NiRRQS91c3IvbGFicmkvYm91c3F1ZXQvYWxnb2xpYi9nZnVuNiJROS91c3IvbG9jYWwvbWFwbGUyMDE2L2xpYkYk

NiVRRC91c3IvbGFicmkvYm91c3F1ZXQvYWxnb2xpYi9hbGdvbGliNiJRQS91c3IvbGFicmkvYm91c3F1ZXQvYWxnb2xpYi9nZnVuRiRROS91c3IvbG9jYWwvbWFwbGUyMDE2L2xpYkYk

NiVRSC91c3IvbGFicmkvYm91c3F1ZXQvYWxnb2xpYi9OZXd0b25HRjEuMzYiUUEvdXNyL2xhYnJpL2JvdXNxdWV0L2FsZ29saWIvZ2Z1bkYkUTkvdXNyL2xvY2FsL21hcGxlMjAxNi9saWJGJA==

In this session, we determine decoupling pairs for the eight algebraic models (those with vanishing alternating orbit sum). We use the general construction of Theorem 12.Then, we check the correctness of the decoupling functions given in the paper for the other 9 decoupled models (with infinite group).Finally, we consider a starting point different from (0,0) (Section 7): for the 8 models with a finite froup, this is done in the corresponding section, while for models with an infinite group, this is done at the end of the session.

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwoKiRJInhHRighIiIiIiIqJEkieUdGKEYxRjIqJkYwRjJGNEYyRjI3I0Yu

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCoqKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqJkYwRjFGM0YxRjEqJkYwRjFGNkYxRjEqJkYzRjFGNkYxISIiNyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ0R0YoISIiKUkieUdGKEYzRjUsKEY0RjVGN0YyKiQpLCoqKCIiJUYyKUY0RjNGMilGNyIiJEYyRjUqJEY+RjJGMiooRjNGMkY0RjJGN0YyRjUqJEY2RjJGMkYxRjJGMkYyRjIsJCoqRjFGMiwoRjRGMkY3RjVGOUYyRjJGNEY1RjZGNUY1NyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ0R0YoISIiKUkieEdGKEYzRjUsKEY0RjVGN0YyKiQpLCoqKCIiJUYyKUY0RjNGMilGNyIiJEYyRjUqJEY+RjJGMiooRjNGMkY0RjJGN0YyRjUqJEY2RjJGMkYxRjJGMkYyRjIsJCoqRjFGMiwoRjRGMkY3RjVGOUYyRjJGNEY1RjZGNUY1NyNGLg==

qy:=coeff(K,x,0)/coeff(K,x,2); qx:=coeff(K,y,0)/coeff(K,y,2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeUdGKCokSSJ5R0YoISIiNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeEdGKCokSSJ4R0YoISIiNyNGLg==

phi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([subs(y=v,qy)/u,v]): end: psi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([u,subs(x=u,qx)/v]): end: [x,y ]; phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);

NyRJInhHNiJJInlHRiQ=

NyQqJkkieUc2IiEiIkkieEdGJUYmRiQ=

NyQqJkkieUc2IiEiIkkieEdGJUYmRic=

NyRJInlHNiJJInhHRiQ=

NyRJInlHNiIqJkYjISIiSSJ4R0YkRiY=

NyRJInhHNiIqJkkieUdGJCEiIkYjRic=

NyRJInhHNiJJInlHRiQ=

theta:=proc(p) psi(phi(p)): end: theta_itere:=proc(p,k) if k= 0 then p else theta_itere(theta(p),k-1): fi: end: We apply the construction of Theorem 12, with H=xy n:=3: [seq(theta_itere([x,y],i),i=1..n-1)];

NyQ3JComSSJ5RzYiISIiSSJ4R0YmRidGKDckRiVGJA==

li:=map(p->p[1]*p[2],%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNsaUdGKDckKiRJInlHRighIiIqJEkieEdGKEYxNyNGLg==

Htau:=-1/n*add(i*li[i],i=1..n-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIdGF1R0YoLCYqJiMiIiIiIiRGMUkieUdGKCEiIkY0KiYjIiIjRjJGMUkieEdGKEY0RjQ3I0Yu

The function F(x) Fx:=rationalize(subs(y=Ysol[1],Htau)+ subs(y=Ysol[2],Htau));expand(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiojIiIiIiIkRjFJInRHRighIiJJInhHRihGNCwmKiZGMkYxRjNGMUYxRjVGMUYxRjQ3I0Yu

LCYqJEkieEc2IiEiIkYmKiYjIiIiIiIkRilJInRHRiVGJkYm

Computing G(y) via F(x) Gy:=rationalize(subs(x=Xsol[1],x*y-Fx));expand(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiojIiIiIiIkRjEsJiomRjJGMUkidEdGKEYxISIiKiYiIiVGMUkieUdGKEYxRjFGMUY5RjZGNUY2RjE3I0Yu

LCYqJEkieUc2IiEiIkYmKiYjIiIlIiIkIiIiSSJ0R0YlRiZGKw==

let us check the formula given in the paper for G(y). First, define tau-tilda thetat:=proc(p) phi(psi(p)): end: thetat_itere:=proc(p,k) if k= 0 then p else thetat_itere(thetat(p),k-1): fi: end: [seq(thetat_itere([x,y],i),i=1..n-1)];

NyQ3JEkieUc2IiomRiQhIiJJInhHRiVGJzckRiZGKA==

lit:=map(p->p[1]*p[2],%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRsaXRHRig3JCokSSJ4R0YoISIiKiRJInlHRihGMTcjRi4=

Htaut:=-1/n*add(i*lit[i],i=1..n-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIdGF1dEdGKCwmKiYjIiIiIiIkRjFJInhHRighIiJGNComIyIiI0YyRjFJInlHRihGNEY0NyNGLg==

The invariant associated with H(x,y)=xy. We first form the list of elements of the group. L:=[[x,y]]: for i from 1 to 2*n-1 do if i mod 2=1 then L:=[op(L), phi(L[i])]; else L:=[op(L), psi(L[i])]; fi: od: L;

Nyg3JEkieEc2IkkieUdGJTckKiZGJiEiIkYkRilGJjckRihGJDckRiZGJDckRiZGKDckRiRGKA==

Hsig:=convert(map(p->p[1]*p[2], L),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIc2lnR0YoLCgqKCIiIyIiIkkieEdGKEYxSSJ5R0YoRjFGMSomRjBGMUYyISIiRjEqJkYwRjFGM0Y1RjE3I0Yu

Ix:=rationalize(subs(y=Ysol[1],Hsig));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeEdGKCwkKiYiIiMiIiJJInRHRighIiJGMTcjRi4=

Iy:=rationalize(subs(x=Xsol[1],Hsig));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeUdGKCwkKiYiIiMiIiJJInRHRighIiJGMTcjRi4=

Expected value for G(y) rationalize(subs(x=Xsol[1],Htaut)+subs(x=Xsol[2],Htaut)+(1-1/2/n)*Iy);

LCQqKiMiIiIiIiRGJSwmKiZGJkYlSSJ0RzYiRiVGJSomIiIlRiVJInlHRipGJSEiIkYlRi1GLkYpRi5GLg==

factor(%-Gy);

IiIh

expand(Fx+Gy);

LCgqJEkieEc2IiEiIkYmKiRJInRHRiVGJiIiIiokSSJ5R0YlRiZGJg==

Starting point (a,a) (Section 7). let us check that the x-decoupling function is what we claim for a from 0 to 4 do Fx:=-1/x^(a+1): normal(subs(x=Xsol[1],Fx-(x*y)^(a+1))-subs(x=Xsol[2],Fx-(x*y)^(a+1)));od;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiRJInhHRighIiJGMTcjRi4=

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiQpSSJ4R0YoIiIjISIiRjM3I0Yu

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiQpSSJ4R0YoIiIkISIiRjM3I0Yu

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiQpSSJ4R0YoIiIlISIiRjM3I0Yu

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiQpSSJ4R0YoIiImISIiRjM3I0Yu

IiIh

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwoSSJ4R0YoIiIiSSJ5R0YoRjAqJkYxISIiRi9GM0YwNyNGLg==

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCoqKEkidEdGKCIiIilJInhHRigiIiNGMUkieUdGKEYxRjEqKEYwRjFGM0YxKUY1RjRGMUYxKiZGM0YxRjVGMSEiIkYwRjE3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ5R0YoISIiSSJ0R0YoRjUsKComRjZGMilGNEYzRjJGNUY0RjIqJCkqJkY0RjIsKiomKUY2RjNGMilGNCIiJEYyRjIqKEYzRjJGNkYyRjlGMkY1KiYiIiVGMkY/RjJGNUY0RjJGMkYxRjJGMkYyRjIsJCoqRjFGMiwoRjhGMkY6RjJGNEY1RjJGNEY1RjZGNUY1NyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ4R0YoISIiSSJ0R0YoRjUsKComRjZGMilGNEYzRjJGNUY0RjIqJCkqJkY0RjIsKiomKUY2RjNGMilGNCIiJEYyRjIqKEYzRjJGNkYyRjlGMkY1KiYiIiVGMkY/RjJGNUY0RjJGMkYxRjJGMkYyRjIsJCoqRjFGMiwoRjhGMkY6RjJGNEY1RjJGNEY1RjZGNUY1NyNGLg==

qy:=coeff(K,x,0)/coeff(K,x,2); qx:=coeff(K,y,0)/coeff(K,y,2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeUdGKCokSSJ5R0YoISIiNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeEdGKCokSSJ4R0YoISIiNyNGLg==

phi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([subs(y=v,qy)/u,v]): end: psi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([u,subs(x=u,qx)/v]): end: [x,y ]; phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);

