KylJInhHNiIsJCooIiIkIiIiKUYnI0YnIiIlRigsJiokKSIiIyNGKEYvRihGKEYoISIiRihGKEYoLCQqKEYqRigpRicjRihGK0YoLCoqKEYrRigpRidGMEYoRi5GKEYoKiZGK0YoRjhGKEYxKiYiIipGKEYuRihGKCIjPUYxRihGMUYvLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGJw==

KylJInhHNiIkIis4IzNFJEchIioiIiIkIitrYEFyQkYnIiIjLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIiQ=

res:=map(simplify,series(subs(t=tN,N=Nsing,Qx0),epsilon,2)); nops(%)

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

IiIm

c1:=simplify(factor(expand(rationalize(subs(epsilon=1,op(4,res))))))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjMUdGKCwkKiojIiIiIiIjRjEsJiomLCgqJiIiJEYxSSJ4R0YoRjFGMSomRjdGMSlGN0YwRjEhIiIiIidGMUYxKSomLChGOEYxIiIoRjsqJiIiJUYxRjpGMUY7RjEsJkY4RjFGMUY7RjFGMEYxRjEqJkY3RjEpRkNGMkYxRjFGMUZFRjssJkY4RjFGMUYxRjtGMTcjRi4=

Hx:=c1

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNIeEdGKCwkKiojIiIiIiIjRjEsJiomLCgqJiIiJEYxSSJ4R0YoRjFGMSomRjdGMSlGN0YwRjEhIiIiIidGMUYxKSomLChGOEYxIiIoRjsqJiIiJUYxRjpGMUY7RjEsJkY4RjFGMUY7RjFGMEYxRjEqJkY3RjEpRkNGMkYxRjFGMUZFRjssJkY4RjFGMUYxRjtGMTcjRi4=

Hxpap:=3/2/(1+x)*((2-sqrt(3)+x)/(1-x)*sqrt((7+4*sqrt(3)-x)/(1-x))+1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIeHBhcEdGKCwkKigjIiIkIiIjIiIiLCZJInhHRihGM0YzRjMhIiIsJiooLChGNUYzRjJGMyokKUYxI0YzRjJGM0Y2RjMsJkY1RjZGM0YzRjYpKiYsKEY1RjYiIihGMyomIiIlRjNGO0YzRjNGM0Y9RjZGPEYzRjNGM0YzRjNGMzcjRi4=

simplify(series(Hx/Hxpap,x,10))

KydJInhHNiIiIiIiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNQ==

Let us check Qser(5)

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

N:=50: i:=0: ser:=coeff(subs(y=0,convert(Qser(N),polynom)), x,i): plot([seq([1/n,coeff(ser,t,n)/6^n*n^(5/2)*GAMMA(-3/2)],n=1..N)],a=0..0.1,b=1..4,style=point);

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Ig==

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

KydJInhHNiIiIiQiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGLQ==

KydJInhHNiIkIiIkIiIhRictSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiJGLg==

N:=70: i:=1: ser:=coeff(subs(y=0,convert(Qser(N),polynom)), x,i): plot([seq([1/n,coeff(ser,t,n)/6^n*n^(5/2)*GAMMA(-3/2)],n=1..N)],a=0..0.1,b=3..5,style=point);

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Ig==

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

KylJInhHNiIiIiQiIiEqJiwmKiYiI1IiIiIpRiUjRisiIiNGK0YrIiNtRitGKywmIiM5RisqJiIiKUYrRixGK0YrISIiRistSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGK0Yu

KylJInhHNiIkIiIkIiIhRickIitNJ0dVeiUhIioiIiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKyIiIw==

N:=100: i:=2: ser:=coeff(subs(y=0,convert(Qser(N),polynom)), x,i): plot([seq([1/n,coeff(ser,t,n)/6^n*n^(5/2)*GAMMA(-3/2)],n=1..N)],a=0..0.1,b=4..6.5,style=point);

NistJSdDVVJWRVNHNiQ3YHE3JCQiIzUhIiIkIiIhISIiNyQkIiImISIiJCIrb0xfOFAhIzU3JCQiK0xMTExMISM1JCIrVk5qO14hIzU3JCQiI0QhIiMkIitEaiEqPioqISM1NyQkIiIjISIiJCIrUChRLUMiISIqNyQkIitubW1tOyEjNSQiKjRPLGYiISIpNyQkIitIOWRHOSEjNSQiKUN0XT0hIig3JCQiJEQiISIkJCIrL2NBN0AhIio3JCQiKzY2NjY2ISM1JCIrNCYpPlFCISIqNyQkIiIiISIiJCIrInkob1hEISIqNyQkIisiNDQ0NCohIzYkIisnKXlvS0YhIio3JCQiK0xMTEwkKSEjNiQiKyUqPUctSCEiKjckJCIrI3AyQnAoISM2JCIqPCkqbzAkISIpNyQkIitWciZHOSghIzYkIislZS15PiQhIio3JCQiK25tbW1tISM2JCIreVQsRkwhIio3JCQiJEQnISIlJCIqeiJmWE0hIik3JCQiMjErKzUlSE4jKWUhIz0kIitpQyFcYiQhIio3JCQiK2NiYmJiISM2JCIrVFQmZWwkISIqNyQkIismKnk6al8hIzYkIittWVJcUCEiKjckJCIiJiEiIyQiK3JzRE9RISIqNyQkIitpWiE+dyUhIzYkIiswOjg8UiEiKjckJCIrWFhYWFghIzYkIis6SmYjKlIhIio3JCQiKygzRXlNJSEjNiQiKyUpPjtqUyEiKjckJCIrbm1tbVQhIzYkIitBekdIVCEiKjckJCIiJSEiIyQiKkdzOD4lISIpNyQkIitZUTpZUSEjNiQiKlZvKFxVISIpNyQkIisvUHEuUCEjNiQiKyhcInovViEiKjckJCIrciZHOWQkISM2JCIrRzxzY1YhIio3JCQiK2llRltNISM2JCIrLyo0ZVMlISIqNyQkIitMTExMTCEjNiQiKz4xR19XISIqNyQkIitfayFlQSQhIzYkIit3ZEwnXCUhIio3JCQiJURKISImJCIrRHM6UVghIio3JCQiKi4uLi4kISM1JCIrRiM0emQlISIqNyQkIityazxUSCEjNiQiK2EvdTpZISIqNyQkIitkRzlkRyEjNiQiMjArKyE9ZXleWSEjOzckJCIreXh4eEYhIzYkIislKXo7J28lISIqNyQkIisuRnEtRiEjNiQiKzgoKSoqPVohIio3JCQiK1oqeTpqIyEjNiQiK2krUV1aISIqNyQkIitrRDVrRCEjNiQiK3NgUyF5JSEiKjckJCIjRCEiJCQiKzEtOzRbISIqNyQkIipSQyFSQyEjNSQiKzRKc09bISIqNyQkIisiUV80USMhIzYkIitFajtqWyEiKjckJCIrJlIiZURCISM2JCIrLWtiKSlbISIqNyQkIit0c3NzQSEjNiQiKnZhSCJcISIpNyQkIitBQUFBQSEjNiQiKzoiPWskXCEiKjckJCIrVkkiUjwjISM2JCIrOyEqKiplXCEiKjckJCIrdSZmdzcjISM2JCIrRGh1ISlcISIqNyQkIitMTEwkMyMhIzYkIis9WXEsXSEiKjckJCIrRmoiMy8jISM2JCIrd2siPi0mISIqNyQkIiIjISIjJCIrajJVVF0hIio3JCQiKzlWeWc+ISM2JCIrIylRRGddISIqNyQkIitCcDJCPiEjNiQiKnlcJXldISIpNyQkIitgQ3onKT0hIzYkIitqLC8nNCYhIio3JCQiK189Jj0mPSEjNiQiK3ZZMDheISIqNyQkIis9PT09PSEjNiQiK281X0heISIqNyQkIisnRzlkeSIhIzYkIitLYFlYXiEiKjckJCIrbGZRYTwhIzYkIitgPSI0OyYhIio3JCQiK0p6OEM8ISM2JCIrRE4pZTwmISIqNyQkIithXyJccCIhIzYkIitmPVMhPiYhIio3JCQiK25tbW07ISM2JCIrJDQoWy9fISIqNyQkIitpVU1SOyEjNiQiKyRHZSI9XyEiKjckJCIrRUshSGgiISM2JCIrdUxWSl8hIio3JCQiKyhlLHRlIiEjNiQiKzEkSFZDJiEiKjckJCImRGMiISInJCIqMWlvRCYhIik3JCQiK1E6WVE6ISM2JCIrTm4vcF8hIio3JCQiKzo6Ojo6ISM2JCIrN3cqM0cmISIqNyQkIis4dGAjXCIhIzYkIjIwKytxPUdDSCYhIzs3JCQiK04jKWVxOSEjNiQiMjArKytFXk9JJiEjOzckJCIraWBGXDkhIzYkIitVKnlYSiYhIio3JCQiK0g5ZEc5ISM2JCIrSkZBRGAhIio3JCQiKy8yWDM5ISM2JCIrQ05mTmAhIio3JCQiKyopKSkpKSlRIiEjNiQiKyNvLGRNJiEiKjckJCIrOUknKXA4ISM2JCIrTHFiYmAhIio3JCQiK144Tl44ISM2JCIrUipvXk8mISIqNyQkIitMTExMOCEjNiQiKyJIWVhQJiEiKjckJCIrdSUqeTo4ISM2JCIranZwJFEmISIqNyQkIisqSCwoKUgiISM2JCIrPjNqI1ImISIqNyQkIisjR14/RyIhIzYkIip2YDhTJiEiKTckJCIrJnlBZUUiISM2JCIrKm90KTRhISIqNyQkIiREIiEiJSQiK0t3Pj1hISIqNyQkIissemNNNyEjNiQiK2JBTEVhISIqNyQkIismPjcmPjchIzYkIitPUkdNYSEiKjckJCIreCM+Wz8iISM2JCIrdChlP1cmISIqNyQkIio+dy8+IiEjNSQiKydmaSdcYSEiKjckJCIrKWVxazwiISM2JCIrdDQ1ZGEhIio3JCQiKylwIXppNiEjNiQiK1UjelZZJiEiKjckJCIrKEdEJVw2ISM2JCIrPURdcmEhIio3JCQiK09PT082ISM2JCIrKm92JXlhISIqNyQkIisxYmZCNiEjNiQiK1xNSSZbJiEiKjckJCIrNjY2NjYhIzYkIipIISo+XCYhIik3JCQiKyo0LCopNCIhIzYkIipgUyYpXCYhIik3JCQiK0FsJnAzIiEjNiQiKzQkZVxdJiEiKjckJCIrPClvXzIiISM2JCIrKWZaN14mISIqNyQkIisoeUhRMSIhIzYkIis6QVQ8YiEiKjckJCIrejpqXzUhIzYkIiokZVhCYiEiKTckJCIrbm1tVDUhIzYkIitsPlFIYiEiKjckJCIrTnkjNC4iISM2JCIrN1M+TmIhIio3JCQiK2oiMy8tIiEjNiQiK0hfKjNhJiEiKjckJCIqLCwsLCIhIzUkIitlKClbWWIhIio3JCQiIiIhIiMkIis5dyg+YiYhIiotJSZDT0xPUkc2JyUkUkdCRyQiKUMpZXElISIpJCIiISEiIiQiKCc+IVwmISIpLSUrX0FUVFJJQlVURUc2Iy8lJ3NvdXJjZUclLG1hdGhkZWZhdWx0Ry0lJVZJRVdHNiQ7JCIiISEiIiQiIiIhIiI7JCIjUyEiIiQiI2whIiItJiUmX0FYSVNHNiMiIiI2Ji0lJkNPTE9SRzYmJSRSR0JHJCIiISEiIiQiIiEhIiIkIiIhISIiLSUqTElORVNUWUxFRzYjIiIhLSUqVEhJQ0tORVNTRzYjIiIhLSUtVFJBTlNQQVJFTkNZRzYjJCIiISEiIi0mJSZfQVhJU0c2IyIiIzYmLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSpMSU5FU1RZTEVHNiMiIiEtJSpUSElDS05FU1NHNiMiIiEtJS1UUkFOU1BBUkVOQ1lHNiMkIiIhISIiLSUrQVhFU0xBQkVMU0c2JC1JI21pRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNjVRImE2Ii8lJ2ZhbWlseUdRITYiLyUlc2l6ZUdRIzEwNiIvJSVib2xkR1EmZmFsc2U2Ii8lJ2l0YWxpY0dRJXRydWU2Ii8lKnVuZGVybGluZUdRJmZhbHNlNiIvJSpzdWJzY3JpcHRHUSZmYWxzZTYiLyUsc3VwZXJzY3JpcHRHUSZmYWxzZTYiLyUrZm9yZWdyb3VuZEdRKFswLDAsMF02Ii8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdNiIvJSdvcGFxdWVHUSZmYWxzZTYiLyUrZXhlY3V0YWJsZUdRJmZhbHNlNiIvJSlyZWFkb25seUdRJmZhbHNlNiIvJSljb21wb3NlZEdRJmZhbHNlNiIvJSpjb252ZXJ0ZWRHUSZmYWxzZTYiLyUraW1zZWxlY3RlZEdRJmZhbHNlNiIvJSxwbGFjZWhvbGRlckdRJmZhbHNlNiIvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSZmYWxzZTYiLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWM2Ii1JI21pRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNjVRImI2Ii8lJ2ZhbWlseUdRITYiLyUlc2l6ZUdRIzEwNiIvJSVib2xkR1EmZmFsc2U2Ii8lJ2l0YWxpY0dRJXRydWU2Ii8lKnVuZGVybGluZUdRJmZhbHNlNiIvJSpzdWJzY3JpcHRHUSZmYWxzZTYiLyUsc3VwZXJzY3JpcHRHUSZmYWxzZTYiLyUrZm9yZWdyb3VuZEdRKFswLDAsMF02Ii8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdNiIvJSdvcGFxdWVHUSZmYWxzZTYiLyUrZXhlY3V0YWJsZUdRJmZhbHNlNiIvJSlyZWFkb25seUdRJmZhbHNlNiIvJSljb21wb3NlZEdRJmZhbHNlNiIvJSpjb252ZXJ0ZWRHUSZmYWxzZTYiLyUraW1zZWxlY3RlZEdRJmZhbHNlNiIvJSxwbGFjZWhvbGRlckdRJmZhbHNlNiIvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSZmYWxzZTYiLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWM2Ii0lJlNUWUxFRzYjJSZQT0lOVEctJSlfVklTSUJMRUc2IyIiIi0lJVJPT1RHNictJSlCT1VORFNfWEc2IyQiJCsnISIiLSUpQk9VTkRTX1lHNiMkIiRTIiEiIi0lLUJPVU5EU19XSURUSEc2IyQiJSE9JCEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiU/TSEiIi0lKUNISUxEUkVORzYiLSUrQU5OT1RBVElPTkc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZRzYjJCIiISEiIi0lLUJPVU5EU19XSURUSEc2IyQiJStTISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJStTISIiLSUpQ0hJTERSRU5HNiI=NiI=