NyRJInhHNiJJInlHRiQ=

NyQqJkkieUc2IiEiIkkieEdGJUYmRiQ=

NyQqJkkieUc2IiEiIkkieEdGJUYmRic=

NyRJInlHNiJJInhHRiQ=

NyRJInlHNiIqJkYjISIiSSJ4R0YkRiY=

NyRJInhHNiIqJkkieUdGJCEiIkYjRic=

NyRJInhHNiJJInlHRiQ=

theta:=proc(p) psi(phi(p)): end: theta_itere:=proc(p,k) if k= 0 then p else theta_itere(theta(p),k-1): fi: end: n:=3: [seq(theta_itere([x,y],i),i=1..n-1)];

NyQ3JComSSJ5RzYiISIiSSJ4R0YmRidGKDckRiVGJA==

li:=map(p->p[1]*p[2],%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNsaUdGKDckKiRJInlHRighIiIqJEkieEdGKEYxNyNGLg==

Htau:=-1/n*add(i*li[i],i=1..n-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIdGF1R0YoLCYqJiMiIiIiIiRGMUkieUdGKCEiIkY0KiYjIiIjRjJGMUkieEdGKEY0RjQ3I0Yu

The function F(x) Fx:=rationalize(subs(y=Ysol[1],Htau)+ subs(y=Ysol[2],Htau));expand(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiojIiIiIiIkRjFJInRHRighIiIsKComRjNGMSlJInhHRihGMkYxRjEqJClGOCIiI0YxRjQqJiIiJUYxRjNGMUY0RjFGOEY0RjE3I0Yu

LCgqJiMiIiIiIiRGJSlJInhHNiIiIiNGJUYlKihGJEYlSSJ0R0YpISIiRihGJUYtKiYjIiIlRiZGJUYoRi1GLQ==

Computing G(y) from F(x) Gy:=factor(simplify(expand(rationalize(subs(x=Xsol[1],x*y-Fx))),symbolic));expand(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiojIiIiIiIkRjFJInlHRighIiJJInRHRihGNCwoKigiIiVGMUY1RjEpRjNGMkYxRjEqJkY4RjEpRjMiIiNGMUY0RjVGNEYxRjQ3I0Yu

LCgqJiMiIiUiIiQiIiIpSSJ5RzYiIiIjRichIiIqKEYkRidJInRHRipGLEYpRidGJyomI0YnRiZGJ0YpRixGJw==

let us check the formula given in the paper for G(y) thetat:=proc(p) phi(psi(p)): end: thetat_itere:=proc(p,k) if k= 0 then p else thetat_itere(thetat(p),k-1): fi: end: [seq(thetat_itere([x,y],i),i=1..n-1)];

NyQ3JEkieUc2IiomRiQhIiJJInhHRiVGJzckRiZGKA==

lit:=map(p->p[1]*p[2],%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRsaXRHRig3JCokSSJ4R0YoISIiKiRJInlHRihGMTcjRi4=

Htaut:=-1/n*add(i*lit[i],i=1..n-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIdGF1dEdGKCwmKiYjIiIiIiIkRjFJInhHRighIiJGNComIyIiI0YyRjFJInlHRihGNEY0NyNGLg==

The invariant associated with H(x,y)=xy L:=[[x,y ]]: for i from 1 to 2*n-1 do if i mod 2 =1 then L:=[op(L), phi(L[i])]; else L:=[op(L), psi(L[i])]; fi: od: L;

Nyg3JEkieEc2IkkieUdGJTckKiZGJiEiIkYkRilGJjckRihGJDckRiZGJDckRiZGKDckRiRGKA==

Hsig:=convert(map(p->p[1]*p[2], L),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIc2lnR0YoLCgqKCIiIyIiIkkieEdGKEYxSSJ5R0YoRjFGMSomRjBGMUYyISIiRjEqJkYwRjFGM0Y1RjE3I0Yu

Ix:=rationalize(subs(y=Ysol[1],Hsig));expand(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeEdGKCooLCgqKCIiIyIiIkkidEdGKEYyKUkieEdGKCIiJEYyISIiKiZGMUYyKUY1RjFGMkYyKiZGMUYyRjNGMkYyRjJGNUY3RjNGNzcjRi4=

LCgqJiIiIyIiIilJInhHNiJGJEYlISIiKihGJEYlSSJ0R0YoRilGJ0YlRiUqJkYkRiVGJ0YpRiU=

Iy:=rationalize(subs(x=Xsol[1],Hsig));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeUdGKCooLCgqKCIiIyIiIkkidEdGKEYyKUkieUdGKCIiJEYyISIiKiZGMUYyKUY1RjFGMkYyKiZGMUYyRjNGMkYyRjJGNUY3RjNGNzcjRi4=

Expected value for G(y) rationalize(subs(x=Xsol[1],Htaut)+subs(x=Xsol[2],Htaut)+(1-1/2/n)*Iy);

LCQqKiMiIiIiIiRGJUkieUc2IiEiIkkidEdGKEYpLCgqKCIiJUYlRipGJSlGJ0YmRiVGJSomRi1GJSlGJyIiI0YlRilGKkYpRiVGKQ==

factor(%-Gy);

IiIh

We can simplify these expressions by using invariants expand(Gy);expand(Iy);

LCgqJiMiIiUiIiQiIiIpSSJ5RzYiIiIjRichIiIqKEYkRidJInRHRipGLEYpRidGJyomI0YnRiZGJ0YpRixGJw==

LCgqJiIiIyIiIilJInlHNiJGJEYlISIiKihGJEYlSSJ0R0YoRilGJ0YlRiUqJkYkRiVGJ0YpRiU=

factor(Gy-2/3*Iy);

LCQqJEkieUc2IiEiIkYm

factor(Fx+2/3*Ix);expand(%);

LCQqKEkieEc2IiIiIiwmKiZJInRHRiVGJkYkRiZGJkYmISIiRiZGKUYqRio=

LCYqJClJInhHNiIiIiMiIiIhIiIqJkkidEdGJkYpRiVGKEYo

factor(-1/y-x^2+x/t-x*y);

LCQqKCwqKihJInRHNiIiIiIpSSJ4R0YnIiIjRihJInlHRidGKEYoKihGJkYoRipGKClGLEYrRihGKComRipGKEYsRighIiJGJkYoRihGLEYwRiZGMEYw

factor(%/K);

LCQqJkkieUc2IiEiIkkidEdGJUYmRiY=

Starting point (a,a) (Section 7) for a from 0 to 4 do Fx:=-1/x^(a+1): normal(subs(x=Xsol[1],Fx-(x*y)^(a+1))-subs(x=Xsol[2],Fx-(x*y)^(a+1)));od;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiRJInhHRighIiJGMTcjRi4=

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiQpSSJ4R0YoIiIjISIiRjM3I0Yu

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiQpSSJ4R0YoIiIkISIiRjM3I0Yu

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiQpSSJ4R0YoIiIlISIiRjM3I0Yu

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiQpSSJ4R0YoIiImISIiRjM3I0Yu

IiIh

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwuSSJ4R0YoIiIiSSJ5R0YoRjAqJkYxISIiRi9GM0YwKiRGL0YzRjAqJEYxRjNGMComRi9GMEYxRjBGMDcjRi4=

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLDAqKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGMkYxRjZGMUYxKihGMEYxRjNGMUY1RjFGMSomRjBGMUYzRjFGMSomRjBGMUY2RjFGMSomRjNGMUY2RjEhIiJGMEYxNyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCosIyIiIiIiI0YyLCoqJkkidEdGKEYyKUkieUdGKEYzRjIhIiJGNkY5RjhGMiokKSwyKiYpRjZGM0YyKUY4IiIlRjJGMiooRkBGMkY+RjIpRjgiIiRGMkY5KigiIidGMkY+RjJGN0YyRjkqKEYzRjJGNkYyRkJGMkY5KihGQEYyRj5GMkY4RjJGOSokRj5GMkYyKihGM0YyRjZGMkY4RjJGOSokRjdGMkYyRjFGMkYyRjJGOEY5RjZGOSwmRjhGMkYyRjJGOUYyLCQqLEYxRjIsKkY1RjJGOkYyRjZGMkY4RjlGMkY4RjlGNkY5RktGOUY5NyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCosIyIiIiIiI0YyLCoqJkkidEdGKEYyKUkieEdGKEYzRjIhIiJGNkY5RjhGMiokKSwyKiYpRjZGM0YyKUY4IiIlRjJGMiooRkBGMkY+RjIpRjgiIiRGMkY5KigiIidGMkY+RjJGN0YyRjkqKEYzRjJGNkYyRkJGMkY5KihGQEYyRj5GMkY4RjJGOSokRj5GMkYyKihGM0YyRjZGMkY4RjJGOSokRjdGMkYyRjFGMkYyRjJGOEY5RjZGOSwmRjhGMkYyRjJGOUYyLCQqLEYxRjIsKkY1RjJGOkYyRjZGMkY4RjlGMkY4RjlGNkY5RktGOUY5NyNGLg==

qy:=coeff(K,x,0)/coeff(K,x,2); qx:=coeff(K,y,0)/coeff(K,y,2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeUdGKComLCYqJkkidEdGKCIiIkkieUdGKEYyRjJGMUYyRjIsJiomRjFGMilGMyIiI0YyRjJGMEYyISIiNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeEdGKComLCYqJkkidEdGKCIiIkkieEdGKEYyRjJGMUYyRjIsJiomRjFGMilGMyIiI0YyRjJGMEYyISIiNyNGLg==

phi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([subs(y=v,qy)/u,v]): end: psi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([u,subs(x=u,qx)/v]): end: [x,y ]; phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);

NyRJInhHNiJJInlHRiQ=

NyQqJkkieUc2IiEiIkkieEdGJUYmRiQ=

NyQqJkkieUc2IiEiIkkieEdGJUYmRic=

NyRJInlHNiJJInhHRiQ=

NyRJInlHNiIqJkYjISIiSSJ4R0YkRiY=

NyRJInhHNiIqJkkieUdGJCEiIkYjRic=

NyRJInhHNiJJInlHRiQ=

theta:=proc(p) psi(phi(p)): end: theta_itere:=proc(p,k) if k= 0 then p else theta_itere(theta(p),k-1): fi: end: n:=3: [seq(theta_itere([x,y],i),i=1..n-1)];

NyQ3JComSSJ5RzYiISIiSSJ4R0YmRidGKDckRiVGJA==

li:=map(p->p[1]*p[2],%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNsaUdGKDckKiRJInlHRighIiIqJEkieEdGKEYxNyNGLg==

Htau:=-1/n*add(i*li[i],i=1..n-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIdGF1R0YoLCYqJiMiIiIiIiRGMUkieUdGKCEiIkY0KiYjIiIjRjJGMUkieEdGKEY0RjQ3I0Yu

The function F(x) Fx:=factor(rationalize(subs(y=Ysol[1],Htau)+ subs(y=Ysol[2],Htau)));convert(%,parfrac,x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiwjIiIiIiIkRjEsKiomSSJ0R0YoRjEpSSJ4R0YoRjJGMUYxKihGMkYxRjVGMUY3RjEhIiIqJClGNyIiI0YxRjkqJiIiJUYxRjVGMUY5RjFGNUY5LCZGN0YxRjFGMUY5RjdGOUYxNyNGLg==

LCoqJiMiIiIiIiRGJUkieEc2IkYlRiUqKEYkRiUsJkYlRiVJInRHRihGJUYlRishIiJGLComIyIiJUYmRiVGJ0YsRiwqKkYkRiUsJiomIiIjRiVGK0YlRiVGJUYlRiUsJkYnRiVGJUYlRixGK0YsRiU=

Computing G(y) Gy:=factor(simplify(expand(rationalize(subs(x=Xsol[1],x*y-Fx))),symbolic));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiwjIiIiIiIkRjEsKiooIiIlRjFJInRHRihGMSlJInlHRihGMkYxRjEqKEYyRjFGNkYxRjhGMUYxKiZGNUYxKUY4IiIjRjEhIiJGNkY9RjFGNkY9RjhGPSwmRjhGMUYxRjFGPUY9NyNGLg==

let us check the formula given in the paper for G(y) thetat:=proc(p) phi(psi(p)): end: thetat_itere:=proc(p,k) if k= 0 then p else thetat_itere(thetat(p),k-1): fi: end: [seq(thetat_itere([x,y],i),i=1..n-1)];

NyQ3JEkieUc2IiomRiQhIiJJInhHRiVGJzckRiZGKA==

lit:=map(p->p[1]*p[2],%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRsaXRHRig3JCokSSJ4R0YoISIiKiRJInlHRihGMTcjRi4=

Htaut:=-1/n*add(i*lit[i],i=1..n-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIdGF1dEdGKCwmKiYjIiIiIiIkRjFJInhHRighIiJGNComIyIiI0YyRjFJInlHRihGNEY0NyNGLg==

The invariant associated with H(x,y)=xy L:=[[x,y ]]: for i from 1 to 2*n-1 do if i mod 2 =1 then L:=[op(L), phi(L[i])]; else L:=[op(L), psi(L[i])]; fi: od: L;

Nyg3JEkieEc2IkkieUdGJTckKiZGJiEiIkYkRilGJjckRihGJDckRiZGJDckRiZGKDckRiRGKA==

Hsig:=convert(map(p->p[1]*p[2], L),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIc2lnR0YoLCgqKCIiIyIiIkkieEdGKEYxSSJ5R0YoRjFGMSomRjBGMUYyISIiRjEqJkYwRjFGM0Y1RjE3I0Yu

Ix:=rationalize(subs(y=Ysol[1],Hsig));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeEdGKComLCgqKCIiIyIiIkkidEdGKEYyKUkieEdGKCIiJEYyISIiKiZGMUYyKUY1RjFGMkYyKiZGMUYyRjNGMkYyRjIsJiomRjNGMkY5RjJGMiomRjNGMkY1RjJGMkY3NyNGLg==

Iy:=rationalize(subs(x=Xsol[1],Hsig));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeUdGKComLCgqKCIiIyIiIkkidEdGKEYyKUkieUdGKCIiJEYyISIiKiZGMUYyKUY1RjFGMkYyKiZGMUYyRjNGMkYyRjIsJiomRjNGMkY5RjJGMiomRjNGMkY1RjJGMkY3NyNGLg==

Expected value for G(y) rationalize(subs(x=Xsol[1],Htaut)+subs(x=Xsol[2],Htaut)+(1-1/2/n)*Iy);

LCQqKiIiJSIiIiwwKihGJEYlSSJ0RzYiRiUpSSJ5R0YpRiRGJUYlKihGJEYlRihGJSlGKyIiJEYlRiUqKEYuRiVGKEYlKUYrIiIjRiVGJSomRiRGJUYtRiUhIiIqKEYxRiVGKEYlRitGJUYlKiZGJEYlRjBGJUYzRihGM0YlRihGJSwoKigiIzdGJSlGKEYxRiVGLUYlRjMqKCIjQ0YlRjlGJUYwRiVGMyooRjhGJUY5RiVGK0YlRjNGM0Yl

factor(%-Gy);

IiIh

convert(Gy,parfrac,y); convert(Iy,parfrac,y);

LCoqJiMiIiUiIiQiIiJJInlHNiJGJyEiIiooRiRGJywmRidGJ0kidEdGKUYnRidGLUYqRicqKiNGJ0YmRicsJiomIiIpRidGLUYnRipGJUYqRicsJkYoRidGJ0YnRipGLUYqRicqJkYvRidGKEYqRic=

LCoqJiIiIyIiIkkieUc2IkYlISIiKihGJEYlLCZGJUYlSSJ0R0YnRiVGJUYrRihGJSooLCYqJiIiJUYlRitGJUYoRiRGKEYlLCZGJkYlRiVGJUYoRitGKEYlKiZGJEYlRiZGKEYl

simplified expressions factor(Gy-2/3*Iy);

LCQqJEkieUc2IiEiIkYm

factor(Fx+2/3*Ix);

LCQqKCwoKiZJInRHNiIiIiIpSSJ4R0YnIiIjRihGKEYmRihGKiEiIkYoRiZGLCwmRipGKEYoRihGLEYs

Starting point (a,a) (Section 7) for a from 0 to 4 do Fx:=-1/x^(a+1): normal(subs(x=Xsol[1],Fx-(x*y)^(a+1))-subs(x=Xsol[2],Fx-(x*y)^(a+1))); od;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiRJInhHRighIiJGMTcjRi4=

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiQpSSJ4R0YoIiIjISIiRjM3I0Yu

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiQpSSJ4R0YoIiIkISIiRjM3I0Yu

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiQpSSJ4R0YoIiIlISIiRjM3I0Yu

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiQpSSJ4R0YoIiImISIiRjM3I0Yu

IiIh

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwqSSJ4R0YoIiIiKiRGLyEiIkYwKiZJInlHRihGMkYvRjJGMComRi9GMEY0RjBGMDcjRi4=