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

KytJInhHNiIiIiQiIiEqJiwmKiYiI1IiIiIpRiUjRisiIiNGK0YrIiNtRitGKywmIiM5RisqJiIiKUYrRixGK0YrISIiRissJCooI0YlIiIlRissJiIkJHpGKyomIiRlJUYrRixGK0YrRispLCYiIihGKyomRjhGK0YsRitGK0YuRjRGK0YuLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRitGJQ==

KytJInhHNiIkIiIkIiIhRickIitNJ0dVeiUhIioiIiIkIisvZm9LaEYqIiIjLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRitGJg==

Here we show how, starting from the equation between A1 and N that we have obtained, one can reconstruct the structure of the extension Q(t,A1) We first review the structure of Q(N), and how one can naturally introduce the series M

The extension Q(N), of degree 4 algN;tN

LCYqJkkiTkc2IiIiIiksJkYmRiZGJCEiIiIiI0YmRikqJkkidEdGJUYmLCwqJClGJCIiJUYmRiYqJkYqRiYpRiQiIiRGJkYpKiYiIidGJilGJEYqRiZGJiomRipGJkYkRiZGKUYmRiZGJkYm

KihJIk5HNiIiIiIpLCZGJSEiIkYjRiUiIiNGJSwsKiQpRiMiIiVGJUYlKiZGKUYlKUYjIiIkRiVGKComIiInRiUpRiNGKUYlRiUqJkYpRiVGI0YlRihGJUYlRig=

A unique subfield of degree 2 between Q(t) and Q(N) t0:=10000: evala(Subfields(subs(t=t0,algN),2))

PCMtSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmKiQpSSNfWkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IiIiIyIiIiIiIiIrKioqUis/IiIiIg==

Easy to guess which one, using the above special case subs(t=t0,4*t-1+12*t^2)

IisqKipSKz8i

factor(1-4*t-12*t^2)

LCQqJiwmIiIiRiUqJiIiI0YlSSJ0RzYiRiVGJUYlLCZGJSEiIiomIiInRiVGKEYlRiVGJUYr

A taste of Motzkin numbers algM:=-M+t*(1+2*M+4*M^2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGdNR0YoLCZJIk1HRighIiIqJkkidEdGKCIiIiwoKiYiIiVGMylGLyIiI0YzRjMqJkY4RjNGL0YzRjNGM0YzRjNGMzcjRi4=

tM:=factor(solve(algM,t))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN0TUdGKComSSJNR0YoIiIiLCgqJiIiJUYwKUYvIiIjRjBGMComRjVGMEYvRjBGMEYwRjAhIiI3I0Yu

numer(factor(subs(t=tM,algN)))

LCQqJiwqKiZJIk1HNiIiIiIpSSJOR0YnIiIjRihGKCooRitGKEYmRihGKkYoISIiRiZGKEYqRi1GKCwqKigiIiVGKEYmRihGKkYoRigqJEYpRihGLSomRitGKEYqRihGKEYoRi1GKEYt

Choose the right factor, given that N=O(t) algNM:=op(2,%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdOTUdGKCwqKiZJIk1HRigiIiIpSSJOR0YoIiIjRjFGMSooRjRGMUYwRjFGM0YxISIiRjBGMUYzRjY3I0Yu

collect(%,M,factor)

LCYqJiksJiIiIiEiIkkiTkc2IkYmIiIjRiZJIk1HRilGJkYmRihGJw==

factor([solve(%,N)])

NyQsJCooIyIiIiIiI0YmSSJNRzYiISIiLCgqJkYnRiZGKEYmRiZGJkYmKiQpLCYqJiIiJUYmRihGJkYmRiZGJkYlRiZGJkYmRiYsJCooRiVGJkYoRiosKEYsRiZGJkYmRi1GKkYmRiY=

NM:=op(2,%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNOTUdGKCwkKigjIiIiIiIjRjFJIk1HRighIiIsKComRjJGMUYzRjFGMUYxRjEqJCksJiomIiIlRjFGM0YxRjFGMUYxRjBGMUY0RjFGMTcjRi4=

MN:=factor(solve(algNM,M))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNNTkdGKComSSJOR0YoIiIiKSwmRjAhIiJGL0YwIiIjRjM3I0Yu

Mt:=op(2,[solve(algM,M)])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNNdEdGKCwkKigjIiIiIiIpRjEsKComIiIjRjFJInRHRihGMUYxRjEhIiIqJCksKComIiM3RjEpRjZGNUYxRjcqJiIiJUYxRjZGMUY3RjFGMSNGMUY1RjFGMUYxRjZGN0Y3NyNGLg==

Mser:=proc(n) series(Mt,t,n+1): end: Mser(5)

Ky1JInRHNiIiIiJGJSIiI0YmIiIpIiIkIiNLIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIiY=

We return to algA1N. Can we find a simpler generator than A1? Compare the form of algA1N and the denominator of tA2sol algA1N;tA2sol

LCwqKClJIk5HNiIiIiMiIiIpLCZGKCEiIkYlRigiIilGKClJJEFBMUdGJiIiJUYoRigqLEYvRihGJUYoKSwmRiVGKEYoRigiIiRGKClGKiIiKEYoKUYuRjNGKEYrKiwiI0tGKEYlRigpLCgqJEYkRihGKEYlRitGKEYoRjNGKClGKkYvRigpRi5GJ0YoRigqLCIja0YoKUYqRjNGKEYxRihGOUYoRi5GKEYoKiYiJGMjRigpRjoiIidGKEYo

LCQqKiIiIyIiIiwmKigsKiokKUkiTkc2IiIiJEYlRiUqJClGK0YkRiUhIiJGK0YwRiVGMEYlKSwmRiVGMEYrRiUiIiVGJSlJJEFBMUdGLEYkRiVGJSooIiM7RiUsKkYpRiVGLkYlRitGJUYlRjBGJSksKEYuRiVGK0YwRiVGJUYtRiVGMEYlKUYyRiRGMCwmKihGNEYlRitGJUYxRiVGJSomRjdGJUY5RiVGJUYwRjA=

collect(algA1N-(AA1^2*N*(-1 + N)^4 + 16*(N^2 - N + 1)^3)^2,AA1,factor)

LCYqLCIiJSIiIkkiTkc2IkYlKSwmRiZGJUYlRiUiIiRGJSksJkYlISIiRiZGJSIiKEYlKUkkQUExR0YnRipGJUYtKiwiI2tGJSlGLEYqRiVGKEYlKSwoKiQpRiYiIiNGJUYlRiZGLUYlRiVGKkYlRjBGJUYl

factor(%);

LCQqLCIiJSIiIiksJkkiTkc2IkYlRiVGJSIiJEYlKSwmRiUhIiJGKEYlRipGJUkkQUExR0YpRiUsOiomKUYuIiIjRiUpRigiIiZGJUYlKihGJEYlRjFGJSlGKEYkRiVGLSomIiM7RiUpRigiIidGJUYtKihGOkYlRjFGJSlGKEYqRiVGJSomIiNbRiVGM0YlRiUqKEYkRiVGMUYlKUYoRjJGJUYtKiYiIycqRiVGNkYlRi0qJkYxRiVGKEYlRiUqJiIkNyJGJUY8RiVGJSomRkJGJUZARiVGLSomRj5GJUYoRiVGJUY4Ri1GJUYt

collect(op(5,%),AA1,factor)

LCYqKClJJEFBMUc2IiIiIyIiIkkiTkdGJkYoKSwmRighIiJGKUYoIiIlRihGKComIiM7RigpLCgqJClGKUYnRihGKEYpRixGKEYoIiIkRihGLA==

Paper version algpap:=(AA1^2*N*(1-N)^4+16*(N^2-N+1)^3)^2+4*(1+N)^3*(1-N)^3*AA1*(AA1^2*N*(1-N)^4-16*(N^2-N+1)^3)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdwYXBHRigsJiokKSwmKigpSSRBQTFHRigiIiMiIiJJIk5HRihGNiksJkY2RjZGNyEiIiIiJUY2RjYqJiIjO0Y2KSwoKiQpRjdGNUY2RjZGN0Y6RjZGNiIiJEY2RjZGNUY2RjYqLEY7RjYpLCZGN0Y2RjZGNkZCRjYpRjlGQkY2RjRGNiwmRjJGNkY8RjpGNkY2NyNGLg==

factor(algpap/algA1N)

IiIi

In terms of M? same structure in the minimal equation of A1 collect(algA1N/N^2/(-1 + N)^8,AA1,factor)

LCwqJClJJEFBMUc2IiIiJSIiIkYoKixGJ0YoSSJOR0YmISIiKSwmRipGKEYoRigiIiRGKCwmRihGK0YqRihGKylGJUYuRihGKyosIiNLRihGKkYrKSwoKiQpRioiIiNGKEYoRipGK0YoRihGLkYoKUYvRidGKylGJUY3RihGKCouIiNrRigpRi8iIiZGK0YsRihGM0YoRjZGK0YlRihGKCoqIiRjI0YoKUY0IiInRihGNkYrKUYvIiIpRitGKA==

map(c->factor(expand(rationalize(subs(N=NM,c)))),%)

LCwqJClJJEFBMUc2IiIiJSIiIkYoKipGJ0YoKUYlIiIkRigpLCYqJkYnRihJIk1HRiZGKEYoRihGKCNGKyIiI0YoRi8hIiJGKCoqIiNLRigpRiVGMUYoKSwmRi9GKEYoRihGK0YoRi9GMkYoKiwiI2tGKEYlRihGLEYoRjZGKClGL0YxRjJGMiooIiRjI0YoKUY3IiInRihGOkYyRig=

algA1Malg:=collect(M^2*%,AA1,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSphbGdBMU1hbGdHRigsLComKUkiTUdGKCIiIyIiIilJJEFBMUdGKCIiJUYzRjMqKkY2RjMpLCYqJkY2RjNGMUYzRjNGM0YzIyIiJEYyRjNGMUYzKUY1RjxGM0YzKioiI0tGM0YxRjMpLCZGMUYzRjNGM0Y8RjMpRjVGMkYzRjMqKiIja0YzRjhGM0ZARjNGNUYzISIiKiYiJGMjRjMpRkEiIidGM0YzNyNGLg==

factor(algA1Malg- (M*AA1^2+16*(1+M)^3)^2)