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCwqKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGMkYxRjZGMUYxKiZGMEYxRjZGMUYxKiZGM0YxRjZGMSEiIkYwRjE3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCosIyIiIiIiI0YyLCZJInlHRihGMiokKSwkKiZGNUYyLCoqKCIiJUYyKUkidEdGKEYzRjIpRjVGM0YyRjIqKCIiKUYyRj1GMkY1RjJGMiomRjxGMkY9RjJGMkY1ISIiRjJGQ0YxRjJGMkYyRjVGQ0Y+RkMsJkY1RjJGMkYyRkNGMiwkKixGMUYyLCZGNUZDRjZGMkYyRjVGQ0Y+RkNGREZDRkM3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ0R0YoISIiKUkieEdGKEYzRjUsKiomRjRGMkY2RjJGNUY0RjVGN0YyKiQpKiYsKkY5RjIqKEYzRjJGNEYyRjdGMkYyRjRGMkY3RjVGMiwqRjlGMkY+RjVGNEYyRjdGNUYyRjFGMkYyRjJGMiwkKipGMUYyLCpGOUYyRjpGMkY0RjJGN0Y1RjJGNEY1RjZGNUY1NyNGLg==

qy:=coeff(K,x,0)/coeff(K,x,2); qx:=coeff(K,y,0)/coeff(K,y,2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeUdGKComLCYqJkkidEdGKCIiIkkieUdGKEYyRjJGMUYyRjIsJiomRjFGMilGMyIiI0YyRjJGMEYyISIiNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeEdGKCokKUkieEdGKCIiIyEiIjcjRi4=

phi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([subs(y=v,qy)/u,v]): end: psi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([u,subs(x=u,qx)/v]): end: [x,y ]; phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);

NyRJInhHNiJJInlHRiQ=

NyQqJkkieUc2IiEiIkkieEdGJUYmRiQ=

NyQqJkkieUc2IiEiIkkieEdGJUYmKiZGJCIiIilGJyIiI0Yp

NyQqJEkieEc2IiEiIiomSSJ5R0YlIiIiKUYkIiIjRik=

NyQqJEkieEc2IiEiIiokSSJ5R0YlRiY=

NyQqJkkieEc2IiIiIkkieUdGJUYmKiRGJyEiIg==

NyQqJkkieEc2IiIiIkkieUdGJUYmKiYpRiQiIiMhIiJGJ0Yr

NyRJInhHNiIqJilGIyIiIyEiIkkieUdGJEYo

NyRJInhHNiJJInlHRiQ=

theta:=proc(p) psi(phi(p)): end: theta_itere:=proc(p,k) if k= 0 then p else theta_itere(theta(p),k-1): fi: end: n:=4: [seq(theta_itere([x,y],i),i=1..n-1)];

NyU3JComSSJ5RzYiISIiSSJ4R0YmRicqJkYlIiIiKUYoIiIjRio3JCokRihGJyokRiVGJzckKiZGKEYqRiVGKiomRitGJ0YlRic=

li:=map(p->p[1]*p[2],%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNsaUdGKDclSSJ4R0YoKiZJInlHRighIiJGL0YyKiRGL0YyNyNGLg==

Htau:=-1/n*add(i*li[i],i=1..n-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIdGF1R0YoLCgqJiMiIiIiIiVGMUkieEdGKEYxISIiKigjRjEiIiNGMUkieUdGKEY0RjNGNEY0KiYjIiIkRjJGMUYzRjRGNDcjRi4=

The function F(x) Fx:=rationalize(subs(y=Ysol[1],Htau)+ subs(y=Ysol[2],Htau));expand(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiojIiIiIiIjRjFJInRHRighIiJJInhHRihGNCwmKiZGMkYxRjNGMUYxRjVGMUYxRjQ3I0Yu

LCYqJEkieEc2IiEiIkYmKiYjIiIiIiIjRilJInRHRiVGJkYm

Computing G(y) Gy:=rationalize(subs(x=Xsol[1],x*y-Fx));expand(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKComLCYqJiIiJCIiIkkieUdGKEYyRjJGMkYyRjIsJiooIiIjRjJJInRHRihGMkYzRjJGMiomRjZGMkY3RjJGMiEiIjcjRi4=

LCYqKCIiJCIiIiwmKigiIiNGJUkidEc2IkYlSSJ5R0YqRiVGJSomRihGJUYpRiVGJSEiIkYrRiVGJSokRiZGLUYl

let us check the formula given in the paper thetat:=proc(p) phi(psi(p)): end: thetat_itere:=proc(p,k) if k= 0 then p else thetat_itere(thetat(p),k-1): fi: end: [seq(thetat_itere([x,y],i),i=1..n-1)];

NyU3JComSSJ4RzYiIiIiSSJ5R0YmRicqJilGJSIiIyEiIkYoRiw3JCokRiVGLCokRihGLDckKiZGKEYsRiVGLComRihGJ0YqRic=

lit:=map(p->p[1]*p[2],%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRsaXRHRig3JSokSSJ4R0YoISIiKiZJInlHRihGMUYwRjFGMDcjRi4=

Htaut:=-1/n*add(i*lit[i],i=1..n-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIdGF1dEdGKCwoKiYjIiIiIiIlRjFJInhHRighIiJGNCooI0YxIiIjRjFJInlHRihGNEYzRjRGNComIyIiJEYyRjFGM0YxRjQ3I0Yu

The invariant associated with H(x,y)=xy L:=[[x,y ]]: for i from 1 to 2*n-1 do if i mod 2 =1 then L:=[op(L), phi(L[i])]; else L:=[op(L), psi(L[i])]; fi: od: L;

Nyo3JEkieEc2IkkieUdGJTckKiZGJiEiIkYkRilGJjckRigqJkYmIiIiKUYkIiIjRiw3JCokRiRGKUYrNyRGMCokRiZGKTckKiZGJEYsRiZGLEYyNyRGNComRi1GKUYmRik3JEYkRjY=

Hsig:=convert(map(p->p[1]*p[2], L),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIc2lnR0YoLCoqKCIiIyIiIkkieEdGKEYxSSJ5R0YoRjFGMSomRjBGMUYyISIiRjEqJkYwRjFGMkYxRjEqKEYwRjFGM0Y1RjJGNUYxNyNGLg==

Ix:=rationalize(subs(y=Ysol[1],Hsig));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeEdGKCwkKiYiIiMiIiJJInRHRighIiJGMTcjRi4=

Iy:=rationalize(subs(x=Xsol[1],Hsig));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeUdGKCwkKiYiIiMiIiJJInRHRighIiJGMTcjRi4=

Expected value for G(y) rationalize(subs(x=Xsol[1],Htaut)+subs(x=Xsol[2],Htaut)+(1-1/2/n)*Iy);

LCQqLCIiIyIiIiwoKiYiIiRGJSlJInlHNiJGJEYlRiUqJiIiJUYlRipGJUYlRiVGJUYlSSJ0R0YrRiVGKkYlLCgqKEYtRiUpRi5GJEYlKUYqRihGJSEiIiooIiIpRiVGMUYlRilGJUYzKihGLUYlRjFGJUYqRiVGM0YzRjM=

factor(%-Gy);

IiIh

convert(Gy,parfrac,y);

LCYqJiMiIiQiIiMiIiJJInRHNiIhIiJGJyomRihGKiwmSSJ5R0YpRidGJ0YnRipGKg==

factor(Gy-3*Iy/4);

LCQqJkkidEc2IiEiIiwmSSJ5R0YlIiIiRilGKUYmRiY=

expand(Fx+3*Ix/4);

LCYqJEkieEc2IiEiIkYmKiRJInRHRiVGJiIiIg==

Starting point (a,a) (Section 7) for a from 0 to 3 do Fx:=-1/x^(a+1): normal(subs(x=Xsol[1],Fx-(x*y)^(a+1))-subs(x=Xsol[2],Fx-(x*y)^(a+1)));od;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiRJInhHRighIiJGMTcjRi4=

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiQpSSJ4R0YoIiIjISIiRjM3I0Yu

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiQpSSJ4R0YoIiIkISIiRjM3I0Yu

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiQpSSJ4R0YoIiIlISIiRjM3I0Yu

IiIh

Starting point (2b+1,b) for b from 0 to 3 do Gy:=-1/y^(b+1): normal(subs(y=Ysol[1],Gy-x^(2*b+2)*y^(b+1))-subs(y=Ysol[2],Gy-x^(2*b+2)*y^(b+1)));od;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiRJInlHRighIiJGMTcjRi4=

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiQpSSJ5R0YoIiIjISIiRjM3I0Yu

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiQpSSJ5R0YoIiIkISIiRjM3I0Yu

IiIh

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiQpSSJ5R0YoIiIlISIiRjM3I0Yu

IiIh

liste:=1/x+lambda/y+x*y+2*x+1/x/y+x/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwuKiRJInhHRighIiIiIiIqJkknbGFtYmRhR0YoRjJJInlHRihGMUYyKiZGMEYyRjVGMkYyKiYiIiNGMkYwRjJGMiomRjVGMUYwRjFGMiomRjBGMkY1RjFGMjcjRi4=