LCQqKiIiJSIiIkkkQUExRzYiRiUpLCYqJkYkRiVJIk1HRidGJUYlRiVGJSMiIiQiIiNGJSwsKiYpRiZGLkYlRitGJUYlKiYiIztGJSlGK0YtRiUhIiIqJiIjW0YlKUYrRi5GJUY1KiZGN0YlRitGJUY1RjNGNUYlRiU=

algA1Malgalt:=(M*AA1^2+16*(1+M)^3)^2 +4*AA1*(4*M + 1)^(3/2)*(M*AA1^2-16*(1+M)^3)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS1hbGdBMU1hbGdhbHRHRigsJiokKSwmKiZJIk1HRigiIiIpSSRBQTFHRigiIiNGNEY0KiYiIztGNCksJkYzRjRGNEY0IiIkRjRGNEY3RjRGNCoqIiIlRjRGNkY0KSwmKiZGPkY0RjNGNEY0RjRGNCNGPEY3RjQsJkYyRjRGOCEiIkY0RjQ3I0Yu

factor(algA1Malgalt-algA1Malg)

IiIh

This suggests to introduce the following generator P1 P1A1:=M*AA1^2/16/(1+M)^3

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVQMUExR0YoLCQqKiMiIiIiIztGMUkiTUdGKEYxKUkkQUExR0YoIiIjRjEpLCZGM0YxRjFGMSIiJCEiIkYxNyNGLg==

A1P1:=4*sqrt(P1/M*(M + 1))*(M + 1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVBMVAxR0YoLCQqKCIiJSIiIiwmSSJNR0YoRjFGMUYxRjEpKihJI1AxR0YoRjFGMkYxRjMhIiIjRjEiIiNGMUYxNyNGLg==

Let us write the equation for P1 map(c->factor(c/(1+M)^6)/256, subs(AA1=A1P1,algA1Malgalt))

LCYqJCksJiIiIkYmSSNQMUc2IkYmIiIjRiZGJioqKSooRidGJiwmSSJNR0YoRiZGJkYmRiZGLiEiIiNGJkYpRiYpLCYqJiIiJUYmRi5GJkYmRiZGJiMiIiRGKUYmLCZGJkYvRidGJkYmKUYtRilGL0Ym

op(2,%)^2-op(1,%)^2

LCYqLEkjUDFHNiIiIiIpLCZJIk1HRiVGJkYmRiYiIiQhIiJGKUYrKSwmKiYiIiVGJkYpRiZGJkYmRiZGKkYmKSwmRiZGK0YkRiYiIiNGJkYmKiQpLCZGJkYmRiRGJkYvRiZGKw==

algP1M:=map(c->factor(c*M*(1+M)^3),%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdQMU1HRigsJiooSSNQMUdGKCIiIiksJiomIiIlRjFJIk1HRihGMUYxRjFGMSIiJEYxKSwmRjEhIiJGMEYxIiIjRjFGMSooRjZGMSksJkY2RjFGMUYxRjdGMSksJkYxRjFGMEYxRjVGMUY6NyNGLg==

map(c->factor(c/(1-P1)^2/(1+4*M)^3),%)

LCZJI1AxRzYiIiIiKixJIk1HRiRGJSksJkYnRiVGJUYlIiIkRiUpLCZGJUYlRiNGJSIiJUYlKSwmRiUhIiJGI0YlIiIjRjApLCYqJkYtRiVGJ0YlRiVGJUYlRipGMEYw

P1ser:=proc(n) op(3,algeqtoseries(convert(subs(M=Mser(6+n),(algP1M)),polynom),t,P1,n)): end

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZQMXNlckdGKGYqNiNJIm5HRihGKEYoRigtSSNvcEdGJjYkIiIkLV9JJWdmdW5HRidJLmFsZ2VxdG9zZXJpZXNHRig2Ji1JKGNvbnZlcnRHRiY2JC1JJXN1YnNHRiY2JC9JIk1HRigtSSVNc2VyR0YoNiMsJkYwIiIiIiInRkZJJ2FsZ1AxTUdGKEkocG9seW5vbUdGJkkidEdGKEkjUDFHRihGMEYoRihGKDcjRi4=

P1ser(4);P1ser(5)

Ky1JInRHNiIiIiJGJSEiIiIiIyIiJSIiJCEiI0YoLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIiY=

Ky9JInRHNiIiIiJGJSEiIiIiIyIiJSIiJCEiI0YoIiNJIiImLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIic=

algP1:=collect(factor(resultant((algP1M),algM,M)),P1,factor)

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

Now we investigate the extension Q(t,A1) Between Q(M) and Q(t,A1): three subfields of degree 2 algA1M:=collect(factor(resultant(algA1N,algNM,N)),AA1,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdBMU1HRigsLComKUkkQUExR0YoIiIpIiIiKUkiTUdGKCIiJUYzRjMqKiIjO0YzKUY1IiIjRjMsLComRjZGM0Y0RjNGMyomIiNfRjMpRjUiIiRGMyEiIiomIiNPRjNGOUYzRkEqJkYyRjNGNUYzRkFGM0ZBRjMpRjEiIidGM0YzKiwiJDcmRjNGNUYzLCwqJkZARjNGNEYzRjMqJiIjdEYzRj9GM0YzKiYiI2RGM0Y5RjNGMyomIiM6RjNGNUYzRjNGM0YzRjMpLCZGNUYzRjNGM0ZARjMpRjFGNkYzRjMqKiIlJzQlRjNGO0YzKUZSRkZGMylGMUY6RjNGMyomIiZPYidGMylGUiIjN0YzRjM3I0Yu

M0:=10000: evala(Subfields(subs(M=M0,algA1M),2))

PCUtSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmKiQpSSNfWkdGJSIiIyIiIkYvIiYsKyUhIiItRiQ2IywmRitGLyIqKioqKioqKlJGLy1GJDYjLCZGK0YvIi8qKipmKipSK2ciRi8=

One is already known, namely Q(N) =Q(sqrt(1+4M)). What about the other two? Easy to guess subs(M=M0,1-4*M^2)

ISoqKioqKioqUg==

subs(M=M0,-1+M*(-4+M*(4+16*M)))

Ii8qKipmKipSK2ci

factor(-1+M*(-4+M*(4+16*M)))

KigsJiomIiIlIiIiSSJNRzYiRiZGJkYmRiZGJiwmKiYiIiNGJkYnRiZGJkYmRiZGJiwmRipGJkYmISIiRiY=

Let us check this: the degree of AA1 is indeed halved over Q(M,sqrt(1-4*M^2)), that is, equal to 4 factor(algA1M, sqrt(1-4*M^2))

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

and for the other guessed subfield: same behaviour factor(algA1M, sqrt((1-4*M^2)*(1+4*M)))

KiYsQiomKUkiTUc2IiIiIyIiIilJJEFBMUdGJyIiJUYpRikqKCIjS0YpKUYrRihGKSlGJkYsRilGKSomIiRjI0YpKUYmIiInRilGKSooIiQ7JUYpRi9GKSlGJiIiJEYpISIiKioiI2tGKUYvRikpLCoqJiIjO0YpRjdGKUY5KiZGLEYpRiVGKUY5KiZGLEYpRiZGKUYpRilGKSNGKUYoRilGJUYpRikqJiIlTzpGKSlGJiIiJkYpRikqKCIkKUdGKUYlRilGL0YpRjkqKkY/RilGL0YpRjxGKUYmRilGOSomIiVTUUYpRjBGKUYpKihGO0YpRiZGKUYvRilGOSooIiIpRilGL0YpRjxGKUY5KiYiJT9eRilGN0YpRikqJkZORilGL0YpRjkqJkZLRilGJUYpRikqJkZERilGJkYpRilGMkYpRiksQkYkRilGLUYpRjFGKUY1RjlGOkY5RkNGKUZHRjlGSUYpRkpGKUZMRjlGTUYpRk9GKUZRRjlGUkYpRlNGKUYyRilGKQ==

Hence we have identified the three subfields of degree 2 over Q(M) Between Q(t) and Q(t,sqrt(1-4M^2))? Only Q(M) We denote by S1 this square root algS1:=factor(resultant(1-4*M^2-S1^2,algM,M))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdTMUdGKCwwKigiIzsiIiIpSSNTMUdGKCIiJUYxKUkidEdGKCIiI0YxRjEqKCIjW0YxKUYzRjdGMUY1RjEhIiIqKEYwRjFGOkYxRjZGMUY7KiZGNEYxRjpGMUYxKiZGOUYxRjVGMUYxKiZGMEYxRjZGMUYxRjRGOzcjRi4=

genus(algS1,t,S1)

IiIh

factor(parametrization(algS1,t,S1,X))

NyQsJCoqIiIpIiIiLCZJIlhHNiJGJiIjO0YmRiYsKCokKUYoIiIjRiZGJiomIiNLRiZGKEYmRiYiJD8kRiZGJiwsKiQpRigiIiVGJkYmKiYiI1tGJilGKCIiJEYmRiYqJiIlXzZGJkYtRiZGJiomIiZnYCJGJkYoRiZGJiImO2cpRiYhIiJGPywkKigsJkYoRiYiI0NGJkYmLCZGKEYmRiVGJkYmRitGP0Y/

t0:=10000: evala(Subfields(subs(t=t0,algS1),2))

PCMtSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmKiQpSSNfWkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IiIiIyIiIiIiIiIrKioqUis/IiIiIg==

Between Q(t) and Q(t,sqrt((1+4*M)*(1-4M^2))? Only Q(M) as well. We denote by S1 this other square root algS2:=factor(resultant((1+4*M)*(1-4*M^2)-S2^2,algM,M))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdTMkdGKCw0KigiI2siIiIpSSNTMkdGKCIiJUYxKUkidEdGKCIiJEYxRjEqKCIkIz5GMSlGMyIiI0YxRjVGMUYxKigiJEciRjFGOkYxKUY2RjtGMSEiIiooIiMhKUYxRjpGMUY2RjFGPyomIiR3JkYxRjVGMUYxKiYiIztGMUY6RjFGMSomIiQlUUYxRj5GMUYxKiZGRUYxRjZGMUYxRkVGPzcjRi4=

genus(algS2,t,S2)

IiIi

t0:=10000: evala(Subfields(subs(t=t0,algS2),2))

PCMtSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmKiQpSSNfWkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IiIiIyIiIiIiIiIrKioqUis/IiIiIg==

Between Q(t) and Q(N)? Only Q(M) as well. t0:=10000: evala(Subfields(subs(t=t0,algN),2))

PCMtSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmKiQpSSNfWkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IiIiIyIiIiIiIiIrKioqUis/IiIiIg==

Between Q(M) and Q(t,A1): three subfields of degree 4 M0:=10000: evala(Subfields(subs(M=M0,algA1M),4)); nops(%)

PCUtSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywoKiYiMCssKzErNCslIiIiKUkjX1pHRiUiIiVGLUYtKiYiKywrMys7Ri0pRi8iIiNGLSEiIiIoKy8rJUYtLUYkNiMsKComIi4rKy8rMyslRi1GLkYtRi0qJiIyKioqZioqZigqKmYqekYtRjNGLUYtIjYrKy8rOytDKzsrJUYtLUYkNiMsLComIi4rKywrLSsiRi1GLkYtRi0qJiIyKioqPioqUicqeiUqUkYtKUYvIiIkRi1GNSomIjYtK0srVyxnLV8sJ0YtRjNGLUYtKiYiOioqKioqKSpwJSp6JylSJylmJipSRi1GL0YtRjUiPisrLCsxKzorPys6KzErIkYt

IiIk

Between Q(N) and Q(t,A1), degree 2 ? just one of them, which is easy to guess N0:=10000: evala(Subfields(subs(N=N0,algA1N),2))

PCMtSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmKiQpSSNfWkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IiIiIyIiIiIiIiIxLCsnKio+K2cqKiohIiI=

This gives already one algS3M:=factor(resultant(S3^2-(1+N^2)*(1+N*(N-4)),algNM,N))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdTM01HRigsNComKUkiTUdGKCIiJSIiIilJI1MzR0YoRjJGM0YzKigiIilGM0YwRjMpRjUiIiNGM0YzKigiIztGMylGMSIiJEYzRjhGM0YzKiZGO0YzRjBGM0YzKihGOUYzKUYxRjlGM0Y4RjNGMyooRjJGM0YxRjNGOEYzISIiKiZGN0YzRkBGM0ZCKiRGOEYzRkJGM0YzNyNGLg==

This is the obvious extension of the three fields of degree 2 over Q(M) numer(factor(algS3M, {sqrt(1-4*M^2),sqrt(1+4*M)}))

KiosKiooIiIjIiIiKUkiTUc2IkYlRiZJI1MzR0YpRiZGJiooRiVGJiksJiomIiIlRiZGJ0YmISIiRiZGJiNGJkYlRiZGKEYmRiYqJiksJiomRi9GJkYoRiZGJkYmRiZGMUYmRixGJkYmKiRGLEYmRiZGJiwqRiRGJkYrRjBGMkYwRjZGMEYmLCpGJEYmRitGJkYyRjBGNkYmRiYsKkYkRiZGK0YwRjJGJkY2RjBGJg==

nops(%)

IiIl

Hence this one is Q(sqrt(1+4M), sqrt(1-4M^2))=Q(t,S3) Between Q(M) and Q(t,S3): the three known extensions M0:=10000: evala(Subfields(subs(M=M0,algS3M),2))

PCUtSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmKiQpSSNfWkdGJSIiIyIiIkYvIiYsKyUhIiItRiQ2IywmRitGLyIqKioqKioqKlJGLy1GJDYjLCZGK0YvIi8qKipmKipSK2ciRi8=

What are the other two extensions of degree 4 of Q(M)? We know that we have A1^2, or equivalently P1, somewhere. Does it coincide with the above field? No, since we have polynomials of degree 2 below numer(factor(algP1M, {sqrt(1-4*M^2),sqrt(1+4*M)})); nops(%)

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

IiIk

Hence Q(t,P1) = Q(t,A1^2) is another subfield.. Here is the equation for A1s=A1^2 algA1sM:=subs(AA1=sqrt(A1s),algA1M)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShhbGdBMXNNR0YoLCwqJilJJEExc0dGKCIiJSIiIilJIk1HRihGMkYzRjMqKiIjO0YzKUY1IiIjRjMsLComRjJGM0Y0RjNGMyomIiNfRjMpRjUiIiRGMyEiIiomIiNPRjNGOEYzRkAqJiIiKUYzRjVGM0ZARjNGQEYzKUYxRj9GM0YzKiwiJDcmRjNGNUYzLCwqJkY/RjNGNEYzRjMqJiIjdEYzRj5GM0YzKiYiI2RGM0Y4RjNGMyomIiM6RjNGNUYzRjNGM0YzRjMpLCZGNUYzRjNGM0Y/RjMpRjFGOUYzRjMqKiIlJzQlRjNGOkYzKUZRIiInRjNGMUYzRjMqJiImT2InRjMpRlEiIzdGM0YzNyNGLg==

numer(factor(algA1sM, {sqrt(1-4*M^2),sqrt(1+4*M)})); nops(%)

KiYsQiomIiRjIyIiIilJIk1HNiIiIidGJkYmKigiI0tGJkkkQTFzR0YpRiYpRigiIiVGJkYmKiwiI2tGJkYtRiYpLCYqJkYvRiZGKEYmRiZGJkYmI0YmIiIjRiYpLCYqJkYvRiYpRihGNkYmISIiRiZGJkY1RiZGOkYmRjsqJiIlTzpGJilGKCIiJkYmRiYqJilGLUY2RiZGOkYmRiYqKCIkOyVGJkYtRiYpRigiIiRGJkY7KiwiIztGJkYtRiZGMkYmRjdGJkYoRiZGJiomIiVTUUYmRi5GJkYmKigiJClHRiZGLUYmRjpGJkY7KioiIilGJkYtRiZGMkYmRjdGJkYmKiYiJT9eRiZGREYmRiYqKEYxRiZGLUYmRihGJkY7KiZGSUYmRjpGJkYmKiZGTUYmRi1GJkY7KiZGPUYmRihGJkYmRiVGJkYmLEJGJEYmRitGJkYwRiZGPEYmRkBGJkZCRjtGRkY7RkhGJkZKRjtGTEY7Rk5GJkZQRjtGUUYmRlJGO0ZTRiZGJUYmRiY=

IiIj

Between Q(M) and Q(t,A1^2), what do we have? factor(algA1sM, sqrt(1+4*M))

LGhvKiYiJk9iJyIiIilJIk1HNiIiIzdGJUYlKigiJiVRO0YlSSRBMXNHRihGJSlGJyIjNUYlRiUqJiInS2t5RiUpRiciIzZGJUYlKigiJU86RiUpRiwiIiNGJSlGJyIiKUYlRiUqKCInKW85IkYlRixGJSlGJyIiKkYlISIiKiYiKHdgSyVGJUYtRiVGJSooIiNrRiUpRiwiIiRGJSlGJyIiJ0YlRiUqKCImJSk+JUYlRjVGJSlGJyIiKEYlRiUqKCIoWyd6NkYlRixGJUY3RiVGPSomIik/elQ5RiVGO0YlRiUqJilGLCIiJUYlKUYnRlBGJUYlKigiJEspRiVGQkYlKUYnIiImRiVGPSooIic/ZjlGJUY1RiVGREYlRiUqKCIoL1p5JEYlRixGJUZIRiVGPSomIik/LldLRiVGN0YlRiUqKCIkdyZGJUZCRiVGUUYlRj0qKCInJyopMyNGJUY1RiVGVEYlRiUqKCIoQ21VJ0YlRixGJUZERiVGPSomIik3WCE+JkYlRkhGJUYlKigiJEciRiVGQkYlKUYnRkNGJUY9KigiJyFbWyJGJUY1RiVGUUYlRiUqKCIoI3pobEYlRixGJUZURiVGPSomIilrX2JnRiVGREYlRiUqKCIjO0YlRkJGJSlGJ0Y2RiVGPSooIiZnUCZGJUY1RiVGYG9GJUYlKigiKDMtPiVGJUYsRiVGUUYlRj0qJkZdb0YlRlRGJUYlKigiJTsjKkYlRjVGJUZpb0YlRiUqKCIobzZuIkYlRixGJUZgb0YlRj0qJkZlbkYlRlFGJUYlKigiJDcmRiVGNUYlRidGJUYlKigiJy9iU0YlRixGJUZpb0YlRj0qJkZNRiVGYG9GJUYlKigiJld0JkYlRixGJUYnRiVGPSomRj9GJUZpb0YlRiUqJiIlJzQlRiVGLEYlRj0qJkYwRiVGJ0YlRiVGJEYl

factor(algA1sM, sqrt(1-4*M^2))

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

Only one extension factor(algA1sM, sqrt((1-4*M^2)*(1+4*M)))

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

There is also tA2, which, as we have seen, has degre 8 over Q(t) algtA2N;algtA2M

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

LCwqJilJIk1HNiIiIiQiIiIpSSR0QTJHRiYiIiVGKEYoKipGK0YoKUYlIiIjRigpLCYqJkYuRihGJUYoRihGKEYoRi5GKClGKkYnRighIiIqKEYlRigsLComIiMnKkYoKUYlRitGKEYoKiYiJEciRihGJEYoRigqJkY3RihGLUYoRigqJiIjT0YoRiVGKEYoIiImRihGKClGKkYuRihGKCoqRi5GKCwoKiYiI0tGKEY4RihGKComRitGKEYlRihGKEYoRihGKEYvRihGKkYoRjMqKEYrRihGJUYoLDAqJiIja0YoKUYlIiInRihGKComRkhGKClGJUY+RihGKComIiM7RihGOEYoRjMqJiIjP0YoRiRGKEYzKiYiIipGKEYtRihGKComIiIoRihGJUYoRihGKEYoRihGKA==

We examine whether this series is in the two extensions that we have already identified numer(factor(algtA2M, {sqrt(1-4*M^2),sqrt(1+4*M)})); nops(%)

KiYsNiomIiNLIiIiKUkiTUc2IiIiJUYmRiYqKCIjO0YmKUYoIiIkRiZJJHRBMkdGKUYmISIiKioiIilGJilGKCIiI0YmKSwmKiZGKkYmRihGJkYmRiZGJiNGJkY0RiYpLCYqJkYqRiZGM0YmRjBGJkYmRjhGJkYmKihGNEYmRjNGJilGL0Y0RiZGJiooRixGJkYzRiZGL0YmRjAqKkY0RiZGKEYmRjVGJkY5RiZGMCooRipGJkYoRiZGL0YmRjAqJkY1RiZGOUYmRjBGN0YmRiZGJkYmLDZGJEYmRitGMEYxRjBGPEYmRj5GMEY/RiZGQEYwRkFGJkY3RiZGJkYmRiY=

IiIj

Quadratic terms here as well subs(RootOf(algA1sM,A1s)=A1s,numer(factor(algtA2M, RootOf(algA1sM,A1s)))); nops(%)

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

IiIj

So tA2 is a generator of the third extension. We would like to find a simpler generator factor(subs(AA1=A1P1,M=MN,tA2sol)); convert(%,parfrac,P1);

LCQqLCIiIyIiIiwyKiQpSSJORzYiIiIlRiVGJSomKUYpIiIkRiVJI1AxR0YqRiUhIiIqJEYtRiVGJSomKUYpRiRGJUYvRiVGJSokRjNGJUYlKiZGKUYlRi9GJUYlRilGMEYvRiVGJSwmRiVGJUYvRiVGMEYpRjApLCZGJUYwRilGJUYkRjBGJQ==

LCYqKiIiIyIiIiwqKiQpSSJORzYiIiIkRiVGJSokKUYpRiRGJSEiIkYpRi5GJUYuRiVGKUYuKSwmRiVGLkYpRiVGJEYuRi4qKiwqKiZGJEYlRihGJUYlKiYiIidGJUYtRiVGJSomRjVGJUYpRiVGJUYkRiVGJSwmRiVGJUkjUDFHRipGJUYuRilGLkYwRi5GJQ==

tA2P1:=map(c->simplify(factor(expand(rationalize(subs(N=NM,c))))),%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZ0QTJQMUdGKCwmKiYsKiokKSwmKiYiIiUiIiJJIk1HRihGNkY2RjZGNiMiIiQiIiNGNkY2KiZGNUY2KUY3RjpGNkY2RjRGNkY2RjZGNkY3ISIiRjYqKkY6RjZGMkY2RjdGPSwmRjZGNkkjUDFHRihGNkY9Rj03I0Yu

Could this be a simpler generator? factor(tA2P1-(1+2*M)^2/M)

KiopLCYqJiIiJSIiIkkiTUc2IkYnRidGJ0YnIyIiJCIiI0YnLCZGJyEiIkkjUDFHRilGJ0YnRihGLiwmRidGJ0YvRidGLg==

Let us try with this gen:=(4*M + 1)^(3/2)*(1 -P1)/((1 + P1))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRnZW5HRigqKCksJiomIiIlIiIiSSJNR0YoRjNGM0YzRjMjIiIkIiIjRjMsJkYzRjNJI1AxR0YoISIiRjMsJkYzRjNGOUYzRjo3I0Yu

factor(resultant(numer(gen^2-G^2),algP1M,P1))

LCQqKCIjOyIiIiksJiomIiIlRiVJIk1HNiJGJUYlRiVGJSIiJ0YlKSw6KiYiJGMjRiUpRioiIihGJUYlKiYiJGcqRiUpRipGLEYlRiUqKCIja0YlKUkiR0dGKyIiI0YlKUYqIiIkRiUhIiIqJiIlI1IiRiUpRioiIiZGJUYlKiQpRjlGKUYlRiUqKCIjW0YlRjhGJSlGKkY6RiVGPSomIiQhKSpGJSlGKkYpRiVGJSooIiM3RiVGOEYlRipGJUY9KiYiJFskRiVGO0YlRiUqJEY4RiVGPSomIiNnRiVGRkYlRiVGKEYlRjpGJUYl

collect(op(1,op(3,%)),G,factor)

LCgqJClJIkdHNiIiIiUiIiJGKComKSwmKiZGJ0YoSSJNR0YmRihGKEYoRigiIiRGKClGJSIiI0YoISIiKipGJ0YoRi1GKCksJkYtRihGKEYoRi5GKEYqRihGKA==

P2P1:=gen

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVQMlAxR0YoKigpLCYqJiIiJSIiIkkiTUdGKEYzRjNGM0YzIyIiJCIiI0YzLCZGM0YzSSNQMUdGKCEiIkYzLCZGM0YzRjlGM0Y6NyNGLg==

algP2M:=subs(G=P2,%%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdQMk1HRigsKCokKUkjUDJHRigiIiUiIiJGMyomKSwmKiZGMkYzSSJNR0YoRjNGM0YzRjMiIiRGMylGMSIiI0YzISIiKipGMkYzRjhGMyksJkY4RjNGM0YzRjlGM0Y1RjNGMzcjRi4=

P1P2:=solve(P2P1-P2,P1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVQMVAyR0YoKiYsJiokKSwmKiYiIiUiIiJJIk1HRihGNUY1RjVGNSMiIiQiIiNGNUY1SSNQMkdGKCEiIkY1LCZGMEY1RjpGNUY7NyNGLg==

tA2P1;tA2P2:=collect(factor(subs(P1=P1P2,%)),P2,factor)

LCYqJiwqKiQpLCYqJiIiJSIiIkkiTUc2IkYqRipGKkYqIyIiJCIiI0YqRioqJkYpRiopRitGL0YqRipGKEYqRipGKkYqRishIiJGKioqRi9GKkYmRipGK0YyLCZGKkYqSSNQMUdGLEYqRjJGMg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZ0QTJQMkdGKCwmKiZJIk1HRighIiJJI1AyR0YoIiIiRjEqJiksJiomIiIjRjNGMEYzRjNGM0YzRjhGM0YwRjFGMzcjRi4=