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLDAqKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKkY0RjFGMEYxRjJGMUY2RjFGMSooSSdsYW1iZGFHRihGMUYwRjFGM0YxRjEqJkYwRjFGMkYxRjEqJkYwRjFGNkYxRjEqJkYzRjFGNkYxISIiRjBGMTcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YyLCgqJkknbGFtYmRhR0YoRjJJInRHRihGMiEiIkkieUdGKEYyKiQpLDAqKCIiJUYyKUY3RjNGMilGOSIiJEYyRjgqJilGNkYzRjJGP0YyRjIqKCIjN0YyRj9GMilGOUYzRjJGOCoqRjNGMkY2RjJGN0YyRjlGMkY4KihGRUYyRj9GMkY5RjJGOComRj5GMkY/RjJGOCokRkZGMkYyRjFGMkYyRjJGN0Y4KSwmRjlGMkYyRjJGM0Y4RjIsJCoqRjFGMiwoRjVGMkY6RjJGOUY4RjJGN0Y4RktGOEY4NyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ0R0YoISIiKUkieEdGKEYzRjUsKiooRjNGMkY0RjJGNkYyRjVGNEY1RjdGMiokKSwsKioiIiVGMkknbGFtYmRhR0YoRjIpRjRGM0YyKUY3IiIkRjJGNSooRj5GMkY0RjJGQUYyRjUqJEZARjJGMiooRjNGMkY0RjJGN0YyRjUqJEY2RjJGMkYxRjJGMkYyRjIsJCoqRjFGMiwqRjlGMkY6RjJGNEYyRjdGNUYyRjRGNUY2RjVGNTcjRi4=

qy:=coeff(K,x,0)/coeff(K,x,2); qx:=coeff(K,y,0)/coeff(K,y,2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeUdGKComLCYqJkkidEdGKCIiIkkieUdGKEYyRjJGMUYyRjIsKComRjFGMilGMyIiI0YyRjIqKEY3RjJGMUYyRjNGMkYyRjFGMiEiIjcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeEdGKCooLCgqKEknbGFtYmRhR0YoIiIiSSJ0R0YoRjJJInhHRihGMkYyKiZGM0YyKUY0IiIjRjJGMkYzRjJGMkYzISIiRjZGODcjRi4=

phi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([subs(y=v,qy)/u,v]): end: psi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([u,subs(x=u,qx)/v]): end: [x,y ]; phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);

NyRJInhHNiJJInlHRiQ=

NyQqJkkieEc2IiEiIiwmSSJ5R0YlIiIiRilGKUYmRig=

NyQqJkkieEc2IiEiIiwmSSJ5R0YlIiIiRilGKUYmKiYsLiomKUYkIiIjRikpRihGLkYpRikqKEknbGFtYmRhR0YlRilGJEYpRihGKUYpKihGLkYpRi1GKUYoRilGKSomRjFGKUYkRilGKSokRi1GKUYpRilGKUYpRihGJg==

NyQqKEkieEc2IiIiIkkieUdGJUYmLCoqJilGJCIiI0YmRidGJkYmKiZJJ2xhbWJkYUdGJUYmRiRGJkYmKiRGKkYmRiZGJkYmISIiKiYsLiomRipGJilGJ0YrRiZGJiooRi1GJkYkRiZGJ0YmRiYqKEYrRiZGKkYmRidGJkYmRixGJkYuRiZGJkYmRiZGJ0Yv

NyQqKEkieEc2IiIiIkkieUdGJUYmLCoqJilGJCIiI0YmRidGJkYmKiZJJ2xhbWJkYUdGJUYmRiRGJkYmKiRGKkYmRiZGJkYmISIiKigsKEYsRiZGLkYmRiZGJkYmRipGL0YnRi8=

NyRJInhHNiIqKCwoKiZJJ2xhbWJkYUdGJCIiIkYjRilGKSokKUYjIiIjRilGKUYpRilGKUYrISIiSSJ5R0YkRi0=

NyRJInhHNiJJInlHRiQ=

theta:=proc(p) psi(phi(p)): end: theta_itere:=proc(p,k) if k= 0 then p else theta_itere(theta(p),k-1): fi: end: n:=3: [seq(theta_itere([x,y],i),i=1..n-1)];

NyQ3JComSSJ4RzYiISIiLCZJInlHRiYiIiJGKkYqRicqJiwuKiYpRiUiIiNGKilGKUYvRipGKiooSSdsYW1iZGFHRiZGKkYlRipGKUYqRioqKEYvRipGLkYqRilGKkYqKiZGMkYqRiVGKkYqKiRGLkYqRipGKkYqRipGKUYnNyQqKEYlRipGKUYqLCoqJkYuRipGKUYqRipGNEYqRjVGKkYqRipGJyooLChGNEYqRjVGKkYqRipGKkYuRidGKUYn

li:=map(p->p[1]*p[2],%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNsaUdGKDckKipJInhHRighIiIsJkkieUdGKCIiIkY0RjRGMSwuKiYpRjAiIiNGNClGM0Y4RjRGNCooSSdsYW1iZGFHRihGNEYwRjRGM0Y0RjQqKEY4RjRGN0Y0RjNGNEY0KiZGO0Y0RjBGNEY0KiRGN0Y0RjRGNEY0RjRGM0YxKihGMEYxLCoqJkY3RjRGM0Y0RjRGPUY0Rj5GNEY0RjRGMSwoRj1GNEY+RjRGNEY0RjQ3I0Yu

Htau:=-1/n*add(i*li[i],i=1..n-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIdGF1R0YoLCYqLCMiIiIiIiRGMUkieEdGKCEiIiwmSSJ5R0YoRjFGMUYxRjQsLiomKUYzIiIjRjEpRjZGOkYxRjEqKEknbGFtYmRhR0YoRjFGM0YxRjZGMUYxKihGOkYxRjlGMUY2RjFGMSomRj1GMUYzRjFGMSokRjlGMUYxRjFGMUYxRjZGNEY0KiojRjpGMkYxRjNGNCwqKiZGOUYxRjZGMUYxRj9GMUZARjFGMUYxRjQsKEY/RjFGQEYxRjFGMUYxRjQ3I0Yu

The function F(x) Fx:=factor(rationalize(subs(y=Ysol[1],Htau)+ subs(y=Ysol[2],Htau)));expand(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiwjIiIiIiIkRjEsKioqRjJGMUknbGFtYmRhR0YoRjEpSSJ0R0YoIiIjRjFJInhHRihGMUYxKihGNUYxRjdGMSlGOUY4RjFGMSomRjJGMUY2RjFGMSokRjtGMUYxRjFGNyEiIkY7Rj4sJiomRjVGMUY3RjFGMUYxRjFGPkY+NyNGLg==

LCoqKkkidEc2IiIiIkkieEdGJSEiIiwmKiZJJ2xhbWJkYUdGJUYmRiRGJkYmRiZGJkYoRitGJkYoKigjRiYiIiRGJkYpRihGK0YmRigqKEYkRiYpRiciIiNGKEYpRihGKCooRi1GJkYkRihGKUYoRig=

Computing G(y) Gy:=factor(simplify(expand(rationalize(subs(x=Xsol[1],x*y-Fx))),symbolic));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiwjIiIiIiIkRjEsOCoqRjJGMSlJJ2xhbWJkYUdGKCIiI0YxKUkidEdGKEY3RjFJInlHRihGMUYxKihGMkYxRjhGMSlGOkYyRjFGMSoqIiIlRjFGNkYxRjlGMSlGOkY3RjEhIiIqKCIiKkYxRjhGMUY/RjFGMSooRjZGMUY5RjFGOkYxRjEqKEZCRjFGOEYxRjpGMUYxKiZGNkYxRjlGMUZAKiZGMkYxRjhGMUYxKiZGPkYxRj9GMUZAKiZGN0YxRjpGMUZARjFGQEYxRjlGQCksJkY6RjFGMUYxRjdGQCwmRkVGMUYxRjFGQEZANyNGLg==

let us check the formula given in the paper for G(y) thetat:=proc(p) phi(psi(p)): end: thetat_itere:=proc(p,k) if k= 0 then p else thetat_itere(thetat(p),k-1): fi: end: [seq(thetat_itere([x,y],i),i=1..n-1)];

NyQ3JCooSSJ4RzYiIiIiSSJ5R0YmRicsKiomKUYlIiIjRidGKEYnRicqJkknbGFtYmRhR0YmRidGJUYnRicqJEYrRidGJ0YnRichIiIqKCwoRi1GJ0YvRidGJ0YnRidGK0YwRihGMDckKiZGJUYwLCZGKEYnRidGJ0YwKiYsLiomRitGJylGKEYsRidGJyooRi5GJ0YlRidGKEYnRicqKEYsRidGK0YnRihGJ0YnRi1GJ0YvRidGJ0YnRidGKEYw

lit:=map(p->p[1]*p[2],%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRsaXRHRig3JCooSSJ4R0YoISIiLCoqJilGMCIiIyIiIkkieUdGKEY2RjYqJkknbGFtYmRhR0YoRjZGMEY2RjYqJEY0RjZGNkY2RjZGMSwoRjhGNkY6RjZGNkY2RjYqKkYwRjEsJkY3RjZGNkY2RjEsLiomRjRGNilGN0Y1RjZGNiooRjlGNkYwRjZGN0Y2RjYqKEY1RjZGNEY2RjdGNkY2RjhGNkY6RjZGNkY2RjZGN0YxNyNGLg==