P2ser:=proc(n) series(subs(M=Mser(n),P1=P1ser(n),P2P1),t,n+1): end: P2ser(5)

Ky9JInRHNiIiIiIiIiEiIiVGJSIjNSIiIyIjVSIiJCIkdyJGJy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIm

P2 in terms of M and A1 ? Using the equation of P1 algP1M; map(c->factor(c/(1+P1)^2/P1),%)

LCYqKEkjUDFHNiIiIiIpLCYqJiIiJUYmSSJNR0YlRiZGJkYmRiYiIiRGJiksJkYmISIiRiRGJiIiI0YmRiYqKEYrRiYpLCZGK0YmRiZGJkYsRiYpLCZGJkYmRiRGJkYqRiZGLw==

LCYqKCksJiomIiIlIiIiSSJNRzYiRihGKEYoRigiIiRGKCksJkYoISIiSSNQMUdGKkYoIiIjRigpLCZGKEYoRi9GKEYwRi5GKCoqRilGKCksJkYpRihGKEYoRitGKEYxRihGL0YuRi4=

P2P1

KigpLCYqJiIiJSIiIkkiTUc2IkYnRidGJ0YnIyIiJCIiI0YnLCZGJ0YnSSNQMUdGKSEiIkYnLCZGJ0YnRi5GJ0Yv

P2P1alt:=simplify(sqrt(-op(2,%%)),symbolic)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShQMlAxYWx0R0YoKiopLCZJIk1HRigiIiJGMkYyIyIiJCIiI0YyLCZGMkYySSNQMUdGKEYyRjIpRjcjRjJGNSEiIilGMUY5RjI3I0Yu

P2A1:=collect(simplify(factor(subs(P1=P1A1,P2P1alt)),symbolic),AA1,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVQMkExR0YoLCYqKCMiIiIiIiVGMUkiTUdGKEYxSSRBQTFHRihGMUYxKihGMkYxKSwmRjNGMUYxRjEiIiRGMUY0ISIiRjE3I0Yu

tA2P2; collect(subs(P2=P2A1,%),AA1,factor)

LCYqJkkiTUc2IiEiIkkjUDJHRiUiIiJGJiomKSwmKiYiIiNGKEYkRihGKEYoRihGLUYoRiRGJkYo

LCgqJiksJiomIiIjIiIiSSJNRzYiRihGKEYoRihGJ0YoRikhIiJGKComI0YoIiIlRihJJEFBMUdGKkYoRisqKkYuRigpLCZGKUYoRihGKCIiJEYoRilGK0YvRitGKw==

We recover the expression announced in the paper and checked earlier tA2pap

LCgqJiksJiomIiIjIiIiSSJNRzYiRihGKEYoRihGJ0YoRikhIiJGKComI0YoIiIlRihJJEFBMUdGKkYoRisqKkYuRigpLCZGKUYoRihGKCIiJEYoRilGK0YvRitGKw==

Between Q(M) and Q(t,P2), what do we have? factor(algP2M, sqrt(1+4*M))

LDoqJiIkYyMiIiIpSSJNRzYiIiIoRiVGJSomIiRnKkYlKUYnIiInRiVGJSomIiUjUiJGJSlGJyIiJkYlRiUqKCIja0YlKUYnIiIkRiUpSSNQMkdGKCIiI0YlISIiKiYiJCEpKkYlKUYnIiIlRiVGJSooIiNbRiUpRidGOEYlRjZGJUY5KiQpRjdGPUYlRiUqJiIkWyRGJUY0RiVGJSooIiM3RiVGJ0YlRjZGJUY5KiYiI2dGJUZARiVGJSokRjZGJUY5KiZGPUYlRidGJUYl

factor(algP2M, sqrt(1-4*M^2))

LDoqJiIkYyMiIiIpSSJNRzYiIiIoRiVGJSomIiRnKkYlKUYnIiInRiVGJSomIiUjUiJGJSlGJyIiJkYlRiUqKCIja0YlKUYnIiIkRiUpSSNQMkdGKCIiI0YlISIiKiYiJCEpKkYlKUYnIiIlRiVGJSooIiNbRiUpRidGOEYlRjZGJUY5KiQpRjdGPUYlRiUqJiIkWyRGJUY0RiVGJSooIiM3RiVGJ0YlRjZGJUY5KiYiI2dGJUZARiVGJSokRjZGJUY5KiZGPUYlRidGJUYl

Only one extension factor(algP2M, sqrt((1-4*M^2)*(1+4*M)))

LCQqKCMiIiIiIiVGJSwyKiYiI2tGJSlJIk1HNiIiIiRGJUYlKigiIilGJSlGKyIiI0YlKSwqKiYiIztGJUYqRiUhIiIqJkYmRiVGMEYlRjYqJkYmRiVGK0YlRiVGJUYlI0YlRjFGJUY2KiYiI1tGJUYwRiVGJSooRjFGJUYrRiVGMkYlRiUqJkYxRiUpSSNQMkdGLEYxRiVGNiomIiM3RiVGK0YlRiUqJEYyRiVGJUYlRiVGJSwyRihGJUYuRiVGOkYlRjxGNkY9RjZGQEYlRkJGNkYlRiVGJUYl

How can we write tA0 collect(subs(tA2=tA2pap,M=MN,tA0sol),AA1,factor)

LCoqJiMiIiIiIilGJSlJJEFBMUc2IiIiI0YlRiUqKiNGJSIiJUYlLCoqJClJIk5HRikiIiRGJUYlKiQpRjFGKkYlRiVGMUYlRiUhIiJGJSksJkYlRjVGMUYlRipGNUYoRiVGNSooLDAqJkYyRiUpRjEiIidGJUYlKiZGPEYlKUYxIiImRiVGNSomIiM2RiUpRjFGLUYlRiUqJiIjPUYlRjBGJUY1KiZGQUYlRjRGJUYlKiZGPEYlRjFGJUY1RjJGJUYlRjFGNSlGN0YtRjVGNSouRi1GJUYuRiUpLChGM0YlRjFGNUYlRiVGMkYlKUY3RjxGNUYxRjVGKEY1RjU=

map(c->factor(simplify(factor(expand(rationalize(subs(N=NM,c)))))),%)

LCoqJiMiIiIiIilGJSlJJEFBMUc2IiIiI0YlRiUqKkYkRiVGKEYlSSJNR0YpISIiLCwqJiIiJUYlKUYsRipGJUYlKihGMEYlRixGJSksJiomRjBGJUYsRiVGJUYlRiUjRiVGKkYlRi1GNUYlKiRGM0YlRi1GJUYlRiVGLSomLCoqJkYqRiUpRiwiIiRGJUYlKiYiIzlGJUYxRiVGLSomIiM3RiVGLEYlRi1GPEYtRiVGLEYtRiUqLEYqRiUpLCZGLEYlRiVGJUY8RiVGMUYtRi5GJUYoRi1GLQ==

collect(subs(sqrt(1+4*M)=sqrt(delta),%),[AA1,delta],factor)

LCoqJiMiIiIiIilGJSlJJEFBMUc2IiIiI0YlRiUqJiwmKipGJEYlLCYqJiIiJUYlSSJNR0YpRiVGJUYlRiVGJUYxISIiKUkmZGVsdGFHRikjRiVGKkYlRiUqKEYkRiUpLCYqJkYqRiVGMUYlRiVGJUYlRipGJUYxRjJGMkYlRihGJUYlKiYsKiomRipGJSlGMSIiJEYlRiUqJiIjOUYlKUYxRipGJUYyKiYiIzdGJUYxRiVGMkY+RjJGJUYxRjJGJSomLCYqLEYqRiVGLkYlKSwmRjFGJUYlRiVGPkYlRkFGMkYzRiVGJSoqRipGJUZHRiVGN0YlRkFGMkYyRiVGKEYyRiU=

Compare with the paper factor(subs(delta=(1+4*M),tA0pap-%))

IiIh

and tC00 C00timestpap:=subs(tA2=tA2pap,C00timest)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS1DMDB0aW1lc3RwYXBHRigsKiIiIkYvKigjRi8iIiNGLyksJiomRjJGL0kiTUdGKEYvRi9GL0YvRjJGL0Y2ISIiRi8qJiMiIiQiIilGL0kkQUExR0YoRi9GNyoqRjJGLyksJkY2Ri9GL0YvRjpGL0Y2RjdGPEY3Rjc3I0Yu

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv

We begin with U(x,0). The degree is at most 64 Ix; Ixform;Deltapx;

KiQpLC4qKiIiIyIiIkkidEc2IkYnLCZJInhHRilGJ0YnRidGJy1JIlVHRik2JEYrIiIhRidGJ0YnRidGK0YnKiRGKyEiIkYnKigsJkYnRicqJkYmRidGKEYnRidGJ0YoRjFGKkYxRjEqJkYoRictSSNEREdGKTYjRi9GJ0YnRiZGJw==

LCoqMiMiIiIiIiNGJUkkQUExRzYiRiUsLComSSJOR0YoRiUpSSJ4R0YoRiZGJSEiIiokKUYrRiZGJUYlKihGJkYlRitGJUYtRiVGLiokRixGJUYlRitGLkYlLCZGLUYlRiVGJUYuRi1GLkYrRi4sJkYlRi5GK0YlRi4pSSREcHhHRihGJEYlRiUqJCksKCokRi1GLkYlRi1GLiooLCZGJUYlKiZGJkYlSSJ0R0YoRiVGJUYlRj5GLkYzRi5GLkYmRiVGJSomRjlGJUkkdEEyR0YoRiVGJUkkdEEwR0YoRiU=

LCgiIiJGIyosIiIjRiNJIk5HNiJGIywmKiQpRiZGJUYjRiNGI0YjRiMpLCZGI0YjRiYhIiJGJUYtSSJ4R0YnRiNGLSomKUYuRiVGI0YqRiNGIw==

Eliminate Delta+ res1:=numer(factor(subs(t=tN,resultant(II-subs(sqrt(Dpx)=sDpx,Ixform),sDpx^2-Deltapx,sDpx)))): nops(res1);

IiRkKg==

No linear term in A1, as expected coeff(res1,AA1,1);degree(res1,AA1);

IiIh

IiIj

degree(res1,II);indets(res1);

IiIj

PChJJEFBMUc2IkkjSUlHRiRJIk5HRiRJJHRBMEdGJEkkdEEyR0YkSSJ4R0Yk

Express all tA_i in terms of A1^2 and N, and eliminate A1^2 tA2sol;tA0sol;

LCQqKiIiIyIiIiwmKigsKiokKUkiTkc2IiIiJEYlRiUqJClGK0YkRiUhIiJGK0YwRiVGMEYlKSwmRiVGMEYrRiUiIiVGJSlJJEFBMUdGLEYkRiVGJSooIiM7RiUsKkYpRiVGLkYlRitGJUYlRjBGJSksKEYuRiVGK0YwRiVGJUYtRiVGMEYlKUYyRiRGMCwmKihGNEYlRitGJUYxRiVGJSomRjdGJUY5RiVGJUYwRjA=

LCgqKCwqKiQpSSJORzYiIiIkIiIiRioqJClGJyIiI0YqRipGJ0YqRiohIiJGKiksJkYqRi5GJ0YqRi1GLkkkdEEyR0YoRipGKiomI0YqIiIpRiopSSRBQTFHRihGLUYqRioqKCwyKiQpRiciIihGKkYqKiYiIiVGKilGJyIiJ0YqRioqJkYpRiopRiciIiZGKkYuKiYiIzdGKilGJ0Y9RipGKiomIiM6RipGJkYqRi4qJiIjNUYqRixGKkYqKiZGQkYqRidGKkYuRi1GKkYqRidGLilGMEY9Ri5GLg==

res2:=numer(factor(subs(tA0=tA0sol,tA2=tA2sol,res1))): nops(res2)

IiUvPw==

algA1N

LCwqKClJIk5HNiIiIiMiIiIpLCZGKCEiIkYlRigiIilGKClJJEFBMUdGJiIiJUYoRigqLEYvRihGJUYoKSwmRiVGKEYoRigiIiRGKClGKiIiKEYoKUYuRjNGKEYrKiwiI0tGKEYlRigpLCgqJEYkRihGKEYlRitGKEYoRjNGKClGKkYvRigpRi5GJ0YoRigqLCIja0YoKUYqRjNGKEYxRihGOUYoRi5GKEYoKiYiJGMjRigpRjoiIidGKEYo

res3:=numer(factor(resultant(res2,algA1N,AA1))): nops(%)

IiIn

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

Nyg8IjwjSSJORzYiRiRGJEYkPCVJI0lJR0YmRiVJInhHRiY=

res4:=op(6,res3): nops(%)

IiZITSI=

This is the same equation for J(x) algIJNx:=res4: degree(res4,II)

IiIp

From now on we specialize to x=2 (later we'll do the same with x=3) Ix;tC00pap

KiQpLC4qKiIiIyIiIkkidEc2IkYnLCZJInhHRilGJ0YnRidGJy1JIlVHRik2JEYrIiIhRidGJ0YnRidGK0YnKiRGKyEiIkYnKigsJkYnRicqJkYmRidGKEYnRidGJ0YoRjFGKkYxRjEqJkYoRictSSNEREdGKTYjRi9GJ0YnRiZGJw==

LCoiIiJGIyooI0YjIiIjRiMpLCYqJkYmRiNJIk1HNiJGI0YjRiNGI0YmRiNGKiEiIkYjKiYjIiIkIiIpRiNJJEFBMUdGK0YjRiwqKkYmRiMpLCZGKkYjRiNGI0YvRiNGKkYsRjFGLEYs

res5:=numer(factor(subs(II=Ix,U(x,0)=Ux0,DD(0)=tD0/t,M=MN,t=tN,x=2,res4))): nops(%)

IiIk

map(indets,[seq(op(i,res5),i=1..nops(res5))])

NyU8IjwlSSJORzYiSSRVeDBHRiZJJHREMEdGJjwjRiU=

res6:=op(2,res5): nops(%)

IiZJWyI=

algtD0N

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

Eliminate D0 res7:=factor(resultant(res6,algtD0N,tD0)): nops(%)

IiIn

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

Nyg8IjwjSSJORzYiRiQ8JEYlSSRVeDBHRiZGJ0Yn

n:=2: series(subs(Ux0=subs(x=2,y=0,User(n)),N=Nser(n),op(4,res7)),t,n); series(subs(Ux0=subs(x=2,y=0,User(n)),N=Nser(n),op(5,res7)),t,n); series(subs(Ux0=subs(x=2,y=0,User(n)),N=Nser(n),op(6,res7)),t,n)

KylJInRHNiIhJSU9JiIiISEnP141IiIiLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIiM=

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiM=

KylJInRHNiIiK0M9NUg9IiIhIS5zVVh5VWAiIiIiLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIiM=

Choose the right factor res8:=op(5,res7): nops(%)

IiVNOw==

algU20N:=res8: degree(%,Ux0)

IiM7

Lats effort: eliminate N res9:=factor(resultant(res8,algN,N)): nops(%)

IiU/QA==

algU20:=collect(res9,Ux0,factor): degree(%,Ux0)

IiNr

Let us check n:=30: series(subs(Ux0=subs(x=2,y=0,User(n)),algU20),t,n+1)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0o=

save(algU20,'algU20'):save(algU20N,'algU20N'): We will also need a different value of x to determine the degree of C(x,y). Take x=3 now. res5:=numer(factor(subs(II=Ix,U(x,0)=Ux0,DD(0)=tD0/t,M=MN,t=tN,x=3,res4))): nops(%)

IiIk

map(indets,[seq(op(i,res5),i=1..nops(res5))])

NyU8IjwlSSJORzYiSSRVeDBHRiZJJHREMEdGJjwjRiU=

res6:=op(2,res5): nops(%)

IiYtWyI=

res7:=factor(resultant(res6,algtD0N,tD0)): nops(%)

IiIn

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

Nyg8IjwjSSJORzYiRiQ8JEYlSSRVeDBHRiZGJ0Yn

n:=2: series(subs(Ux0=subs(x=3,y=0,User(n)),N=Nser(n),op(4,res7)),t,n); series(subs(Ux0=subs(x=3,y=0,User(n)),N=Nser(n),op(5,res7)),t,n); series(subs(Ux0=subs(x=3,y=0,User(n)),N=Nser(n),op(6,res7)),t,n)

KylJInRHNiIhMUMtNVkpcEUkRyIiISIzU1FrYihmdjVfJiIiIi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIj

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiM=

KylJInRHNiIhJlwhZiIiISEoPERIIiIiIi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIj

Choose the right factor res8:=op(5,res7): nops(%)

IiVMOw==

algU30N:=res8: degree(%,Ux0)

IiM7

Eliminate N res9:=factor(resultant(res8,algN,N)): nops(%)

IiU/QA==

algU30:=collect(res9,Ux0,factor): degree(%,Ux0)

IiNr

Let us check n:=30: series(subs(Ux0=subs(x=3,y=0,User(n)),algU30),t,n+1)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0o=

save(algU30,'algU30'):save(algU30N,'algU30N'): save (algtD0N,'algtD0N') Now D(y). First, Ix and Jx satisfy the same equation over Q(N,x), called res4 so far.We take y=3 indets(res4)

PCVJI0lJRzYiSSJOR0YkSSJ4R0Yk

Jy;

KiYsJiokKSwmKiYsJkkieUc2IiIiIiokRikhIiJGK0YrSSJ0R0YqRitGLUYrRisiIiNGK0YrKioiIiVGKylGLkYvRissJkYpRitGK0YrRissJkYrRitGLEYrRitGLUYrKSwmKiZGKUYrLUkjRERHRio2I0YpRitGKyomRjNGLUYuRi1GK0YvRis=

res10:=numer(factor(subs(x=y,II=Delta*(y*Dy+1/t/(y+1))^2,y=3,t=tN,res4))): nops(%);

IiIl

indets(res10)

PCRJI0R5RzYiSSJOR0Yk

map(indets,[seq(op(i,res10),i=1..4)]);

NyY8IjwjSSJORzYiPCRJI0R5R0YmRiVGJA==

res11:=op(3,res10): degree(%,Dy);nops(%%)

IiM7

IiVUOw==

res12:=factor(resultant(res11,algN,N)): nops(%)

IiV6Sg==

indets(res12)

PCRJI0R5RzYiSSJ0R0Yk

algDy3:=res12: Let us check n:=30:series(subs(Dy=subs(y=3,Dser(n)), res12),t,n);

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0k=

The degree of C(x,y) is at most 256 eqC;

LCwqJiwmIiIiRiUqJkkidEc2IkYlLC5JInhHRihGJUkieUdGKEYlKiZGKiEiIkYrRi1GJSokRipGLUYlKiRGK0YtRiUqJkYqRiVGK0YlRiVGJUYtRiUtSSJDR0YoNiRGKkYrRiVGLUYlRiUqKkYnRiUsJkYlRiVGLkYlRiVGK0YtLUkjQ21HRig2I0YuRiVGLSoqRidGJUYqRi0sJkYlRiVGL0YlRiUtRjc2I0YvRiVGLSoqRidGJUYqRi1GK0YtSSRDMDBHRihGJUYt

It suffices to determine the degree of (1+x)Cm(1/x)+(1+y)Cm(1/y)+C00 or equivalently, of (1+x)U(x,0)+ (1+y)U(y,0)+ C00 We do this for specific values of x and y (x=2,y=3) algtD0N

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

subs(t=tN,x=2,y=3,Ux0=U20,Uy0=U30,sum-(1+x)*Ux0-(1+y)*Uy0-tD0/tN)

LCpJJHN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IiIiIiomIiIkRihJJFUyMEdGJ0YoISIiKiYiIiVGKEkkVTMwR0YnRihGLCoqSSR0RDBHRidGKEkiTkdGJ0YsKSwmRihGLEYyRigiIiNGLCwsKiQpRjJGLkYoRigqJkY1RigpRjJGKkYoRiwqJiIiJ0YoKUYyRjVGKEYoKiZGNUYoRjJGKEYsRihGKEYoRiw=

res13:=numer(factor(resultant((%),algtD0N,tD0))): nops(res13)

IiRMKA==

indets(res13);degree(res13,sum)

PCZJIk5HNiJJJFUyMEdGJEkkVTMwR0YkSSRzdW1HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ=

IiIl

Warning: Heavy computations from here on res14:=numer(factor(resultant(res13,subs(Ux0=U30,algU30N),U30))):nops(res14);

IiIn

map(indets,[seq(op(i,res14),i=1..nops(res14))]);

Nyg8IjwjSSJORzYiRiQ8JUYlSSRVMjBHRiZJJHN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJkYnRic=

n:=2: series(subs(N=Nser(n), sum = (1+2)*U20+(1+3)*U30+DD0, U20=subs(x=2,y=0,User(n)), U30=subs(x=3,y=0,User(n)), DD0=subs(y=0,Dser(n)), op(4,res14)),t,n); series(subs(N=Nser(n), sum = (1+2)*U20+(1+3)*U30+DD0, U20=subs(x=2,y=0,User(n)), U30=subs(x=3,y=0,User(n)), DD0=subs(y=0,Dser(n)), op(5,res14)),t,n); series(subs(N=Nser(n), sum = (1+2)*U20+(1+3)*U30+DD0, U20=subs(x=2,y=0,User(n)), U30=subs(x=3,y=0,User(n)), DD0=subs(y=0,Dser(n)), op(6,res14)),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiM=

KylJInRHNiIiLDM1JFxHPiIiISEuIz5LNiwoKT4iIiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIiIw==

KylJInRHNiIiM0QrXiRmIVxpLCMqIiIhITZDO0UzczE7ImVwPiIiIi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIj

res15:=op(4,res14): nops(%)

IiYuSyI=

degree(res15,U20); degree(res15,N);

IiM7

IiRDIg==

degree(algU20N,Ux0);degree(algU20N,N)

IiM7

IiQ/Ig==

save(res15,'algsumU20N') We could go on eliminating U20... But this takes forever #res16:=numer(factor(resultant(res15,subs(Ux0=U20,algU20N),U20))): nops(res16): Now we fix t so that 6txy <1, say t=1/100. We still want to eliminate N and U20. Let us first lower the degree in N res16:=factor(rem(res15,subs(t=1/100,algN),N)): nops(%)

IiQ3Jw==

indets(res16); degree(res16,N)

PCVJIk5HNiJJJFUyMEdGJEkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0Yk

IiIk

res17:=factor(resultant(res16,subs(t=1/100,algN),N)): nops(%); indets(%%)

IiVYQA==

PCRJJFUyMEc2Ikkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0Yk

The equation for U20 algU20Nspecialized:=factor(resultant(algU20N,subs(t=1/100,algN),N)): nops(%); indets(%%); degree(algU20Nspecialized,Ux0)

IiNs

PCNJJFV4MEc2Ig==

IiNr

Elimination of U20 (stopped here!) res18:=numer(factor(resultant(res17,subs(Ux0=U20,algU20Nspecialized),U20))):nops(res18);

IiVHNQ==

indets(res17)

PCRJIk5HNiJJJHN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJA==

degree(res17,N)

IiMnKg==

res18:=factor(rem(res17,subs(t=1/100,algN),N)): nops(%)

IiVHNQ==

res19:=numer(factor(resultant(res18,subs(t=1/100,algN),N))):nops(res19);

IiIi

map(indets,[seq(op(i,res19),i=1..nops(res19))]);

NyY8IjwjSSRzdW1HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiJGJEYk

map(degree,[seq(op(i,res19),i=1..nops(res19))]);

NyYiIiEiJGMjRiQiJDcm

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv

Ix; Ixform;

KiQpLC4qKiIiIyIiIkkidEc2IkYnLCZJInhHRilGJ0YnRidGJy1JIlVHRik2JEYrIiIhRidGJ0YnRidGK0YnKiRGKyEiIkYnKigsJkYnRicqJkYmRidGKEYnRidGJ0YoRjFGKkYxRjEqJkYoRictSSNEREdGKTYjRi9GJ0YnRiZGJw==

LCoqMiMiIiIiIiNGJUkkQUExRzYiRiUsLComSSJOR0YoRiUpSSJ4R0YoRiZGJSEiIiokKUYrRiZGJUYlKihGJkYlRitGJUYtRiVGLiokRixGJUYlRitGLkYlLCZGLUYlRiVGJUYuRi1GLkYrRi4sJkYlRi5GK0YlRi4pSSREcHhHRihGJEYlRiUqJCksKCokRi1GLkYlRi1GLiooLCZGJUYlKiZGJkYlSSJ0R0YoRiVGJUYlRj5GLkYzRi5GLkYmRiVGJSomRjlGJUkkdEEyR0YoRiVGJUkkdEEwR0YoRiU=

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

KylJInhHNiIiIiIhIiMsJCooIiIjRiUsKiooRilGJSlJInRHRiRGKUYlLUkiVUdGJDYkIiIhRjFGJUYlKiZGLEYlLUkjRERHRiQ2I0YxRiVGJUYtISIiRiVGNkYlRi1GNkYlRjYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJUYx

The first coefficient is x^(-2) Mind the sign when taking the square root. We want the series to start with +1/x serrootx:=subs(csgn(1/x)=1,factor(simplify(series(sqrt(subs(Dpx=Deltapx,t=tN,tA0=tA0sol,tA2=tA2pap,M=MN,Ixform)),x,2),symbolic)));