Htaut:=-1/n*add(i*lit[i],i=1..n-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIdGF1dEdGKCwmKiojIiIiIiIkRjFJInhHRighIiIsKiomKUYzIiIjRjFJInlHRihGMUYxKiZJJ2xhbWJkYUdGKEYxRjNGMUYxKiRGN0YxRjFGMUYxRjQsKEY6RjFGPEYxRjFGMUYxRjQqLCNGOEYyRjFGM0Y0LCZGOUYxRjFGMUY0LC4qJkY3RjEpRjlGOEYxRjEqKEY7RjFGM0YxRjlGMUYxKihGOEYxRjdGMUY5RjFGMUY6RjFGPEYxRjFGMUYxRjlGNEY0NyNGLg==

The invariant associated with H(x,y)=xy L:=[[x,y ]]: for i from 1 to 2*n-1 do if i mod 2 =1 then L:=[op(L), phi(L[i])]; else L:=[op(L), psi(L[i])]; fi: od: L;

Nyg3JEkieEc2IkkieUdGJTckKiZGJCEiIiwmRiYiIiJGK0YrRilGJjckRigqJiwuKiYpRiQiIiNGKylGJkYxRitGKyooSSdsYW1iZGFHRiVGK0YkRitGJkYrRisqKEYxRitGMEYrRiZGK0YrKiZGNEYrRiRGK0YrKiRGMEYrRitGK0YrRitGJkYpNyQqKEYkRitGJkYrLCoqJkYwRitGJkYrRitGNkYrRjdGK0YrRitGKUYtNyRGOSooLChGNkYrRjdGK0YrRitGK0YwRilGJkYpNyRGJEY9

Hsig:=convert(map(p->p[1]*p[2], L),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIc2lnR0YoLC4qJkkieEdGKCIiIkkieUdGKEYxRjEqKEYwISIiLCZGMkYxRjFGMUY0RjJGMUYxKipGMEY0RjVGNCwuKiYpRjAiIiNGMSlGMkY6RjFGMSooSSdsYW1iZGFHRihGMUYwRjFGMkYxRjEqKEY6RjFGOUYxRjJGMUYxKiZGPUYxRjBGMUYxKiRGOUYxRjFGMUYxRjFGMkY0RjEqKEYwRjEsKiomRjlGMUYyRjFGMUY/RjFGQEYxRjFGMUY0RjdGMUYxKihGMEY0RkJGNCwoRj9GMUZARjFGMUYxRjFGMSooRjBGNEZFRjFGMkY0RjE3I0Yu

Ix:=factor(simplify(expand(rationalize(subs(y=Ysol[1],Hsig))),symbolic));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeEdGKCwkKiwiIiMiIiIsLiooSSdsYW1iZGFHRihGMSlJInRHRihGMEYxKUkieEdGKCIiJEYxRjEqKEY0RjFGNkYxKUY4RjBGMSEiIiomRjZGMUY3RjFGMSokRjVGMUY8KiZGNkYxRjhGMUYxKiRGO0YxRjxGMUY2RjxGO0Y8LCYqJkY0RjFGNkYxRjFGMUYxRjxGPDcjRi4=

Iy:=factor(simplify(expand(rationalize(subs(x=Xsol[1],Hsig))),symbolic));;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeUdGKCwkKiwiIiMiIiIsOComKUkidEdGKEYwRjEpSSJ5R0YoIiIkRjEhIiIqJilJJ2xhbWJkYUdGKEYwRjFGNEYxRjEqKEY8RjFGNUYxKUY3RjBGMUYxKihGOEYxRjRGMUY+RjFGOSooRjxGMUY1RjFGN0YxRjEqKEY4RjFGNEYxRjdGMUY5KihGMEYxRjxGMUY1RjFGMSokRjRGMUY5KiRGPkYxRjFGN0YxRjFGMUYxRjVGOSksJkY3RjFGMUYxRjBGOSwmKiZGPEYxRjVGMUYxRjFGMUY5RjE3I0Yu

Expected value for G(y) factor(rationalize(subs(x=Xsol[1],Htaut)+subs(x=Xsol[2],Htaut)+(1-1/2/n)*Iy));

LCQqLCMiIiIiIiRGJSw4KipGJkYlKUknbGFtYmRhRzYiIiIjRiUpSSJ0R0YrRixGJUkieUdGK0YlRiUqKEYmRiVGLUYlKUYvRiZGJUYlKioiIiVGJUYqRiVGLkYlKUYvRixGJSEiIiooIiIqRiVGLUYlRjRGJUYlKihGKkYlRi5GJUYvRiVGJSooRjdGJUYtRiVGL0YlRiUqJkYqRiVGLkYlRjUqJkYmRiVGLUYlRiUqJkYzRiVGNEYlRjUqJkYsRiVGL0YlRjVGJUY1RiVGLkY1KSwmRi9GJUYlRiVGLEY1LCZGOkYlRiVGJUY1RjU=

factor(%-Gy);

IiIh

convert(Gy,parfrac,y);

LCgqKiMiIiIiIiRGJSwqKihGJkYlKUkidEc2IiIiI0YlSSJ5R0YrRiUhIiIqKCIiJUYlSSdsYW1iZGFHRitGJUYqRiVGJSomRiZGJUYpRiVGLkYwRiVGJSwmKiZGMUYlRipGJUYlRiVGJUYuRipGLkYlKigsJkY0Ri5GLEYuRiUsJkYtRiVGJUYlRi5GKkYuRiUqKEYzRiUpRjdGLEYuRipGLkYl

convert(Gy,parfrac,y);

LCgqKiMiIiIiIiRGJSwqKihGJkYlKUkidEc2IiIiI0YlSSJ5R0YrRiUhIiIqKCIiJUYlSSdsYW1iZGFHRitGJUYqRiVGJSomRiZGJUYpRiVGLkYwRiVGJSwmKiZGMUYlRipGJUYlRiVGJUYuRipGLkYlKigsJkY0Ri5GLEYuRiUsJkYtRiVGJUYlRi5GKkYuRiUqKEYzRiUpRjdGLEYuRipGLkYl

convert(Iy,parfrac,y);

LCgqKiIiIyIiIiwqKiYpSSJ0RzYiRiRGJUkieUdGKkYlISIiKiZJJ2xhbWJkYUdGKkYlRilGJUYlKiRGKEYlRixGJUYlRiUsJkYtRiVGJUYlRixGKUYsRiUqKEYkRiVGKUYsLCZGK0YlRiVGJUYsRiwqKCwmKihGJEYlRi5GJUYpRiVGJUYkRiVGJSlGMkYkRixGKUYsRiU=

Simplifying the functions factor(Gy-Iy/2-1/3/t);

LCQqKCwmKiZJJ2xhbWJkYUc2IiIiIkkidEdGJ0YoRihGKEYoRihGKSEiIiwmSSJ5R0YnRihGKEYoRipGKg==

expand(factor(Fx+Ix/2+1/3/t));

LChJInhHNiIhIiIqJEkidEdGJEYlIiIiKiRGI0YlRiU=

In Section 7, we look at the orbit when y=z-1 r[0]:=[x,z-1]; r[1]:=phi(%); r[2]:=psi(%);r[3]:=phi(%); r[4]:=psi(%);r[5]:=phi(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIhNyRJInhHRigsJkkiekdGKCIiIkY1ISIiNyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIiNyQqJkkiekdGKCEiIkkieEdGKEY0LCZGM0YwRjBGNDcjRjE=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIjNyQqJkkiekdGKCEiIkkieEdGKEY0KiYsKComKUY1RjAiIiIpRjNGMEY6RjoqKEknbGFtYmRhR0YoRjpGNUY6RjNGOkY6RjpGOkY6LCZGM0Y6RjpGNEY0NyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIkNyQqKCwmSSJ6R0YoIiIiRjUhIiJGNUkieEdGKEY1LCgqJilGNyIiI0Y1RjRGNUY1KiZJJ2xhbWJkYUdGKEY1RjdGNUY1RjVGNUY2KiYsKComRjpGNSlGNEY7RjVGNSooRj1GNUY3RjVGNEY1RjVGNUY1RjVGM0Y2NyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIlNyQqKCwmSSJ6R0YoIiIiRjUhIiJGNUkieEdGKEY1LCgqJilGNyIiI0Y1RjRGNUY1KiZJJ2xhbWJkYUdGKEY1RjdGNUY1RjVGNUY2KigsKEY8RjUqJEY6RjVGNUY1RjVGNUY6RjZGM0Y2NyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiImNyRJInhHRigqKCwoKiZJJ2xhbWJkYUdGKCIiIkYyRjdGNyokKUYyIiIjRjdGN0Y3RjdGN0Y5ISIiLCZJInpHRihGN0Y3RjtGOzcjRjE=