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

Ix; C00timestpap

KiQpLC4qKiIiIyIiIkkidEc2IkYnLCZJInhHRilGJ0YnRidGJy1JIlVHRik2JEYrIiIhRidGJ0YnRidGK0YnKiRGKyEiIkYnKigsJkYnRicqJkYmRidGKEYnRidGJ0YoRjFGKkYxRjEqJkYoRictSSNEREdGKTYjRi9GJ0YnRiZGJw==

LCoiIiJGIyooI0YjIiIjRiMpLCYqJkYmRiNJIk1HNiJGI0YjRiNGI0YmRiNGKiEiIkYjKiYjIiIkIiIpRiNJJEFBMUdGK0YjRiwqKkYmRiMpLCZGKkYjRiNGI0YvRiNGKkYsRjFGLEYs

subs(U(x,0)=add(UU[i]*x^i,i=0..8),DD(0)=C00timestpap/t,Ix);

KiQpLDIqKiIiIyIiIkkidEc2IkYnLCZJInhHRilGJ0YnRidGJyw0KiYpRisiIilGJyZJI1VVR0YpNiNGL0YnRicqJilGKyIiKEYnJkYxNiNGNUYnRicqJilGKyIiJ0YnJkYxNiNGOkYnRicqJilGKyIiJkYnJkYxNiNGP0YnRicqJilGKyIiJUYnJkYxNiNGREYnRicqJilGKyIiJEYnJkYxNiNGSUYnRicqJilGK0YmRicmRjE2I0YmRidGJyomRitGJyZGMTYjRidGJ0YnJkYxNiMiIiFGJ0YnRidGJkYnRitGJyokRishIiJGJyooLCZGJ0YnKiZGJkYnRihGJ0YnRidGKEZXRipGV0ZXKigjRidGJkYnKSwmKiZGJkYnSSJNR0YpRidGJ0YnRidGJkYnRmpuRldGJyomI0ZJRi9GJ0kkQUExR0YpRidGVyoqRiZGJyksJkZqbkYnRidGJ0ZJRidGam5GV0Zdb0ZXRldGJkYn

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

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

factor(series(%,x,5));

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

serrootx;

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

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diffe:=convert(%,polynom): numer(factor(subs(UU[0]=(AA1/4-1)/t,t=tN,M=MN,coeff(diffe,x,0))));

IiIh

A1;

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coeff(coeff(convert(Cser(10),polynom),x,-1),y,0)

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n:=10:series(subs(P1=P1ser(n),M=Mser(n),A1P1/4-1),t,n)

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Jy; Jyform;

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series(Jy,y,2);

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normal(series(subs(Dpy=subs(x=y,Deltapx),Jyform),y,2));

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This is now a series in y with rational coefficients in Q(t,A1) The first coefficient is 1/y^2 -> same properties for the square root; Mind the sign when taking the square root. We want the constant term to be 16, not -16. expand(Delta);

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serrooty:=map(factor,subs(csgn(1/y)=1,csgn(1/t)=1,t=t,factor(simplify(series(sqrt(subs(Dpy=subs(x=y,Deltapx),Jyform)/subs(x=y,Delta)),y,3)))));

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subs(DD(y)=add(DD[i]*y^i,i=0..6),Jy/Delta);

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factor(series(simplify(sqrt(%),symbolic),y,4));

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

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DD0 is already determined C00timestsol; factor(solve(coeff(serdiff,y,1),DD[0]));collect(subs (tA2=tA2sol,%),AA1,factor);factor(t*%-%%%);

LCgqKCwmKigsKiokKUkiTkc2IiIiJCIiIkYsKiQpRikiIiNGLCEiIkYpRjBGLEYwRiwpLCZGLEYwRilGLCIiJUYsKUkkQUExR0YqRi9GLEYsKigiIztGLCwqRidGLEYtRixGKUYsRixGMEYsKSwoRi1GLEYpRjBGLEYsRitGLEYwRiwpRjJGL0YwLCYqKEY0RixGKUYsRjFGLEYsKiZGN0YsRjlGLEYsRjBGMComI0YsRjNGLEY1RixGMEYsRiw=

LCQqKCMiIiIiIiVGJSwoRiYhIiIqJiIiI0YlSSR0QTJHNiJGJUYoSSRBQTFHRixGJUYlSSJ0R0YsRihGKA==

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

IiIh

subs(x=1/x,y=1/x,eqC);

LCoqJiwmIiIiRiUqJkkidEc2IkYlLCoqJiIiI0YlSSJ4R0YoISIiRiUqJClGLEYrRiVGJSomRitGJUYsRiVGJSokRi9GLUYlRiVGLUYlLUkiQ0dGKDYkKiRGLEYtRjVGJUYtRiVGJSosRitGJUYnRiUsJkYsRiVGJUYlRiVGLEYlLUkjQ21HRig2I0YsRiVGLSooRidGJUYvRiVJJEMwMEdGKEYlRi0=

factor(subs(U(x,0)=Cm(x)/x,DD(0)=C00, isolate(%,C00),Ix));

KiwpLCwqJkkidEc2IiIiIilJInhHRiciIiVGKEYoKigiIiNGKEYmRigpRioiIiRGKEYoKihGLUYoRiZGKEYqRihGKCokKUYqRi1GKCEiIkYmRihGLUYoKSwoKigtSSJDR0YnNiQqJEYqRjNGOkYoRiZGKEYqRihGKComRjdGKEYmRihGKEYxRihGLUYoKUYqIiIpRjMpRiZGLUYzKSwmRipGKEYoRihGLUYz

factor(%/(subs(y=x,x=1/x,K)*(C(1/x,1/x)/x^2+1/t/(1+x)))^2);

IiIi

Now when x=1 subs(x=1,y=1,eqC);

LCoqJiwmIiIiRiUqJiIiJ0YlSSJ0RzYiRiUhIiJGJS1JIkNHRik2JEYlRiVGJUYqRiVGJSooIiIlRiVGKEYlLUkjQ21HRik2I0YlRiVGKiomRihGJUkkQzAwR0YpRiVGKg==

II1C:=factor(subs(U(x,0)= Cm(1), x=1, DD(0)=C00,isolate(%,C00),Ix));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVJSTFDR0YoLCQqKiMiIiIiIiVGMSksJkYxISIiKiYiIidGMUkidEdGKEYxRjEiIiNGMSksJiooRjlGMS1JIkNHRig2JEYxRjFGMUY4RjFGMUYxRjFGOUYxKUY4RjlGNUYxNyNGLg==

n:=10: series(subs(C(1,1)=subs(x=1,y=1,Cser(n)),II1C),t,n)

KzdJInRHNiIjIiIiIiIlISIjRighIiIiIiQiIiFGJyIiIyEiJUYqIiM9RichI1siIiYhI1EiIichJVc2IiIoLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiYiIik=

Ix1a:=subs(Dpx=Deltapx,x=1,Ixform);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVJeDFhR0YoLCoqLiMiIiIiIiVGMUkkQUExR0YoRjEsKCokKUkiTkdGKCIiI0YxRjEqJkYyRjFGN0YxISIiRjFGMUYxRjdGOiwmRjFGOkY3RjFGOiksKEYxRjEqKkY4RjFGN0YxLCZGNUYxRjFGMUYxKSwmRjFGMUY3RjpGOEY6RjpGNUYxI0YxRjhGMUYxKihGMEYxKSwmRjFGMSomRjhGMUkidEdGKEYxRjFGOEYxKUZHRjhGOkYxKipGQkYxRkVGMUZHRjpJJHRBMkdGKEYxRjpJJHRBMEdGKEYxNyNGLg==

series(subs(tA0=tA0pap, tA2=tA2pap,AA1=A1ser(n), M=Mser(n),N=Nser(n), %),t,n)

KzVJInRHNiIjIiIiIiIlISIjRighIiIiIiQiIiFGJyIiIyEiJUYqIiM9RichI1siIiYhI1EiIictSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJiIiKA==

Simplification of the square root Delta(1) factor(subs(x=1,Deltapx))

KigsJiokKUkiTkc2IiIiIyIiIkYpRilGKUYpLChGJEYpKiYiIiVGKUYmRikhIiJGKUYpRikpLCZGKUYtRiZGKUYoRi0=

simplify(factor(expand(rationalize(subs(x=1,N=NM,Deltapx/(1-N)^2)))),symbolic);

LCYqJiIiJSIiIilJIk1HNiIiIiNGJSEiIkYlRiU=

Let us check series(subs(x=1,N=Nser(n),Deltapx),t,n)

KzlJInRHNiIiIiIiIiEhIiNGJSEiJCIiIyEjPSIiJCEjZSIiJSEkMyQiIiYhJUo4IiInISVFbyIiKCEmJUhMIiIpISdLQjwiIiotSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNQ==

series(subs(N=Nser(n),M=Mser(n),(1-4*M^2)*(1-N)^2),t,n)

KzlJInRHNiIiIiIiIiEhIiNGJSEiJCIiIyEjPSIiJCEjZSIiJSEkMyQiIiYhJUo4IiInISVFbyIiKCEmJUhMIiIpISdLQjwiIiotSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNQ==

II1:=simplify(subs(Dpx=(1-4*M^2)*(1-N)^2,x=1,Ixform)) assuming N<1;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRJSTFHRigsJCoqIyIiIiIiJUYxLCYqKilJInRHRigiIiNGMUkkQUExR0YoRjEsKCokKUkiTkdGKEY3RjFGMSomRjJGMUY8RjEhIiJGMUYxRjEpLCYqJkYyRjEpSSJNR0YoRjdGMUY+RjFGMSNGMUY3RjFGPiooRjJGMSwoRjBGMSomLChJJHRBMEdGKEYxSSR0QTJHRihGPkYxRjFGMUY1RjFGMSomLCYqJkZERjFGSkYxRj5GMUYxRjFGNkYxRjFGMUY8RjFGMUYxRjxGPkY1Rj5GMTcjRi4=

II1:=collect(II1,[AA1,tA2,tA0],factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRJSTFHRigsKiosIyIiIiIiJUYxLCgqJClJIk5HRigiIiNGMUYxKiZGMkYxRjZGMSEiIkYxRjFGMSksJComLCYqJkY3RjFJIk1HRihGMUYxRjFGOUYxLCZGPkYxRjFGMUYxRjkjRjFGN0YxRjZGOUkkQUExR0YoRjFGOSoqRkFGMSwmRjFGMSomRjdGMUkidEdGKEYxRjFGMUZGRjlJJHRBMkdGKEYxRjlJJHRBMEdGKEYxKihGMEYxKUZERjdGMSlGRkY3RjlGMTcjRi4=

One expects degree 16 for this quantity, but we will see that it has degree 8 only and lies in Q(t,P1)=Q(t,A1^2) II1a:=map(c->factor(expand(rationalize(subs(N=NM,c)))),II1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVJSTFhR0YoLCoqLCMiIiIiIiVGMSksJComLCYqJiIiI0YxSSJNR0YoRjFGMUYxISIiRjEsJkY3RjFGMUYxRjFGOiNGMUY4RjFGNkYxRjlGOkkkQUExR0YoRjFGMSoqRjxGMSwmRjFGMSomRjhGMUkidEdGKEYxRjFGMUZBRjpJJHRBMkdGKEYxRjpJJHRBMEdGKEYxKihGMEYxKUY/RjhGMSlGQUY4RjpGMTcjRi4=

We need to write the square root sqrt(1-4M^2) in terms of M and A1 res:=factor(algA1M,sqrt(1-4*M^2))

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

subs(M=0,op(1,res));subs(M=0,op(2,res));

LCYqJiIjayIiIkkkQUExRzYiRiVGJSIkYyNGJQ==

LCYqJiIjayIiIkkkQUExRzYiRiUhIiIiJGMjRiU=

Choose the correct factor delta:=op(2,factor(isolate(op(2,res),sqrt(-4*M^2 + 1))))

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

tA2pap;tA0pap

LCgqJiksJiomIiIjIiIiSSJNRzYiRihGKEYoRihGJ0YoRikhIiJGKComI0YoIiIlRihJJEFBMUdGKkYoRisqKkYuRigpLCZGKUYoRihGKCIiJEYoRilGK0YvRitGKw==

LCgqJiMiIiIiIilGJSlJJEFBMUc2IiIiI0YlRiUqJiwqKiZGKkYlKUkiTUdGKSIiJEYlRiUqJiIjOUYlKUYvRipGJSEiIiomIiM3RiVGL0YlRjRGMEY0RiVGL0Y0RiUqJiwmKihGJEYlRihGJUYvRjRGJSoqRipGJSksJkYvRiVGJUYlRjBGJUYoRjRGM0Y0RiVGJSwmKiQpLCYqJiIiJUYlRi9GJUYlRiVGJSNGMEYqRiVGJSokKSwmKiZGKkYlRi9GJUYlRiVGJUYqRiVGNEYlRiU=

II1b:=collect(subs(tA0=tA0pap,tA2=tA2pap,sqrt(-(2*M - 1)*(1 + 2*M))=delta,t=tM,II1a),[AA1],factor)

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

We still have roots of (1+4M) isolate(algA1Malg,(4*M + 1)^(3/2))

LyokKSwmKiYiIiUiIiJJIk1HNiJGKEYoRihGKCMiIiQiIiNGKComLCgqJilGKUYtRigpSSRBQTFHRipGJ0YoISIiKioiI0tGKEYpRigpLCZGKUYoRihGKEYsRigpRjNGLUYoRjQqJiIkYyNGKClGOCIiJ0YoRjRGKCwmKihGJ0YoRilGKClGM0YsRihGKCooIiNrRihGN0YoRjNGKEY0RjQ=

II1c:=factor(subs(%,II1b))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVJSTFjR0YoLCQqLCMiIiIiI0tGMSxSKigiIiZGMSlJJEFBMUdGKCIiJ0YxKUkiTUdGKCIiJEYxRjEqKCIkbyRGMSlGNyIiJUYxKUY6RjVGMSEiIiooIiRvKEYxKUY3IiIjRjEpRjoiIihGMUYxKiYiJSc0JUYxKUY6IiIqRjFGQSooIiU7O0YxRj5GMSlGOkY/RjFGQSooIiYrRyJGMUZERjEpRjpGOEYxRjEqJiIma28kRjEpRjoiIilGMUZBKigiJSlbIkYxRj5GMUY5RjFGQSooIiZTQSVGMUZERjFGQEYxRjEqJiInY3U5RjFGRkYxRkEqKCIkayVGMUY+RjEpRjpGRUYxRkEqKCImLypmRjFGREYxRk5GMUYxKiYiJ2tTTUYxRlFGMUZBKihGVUYxRj5GMUY6RjFGQSooIiYhKUclRjFGREYxRjlGMUYxKiYiJyc0OyZGMUZARjFGQSooIiZXZCJGMUZERjFGaG5GMUYxKiZGYW9GMUZORjFGQSooIiUpbyNGMUZERjFGOkYxRjEqJkZcb0YxRjlGMUZBKiYiJEciRjFGREYxRjEqJkZlbkYxRmhuRjFGQSomRlNGMUY6RjFGQUZJRkFGMUZobkZBRkRGQSwsKiZGREYxRjpGMUYxKiYiIztGMUY5RjFGQSomIiNbRjFGaG5GMUZBKiZGYXBGMUY6RjFGQUZfcEZBRkFGMTcjRi4=

This is even in A1 II1d:=subs(AA1=sqrt(A1s),II1c)

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

algA1sMJSFH

LCwqJilJJEExc0c2IiIiJSIiIilJIk1HRiZGJ0YoRigqKiIjO0YoKUYqIiIjRigsLComRidGKEYpRihGKComIiNfRigpRioiIiRGKCEiIiomIiNPRihGLUYoRjUqJiIiKUYoRipGKEY1RihGNUYoKUYlRjRGKEYoKiwiJDcmRihGKkYoLCwqJkY0RihGKUYoRigqJiIjdEYoRjNGKEYoKiYiI2RGKEYtRihGKComIiM6RihGKkYoRihGKEYoRigpLCZGKkYoRihGKEY0RigpRiVGLkYoRigqKiIlJzQlRihGL0YoKUZGIiInRihGJUYoRigqJiImT2InRigpRkYiIzdGKEYo

Normal form for II1 II1e:=collect(factor(subs(RootOf(algA1sM,A1s)=AA1^2,evala(Normal(subs(A1s=RootOf(algA1sM,A1s),II1d))))),AA1,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVJSTFlR0YoLCgqKiMiIiIiJDcmRjFJIk1HRihGMSksJkYzRjFGMUYxIiIkISIiKUkkQUExR0YoIiIlRjFGMSosI0YxIiNLRjFGM0Y3LCwqJiIjNUYxKUYzRjpGMUYxKiYiI01GMSlGM0Y2RjFGNyomIiM9RjEpRjMiIiNGMUY3KiZGSEYxRjNGMUY3RjFGN0YxRjRGNylGOUZIRjFGMSooI0YxRjpGMSwsKiYiIzlGMUZBRjFGMSomIiNBRjFGREYxRjcqJiIiJ0YxRkdGMUY3RklGMUYxRjdGMUZHRjdGNzcjRi4=

n:=10: series(subs(M=Mser(n),A1s=AA1^2, AA1=A1ser(n), II1b),t,n)

KzdJInRHNiIjIiIiIiIlISIjRighIiIiIiQiIiFGJyIiIyEiJUYqIiM9RichI1siIiYhI1EiIichJVc2IiIoLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiYiIik=

series(subs(C(1,1)=subs(x=1,y=1,Cser(n)),II1C),t,n)

KzdJInRHNiIjIiIiIiIlISIjRighIiIiIiQiIiFGJyIiIyEiJUYqIiM9RichI1siIiYhI1EiIichJVc2IiIoLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiYiIik=

Using t ? Does not give nicer expressions factor(expand(rationalize(subs(M=Mt,op(1,II1e))))); factor(expand(rationalize(subs(M=Mt,op(3,II1e)))))

LCQqLCMiIiIiJUM1RiUpSSRBQTFHNiIiIiVGJUkidEdGKUYlLCwqKCIiJ0YlRitGJSksJComLCZGJUYlKiYiIiNGJUYrRiVGJUYlLCZGJSEiIiomRi5GJUYrRiVGJUYlRjYjRiVGNEYlRiUqJiIjT0YlKUYrRjRGJUY2KiRGL0YlRjYqJiIjQ0YlRitGJUY2RiVGNkYlKSwmKiYiIiRGJUYrRiVGJUYlRiVGQkY2RjY=

LCQqKCMiIiIiJSc0JUYlLDwqKCIkWyVGJSlJInRHNiIiIiRGJSksJComLCZGJUYlKiYiIiNGJUYrRiVGJUYlLCZGJSEiIiomIiInRiVGK0YlRiVGJUY1I0YlRjNGJUY1KigiJUdIRiUpRisiIiVGJUYuRiVGJSooIiNrRiUpRi8jRi1GM0YlRitGJUY1KiYiIihGJUY/RiVGJSooIiRXIkYlRj9GJUYqRiVGJSooIiQ7IkYlRj9GJSlGK0YzRiVGJSooIiVLU0YlKUYrIiImRiVGLkYlRjUqKCIjIypGJUYrRiVGLkYlRiUqKCIkKUdGJUZHRiVGLkYlRjUqJkY+RiVGO0YlRiUqJiIlTzpGJUZKRiVGJSomIiUvQkYlKUYrRjdGJUYlKiZGQkYlRi5GJUY1RiVGVUY1RiU=

Equation for C(1,1) II1e;II1C;

LCgqKiMiIiIiJDcmRiVJIk1HNiJGJSksJkYnRiVGJUYlIiIkISIiKUkkQUExR0YoIiIlRiVGJSosI0YlIiNLRiVGJ0YsLCwqJiIjNUYlKUYnRi9GJUYlKiYiI01GJSlGJ0YrRiVGLComIiM9RiUpRiciIiNGJUYsKiZGPUYlRidGJUYsRiVGLEYlRilGLClGLkY9RiVGJSooI0YlRi9GJSwsKiYiIzlGJUY2RiVGJSomIiNBRiVGOUYlRiwqJiIiJ0YlRjxGJUYsRj5GJUYlRixGJUY8RixGLA==

LCQqKiMiIiIiIiVGJSksJkYlISIiKiYiIidGJUkidEc2IkYlRiUiIiNGJSksJiooRi5GJS1JIkNHRi02JEYlRiVGJUYsRiVGJUYlRiVGLkYlKUYsRi5GKUYl

algC11M:=collect(op(1,op(2,numer(factor(resultant(II1e-subs(C(1,1)=C11,t=tM,II1C),algA1M,AA1))))),C11,factor);

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

series(subs(M=Mser(n),C11=subs(x=1,y=1,Cser(n)),%),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

nops(factor(algC11M))

IiRxIg==

algC11:=collect(numer(factor(resultant(algC11M,algM,M))),C11,factor);degree(%,C11)

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

IiM7

n:=20: series(subs(C11=subs(x=1,y=1,Cser(n)),algC11),t,n);

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIz8=

nops(factor(algC11))

IiQnPQ==

Asymptotics II1e

LCgqKiMiIiIiJDcmRiVJIk1HNiJGJSksJkYnRiVGJUYlIiIkISIiKUkkQUExR0YoIiIlRiVGJSosI0YlIiNLRiVGJ0YsLCwqJiIjNUYlKUYnRi9GJUYlKiYiI01GJSlGJ0YrRiVGLComIiM9RiUpRiciIiNGJUYsKiZGPUYlRidGJUYsRiVGLEYlRilGLClGLkY9RiVGJSooI0YlRi9GJSwsKiYiIzlGJUY2RiVGJSomIiNBRiVGOUYlRiwqJiIiJ0YlRjxGJUYsRj5GJUYlRixGJUY8RixGLA==

Msing;A1sing

LDQjIiIiIiIjRiQqKEYjRiQpIiIkRiNGJClJKGVwc2lsb25HNiJGI0YkISIiKiYjRigiIiVGJEYqRiRGJCooIyIiKCIjO0YkRidGJClGKiNGKEYlRiRGLComRi5GJClGKkYlRiRGJCooIyIkNiIiJGMjRiRGJ0YkKUYqIyIiJkYlRiRGLComRi5GJClGKkYoRiRGJCooIyIkKCkpIiVbP0YkRidGJClGKiNGMkYlRiRGLC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRis2IyokKUYqRi9GJEYk

LCwqJiIiJyIiIilGJCNGJSIiI0YlRiUqJkYkRiUpIiIkRidGJSEiIioqRihGJSlJKGVwc2lsb25HNiIjRisiIiVGJSlGKyNGJUYyRiUsJiokRiZGJUYsKiRGKkYlRiVGJUYlKihGK0YlRi9GJUY1RiVGLC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRjA2IyokKUYvIyIiJkYyRiVGJQ==

II1sing:=map(factor,series(subs(M=Msing,AA1=A1sing,II1e),epsilon,5))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShJSTFzaW5nR0YoLCYqKiksJiokKSIiJyMiIiIiIiNGNiEiIiokKSIiJEY1RjZGNkY3RjYpRjsjRjsiIiVGNkYzRjYpSShlcHNpbG9uR0YoRj1GNkY2LUkiT0dGJTYjKiQpRkAjIiImRj5GNkY2NyNGLg==

II1C

LCQqKiMiIiIiIiVGJSksJkYlISIiKiYiIidGJUkidEc2IkYlRiUiIiNGJSksJiooRi5GJS1JIkNHRi02JEYlRiVGJUYsRiVGJUYlRiVGLkYlKUYsRi5GKUYl

series(sqrt(II1sing),epsilon)

LCYqKiIiJCIiIiwmKiQpIiIjI0YlRilGJUYlRiUhIiJGJSkqJilGJCNGJSIiJUYlRihGJUYqRiUpSShlcHNpbG9uRzYiI0YkIiIpRiVGJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRjM2IyokKUYyIyIiKEY1RiVGJQ==

kappa:=subs(epsilon=1,op(1,%))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZrYXBwYUdGKCwkKigiIiQiIiIsJiokKSIiIyNGMUY1RjFGMUYxISIiRjEpKiYpRjAjRjEiIiVGMUY0RjFGNkYxRjE3I0Yu

kappapap:=2^(1/4)*3^(9/8)*(sqrt(2)-1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlrYXBwYXBhcEdGKCwkKioiIiQiIiIpIiIjI0YxIiIlRjEpRjAjRjEiIilGMSwmKiQpRjMjRjFGM0YxRjFGMSEiIkYxRjE3I0Yu

simplify(kappa/kappapap)

IiIi

evalf(kappa/GAMMA(5/8))

JCIrKiozeSI9IiEiKg==

n:=80:plot([seq([1/i,coeff(subs(x=1,y=1,Cser(n)),t,i)/6^(i)*i^(3/8)],i=1..n)],a=0..0.1,b=1.1..1.2,style=point);

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ZTYiLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lLG1hdGh2YXJpYW50R1EnaXRhbGljNiItJSZTVFlMRUc2IyUmUE9JTlRHLSUpX1ZJU0lCTEVHNiMiIiItJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiQ1KCEiIi0lKUJPVU5EU19ZRzYjJCIkUyIhIiItJS1CT1VORFNfV0lEVEhHNiMkIiVxUyEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiU1VyEiIi0lKUNISUxEUkVORzYiLSUrQU5OT1RBVElPTkc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZRzYjJCIiISEiIi0lLUJPVU5EU19XSURUSEc2IyQiJStdISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJStdISIiLSUpQ0hJTERSRU5HNiI=NiI=

JSFH