Now z=x r[0]:=[x,x-1]; r[1]:=phi(%); r[2]:=psi(%);r[3]:=phi(%); r[4]:=psi(%);r[5]:=phi(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIhNyRJInhHRigsJkYyIiIiRjQhIiI3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIiNyQqJClJInhHRigiIiMhIiIsJkY0RjBGMEY2NyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIjNyQqJClJInhHRihGMCEiIiomLCgqJClGNCIiJSIiIkY7KiZJJ2xhbWJkYUdGKEY7RjNGO0Y7RjtGO0Y7LCZGNEY7RjtGNUY1NyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIkNyQqKEkieEdGKCIiIiwmRjNGNEY0ISIiRjQsKCokKUYzRjBGNEY0KiZJJ2xhbWJkYUdGKEY0RjNGNEY0RjRGNEY2KiYsKCokKUYzIiIlRjRGNComRjtGNClGMyIiI0Y0RjRGNEY0RjRGNUY2NyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIlNyQqKEkieEdGKCIiIiwmRjNGNEY0ISIiRjQsKCokKUYzIiIkRjRGNComSSdsYW1iZGFHRihGNEYzRjRGNEY0RjRGNiooLChGO0Y0KiQpRjMiIiNGNEY0RjRGNEY0RjVGNkZARjY3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiImNyRJInhHRigqKCwoKiZJJ2xhbWJkYUdGKCIiIkYyRjdGNyokKUYyIiIjRjdGN0Y3RjdGNywmRjJGN0Y3ISIiRjxGOUY8NyNGMQ==

rseq:=[seq(r[i],i=0..5)];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVyc2VxR0YoNyg3JEkieEdGKCwmRjAiIiJGMiEiIjckKiQpRjAiIiNGM0YxNyRGNSomLCgqJClGMCIiJUYyRjIqJkknbGFtYmRhR0YoRjJGNkYyRjJGMkYyRjJGMUYzNyQqKEYwRjJGMUYyLCgqJClGMCIiJEYyRjIqJkY/RjJGMEYyRjJGMkYyRjNGOTckRkEqKCwoRkZGMiokRjZGMkYyRjJGMkYyRjFGM0Y2RjM3JEYwRkg3I0Yu

Expansion at infinity a:='a': b:='b': asy:=proc(p) local u, v: u:=p[1]: v:=p[2]: map(simplify,expand(asympt(u^(a+1)*v^(b+1),x,2) assuming a>0 assuming b>0),symbolic): end: map(simplify,map(normal,expand(simplify(asympt(x^a*(x-1)^b,x,2) assuming a>0 assuming b>0))),symbolic);

LCgpSSJ4RzYiLCZJImFHRiUiIiJJImJHRiVGKEYoKiYpRiQsKEYnRihGKUYoRighIiJGKEYpRihGLS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyokKUYkIiIjRi1GKA==

map(asy,rseq);

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

%[1]-%[2]+%[3]-%[4]+%[5]-%[6];

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

forces a=b asy:=proc(p) local u,v: u:=p[1]: v:=p[2]: map(simplify,expand(asympt(u^(a+1)*v^(a+1),x,2) assuming a>0),symbolic): end: map(asy,rseq);

NygsKilJInhHNiIsJiomIiIjIiIiSSJhR0YmRipGKkYpRipGKiomKUYlLCZGKEYqRipGKkYqRitGKiEiIkYtRi8tSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiMqJClGJUYpRi9GKiwqKUYlLCZGK0YvRipGL0YqKiYpRiUsJkYrRi9GKUYvRipGK0YqRi9GPEYvLUYxNiMqJClGJSIiJEYvRiosKilGJSwmRitGKkYqRipGKiomKUYlRitGKkYrRipGKkZHRiotRjE2IyokRiVGL0YqLCZGJEYqLUYxNiNGKkYqLCopRiUsJkYoRi9GKUYvRioqKClGJSwmRihGL0ZCRi9GKkYrRipJJ2xhbWJkYUdGJkYqRioqJkZSRipGVEYqRipGMEYqLC5GKkYqKihGJUYvRitGKkZURipGKiomRiVGL0YrRipGKiomRiVGL0ZURipGKkZKRipGMEYq

map(simplify,%[1]-%[2]+%[3]-%[4]+%[5]-%[6],symbolic);

LEoqJilJInhHNiIsJiomIiIjIiIiSSJhR0YmRipGKkYqRipGKkYrRiohIiJGJEYsLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjKiQpRiVGKUYsRiopRiUsJkYrRixGKkYsRiwqJilGJSwmRitGLEYpRixGKkYrRipGKkY4RiotRi42IyokKUYlIiIkRixGKilGJSwmRitGKkYqRipGKiomKUYlRitGKkYrRipGKkZCRiotRi42IyokRiVGLEYqLUYuNiNGKkYqKUYlLCZGKEYsRilGLEYqKigpRiUsJkYoRixGPkYsRipGK0YqSSdsYW1iZGFHRiZGKkYqKiZGS0YqRk1GKkYqRipGLCooRiVGLEYrRipGTUYqRiwqJkYlRixGK0YqRiwqJkYlRixGTUYqRixGRUYs

Forces a=0

liste:=1/x+y/x+2*y+x*y+2*x+x/y+1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwwKiRJInhHRighIiIiIiIqJkkieUdGKEYyRjBGMUYyKiYiIiNGMkY0RjJGMiomRjBGMkY0RjJGMiomRjZGMkYwRjJGMiomRjBGMkY0RjFGMiokRjRGMUYyNyNGLg==

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLDIqKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKkY0RjFGMEYxRjJGMUY2RjFGMSoqRjRGMUYwRjFGM0YxRjVGMUYxKiZGMEYxRjJGMUYxKiZGMEYxRjVGMUYxKiZGMEYxRjNGMUYxKiZGMEYxRjZGMUYxKiZGM0YxRjZGMSEiIjcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YyLCoqKEYzRjJJInRHRihGMilJInlHRihGM0YyISIiRjZGOUY4RjIqJCksMCooIiM3RjIpRjZGM0YyKUY4IiIkRjJGOSooIiIpRjJGP0YyRjdGMkY5KigiIiVGMkY2RjJGQEYyRjkqKEZFRjJGP0YyRjhGMkY5KiRGP0YyRjIqKEYzRjJGNkYyRjhGMkY5KiRGN0YyRjJGMUYyRjJGMkY2RjkpLCZGOEYyRjJGMkYzRjlGMiwkKipGMUYyLCpGNUYyRjpGMkY2RjJGOEY5RjJGNkY5RkpGOUY5NyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCoqIyIiIiIiI0YyLCoqKEYzRjJJInRHRihGMilJInhHRihGM0YyISIiRjZGOUY4RjIqJCksMCooIiM3RjIpRjZGM0YyKUY4IiIkRjJGOSooIiIpRjJGP0YyRjdGMkY5KigiIiVGMkY2RjJGQEYyRjkqKEZFRjJGP0YyRjhGMkY5KiRGP0YyRjIqKEYzRjJGNkYyRjhGMkY5KiRGN0YyRjJGMUYyRjJGMiksJkY4RjJGMkYyRjNGOUY2RjlGMiwkKipGMUYyLCpGNUYyRjpGMkY2RjJGOEY5RjJGSkY5RjZGOUY5NyNGLg==

qy:=coeff(K,x,0)/coeff(K,x,2); qx:=coeff(K,y,0)/coeff(K,y,2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeUdGKComLCYqJkkidEdGKCIiIilJInlHRigiIiNGMkYyKiZGMUYyRjRGMkYyRjIsKEYwRjIqKEY1RjJGMUYyRjRGMkYyRjFGMiEiIjcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeEdGKComLCYqJkkidEdGKCIiIilJInhHRigiIiNGMkYyKiZGMUYyRjRGMkYyRjIsKEYwRjIqKEY1RjJGMUYyRjRGMkYyRjFGMiEiIjcjRi4=

phi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([subs(y=v,qy)/u,v]): end: psi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([u,subs(x=u,qx)/v]): end: [x,y ]; phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);

NyRJInhHNiJJInlHRiQ=

NyQqKEkieEc2IiEiIiwmSSJ5R0YlIiIiRilGKUYmRihGKUYo

NyQqKEkieEc2IiEiIiwmSSJ5R0YlIiIiRilGKUYmRihGKSokLCgqJkYkRilGKEYpRilGJEYpRihGKUYm

NyQqKEkieEc2IiIiIkkieUdGJSEiIiwmRiRGJkYmRiZGKCokLCgqJkYkRiZGJ0YmRiZGJEYmRidGJkYo

NyQqKEkieEc2IiIiIkkieUdGJSEiIiwmRiRGJkYmRiZGKEYk

NyRJInlHNiJJInhHRiQ=

NyRJInlHNiIqKEkieEdGJCEiIiwmRiMiIiJGKUYpRidGI0Yp

NyQqJCwoKiZJInhHNiIiIiJJInlHRidGKEYoRiZGKEYpRighIiIqKEYmRiosJkYpRihGKEYoRipGKUYo

NyQqJCwoKiZJInhHNiIiIiJJInlHRidGKEYoRiZGKEYpRighIiIqKEYmRihGKUYqLCZGJkYoRihGKEYq

NyRJInhHNiIqKEYjIiIiSSJ5R0YkISIiLCZGI0YmRiZGJkYo

NyRJInhHNiJJInlHRiQ=

theta:=proc(p) psi(phi(p)): end: theta_itere:=proc(p,k) if k= 0 then p else theta_itere(theta(p),k-1): fi: end: n:=5: [seq(theta_itere([x,y],i),i=1..n-1)];

NyY3JCooSSJ4RzYiISIiLCZJInlHRiYiIiJGKkYqRidGKUYqKiQsKComRiVGKkYpRipGKkYlRipGKUYqRic3JCooRiVGKkYpRicsJkYlRipGKkYqRidGJTckRilGJDckRitGLw==

li:=map(p->p[1]*p[2],%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNsaUdGKDcmKipJInhHRighIiIsJkkieUdGKCIiIkY0RjRGMUYzRjQsKComRjBGNEYzRjRGNEYwRjRGM0Y0RjEqKClGMCIiI0Y0RjNGMSwmRjBGNEY0RjRGMSooKUYzRjlGNEYwRjFGMkYxKipGNUYxRjBGNEYzRjFGOkYxNyNGLg==

Htau:=-1/n*add(i*li[i],i=1..n-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIdGF1R0YoLCoqLCMiIiIiIiZGMUkieEdGKCEiIiwmSSJ5R0YoRjFGMUYxRjRGNkYxLCgqJkYzRjFGNkYxRjFGM0YxRjZGMUY0RjQqKiMiIiNGMkYxKUYzRjtGMUY2RjQsJkYzRjFGMUYxRjRGNCoqIyIiJEYyRjEpRjZGO0YxRjNGNEY1RjRGNCosIyIiJUYyRjFGN0Y0RjNGMUY2RjRGPUY0RjQ3I0Yu

The function F(x) Fx:=factor(rationalize(subs(y=Ysol[1],Htau)+ subs(y=Ysol[2],Htau)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKi4jIiIiIiImRjEsOiooIiM3RjEpSSJ0R0YoIiIjRjEpSSJ4R0YoRjJGMUYxKigiIiVGMUY3RjFGOUYxRjEqKCIjZ0YxRjZGMSlGOiIiJEYxRjEqKCIiJ0YxRjdGMSlGOkY8RjEhIiIqKCIjOkYxRjZGMSlGOkY4RjFGMSooIiNCRjFGN0YxRj9GMUYxKiZGOEYxRkNGMUZEKihGRkYxRjZGMUY6RjFGMSooIiIoRjFGN0YxRkdGMUZEKiYiIipGMUY2RjFGRCooIiM5RjFGN0YxRjpGMUYxKiZGTUYxRkdGMUZERjFGN0ZERkdGRCksJkY6RjFGMUYxRjhGRCwmKiZGQEYxRjdGMUYxRjFGMUZERjE3I0Yu

Computing G(y) Gy:=factor(simplify(expand(rationalize(subs(x=Xsol[1],x*y-Fx))),symbolic));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKi4jIiIiIiImRjEsOiooIiNVRjEpSSJ0R0YoIiIjRjEpSSJ5R0YoRjJGMUYxKigiIzlGMUY3RjFGOUYxRjEqKCIjSUYxRjZGMSlGOiIiJEYxRjEqKCIjQEYxRjdGMSlGOiIiJUYxISIiKigiIzpGMUY2RjEpRjpGOEYxRkUqKCIjOEYxRjdGMUY/RjFGMSomIiIoRjFGQ0YxRkUqKEZHRjFGNkYxRjpGMUZFKihGOEYxRjdGMUZIRjFGRSomIiIqRjFGNkYxRkUqKEZERjFGN0YxRjpGMUYxKiZGOEYxRkhGMUZFRjFGSEZFKSwmRjpGMUYxRjFGOEZFLCYqJkZARjFGN0YxRjFGMUYxRkVGN0ZFRkU3I0Yu

let us check the formula given in the paper thetat:=proc(p) phi(psi(p)): end: thetat_itere:=proc(p,k) if k= 0 then p else thetat_itere(thetat(p),k-1): fi: end: [seq(thetat_itere([x,y],i),i=1..n-1)];

NyY3JCokLCgqJkkieEc2IiIiIkkieUdGKEYpRilGJ0YpRipGKSEiIiooRidGKUYqRissJkYnRilGKUYpRis3JEYqKihGJ0YrLCZGKkYpRilGKUYrRipGKTckRixGJzckRi9GJA==

lit:=map(p->p[1]*p[2],%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRsaXRHRig3JioqLCgqJkkieEdGKCIiIkkieUdGKEYzRjNGMkYzRjRGMyEiIkYyRjNGNEY1LCZGMkYzRjNGM0Y1KigpRjQiIiNGM0YyRjUsJkY0RjNGM0YzRjUqKClGMkY5RjNGNEY1RjZGNSoqRjJGNUY6RjVGNEYzRjBGNTcjRi4=

Htaut:=-1/n*add(i*lit[i],i=1..n-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIdGF1dEdGKCwqKiwjIiIiIiImRjEsKComSSJ4R0YoRjFJInlHRihGMUYxRjVGMUY2RjEhIiJGNUYxRjZGNywmRjVGMUYxRjFGN0Y3KiojIiIjRjJGMSlGNkY7RjFGNUY3LCZGNkYxRjFGMUY3RjcqKiMiIiRGMkYxKUY1RjtGMUY2RjdGOEY3RjcqLCMiIiVGMkYxRjVGN0Y9RjdGNkYxRjNGN0Y3NyNGLg==

The invariant associated with H(x,y)=xy L:=[[x,y ]]: for i from 1 to 2*n-1 do if i mod 2 =1 then L:=[op(L), phi(L[i])]; else L:=[op(L), psi(L[i])]; fi: od: L;

Nyw3JEkieEc2IkkieUdGJTckKihGJCEiIiwmRiYiIiJGK0YrRilGJkYrRiY3JEYoKiQsKComRiRGK0YmRitGK0YkRitGJkYrRik3JCooRiRGK0YmRiksJkYkRitGK0YrRilGLTckRjFGJDckRiZGJDckRiZGKDckRi1GKDckRi1GMTckRiRGMQ==

Hsig:=convert(map(p->p[1]*p[2], L),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIc2lnR0YoLCwqKCIiIyIiIkkieEdGKEYxSSJ5R0YoRjFGMSoqRjBGMSlGM0YwRjFGMiEiIiwmRjNGMUYxRjFGNkYxKixGMEYxRjJGNkY3RjZGM0YxLCgqJkYyRjFGM0YxRjFGMkYxRjNGMUY2RjEqLEYwRjFGOUY2RjJGMUYzRjYsJkYyRjFGMUYxRjZGMSoqRjBGMSlGMkYwRjFGM0Y2RjxGNkYxNyNGLg==

Ix:=factor(simplify(expand(rationalize(subs(y=Ysol[1],Hsig))),symbolic));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeEdGKCwkKi4iIiMiIiIsNiooIiInRjEpSSJ0R0YoRjBGMSlJInhHRigiIiZGMUYxKihGMEYxRjZGMUY3RjFGMSooIiM1RjFGNUYxKUY4IiIkRjFGMSooRj5GMUY2RjEpRjgiIiVGMSEiIiooRkFGMUY2RjFGPUYxRjEqJEZARjFGQiomRjZGMSlGOEYwRjFGQiomRjBGMUY1RjFGQiooRjBGMUY2RjFGOEYxRjEqJEZGRjFGQkYxRjZGQkZGRkIpLCZGOEYxRjFGMUYwRkIsJiomRj5GMUY2RjFGMUYxRjFGQkZCNyNGLg==

Iy:=factor(simplify(expand(rationalize(subs(x=Xsol[1],Hsig))),symbolic));;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeUdGKCwkKi4iIiMiIiIsNiooIiInRjEpSSJ0R0YoRjBGMSlJInlHRigiIiZGMUYxKihGMEYxRjZGMUY3RjFGMSooIiM1RjFGNUYxKUY4IiIkRjFGMSooRj5GMUY2RjEpRjgiIiVGMSEiIiooRkFGMUY2RjFGPUYxRjEqJEZARjFGQiomRjZGMSlGOEYwRjFGQiomRjBGMUY1RjFGQiooRjBGMUY2RjFGOEYxRjEqJEZGRjFGQkYxRkZGQiksJkY4RjFGMUYxRjBGQiwmKiZGPkYxRjZGMUYxRjFGMUZCRjZGQkZCNyNGLg==

Expected value for G(y) factor(rationalize(subs(x=Xsol[1],Htaut)+subs(x=Xsol[2],Htaut)+(1-1/2/n)*Iy));

LCQqLiMiIiIiIiZGJSw6KigiI1VGJSlJInRHNiIiIiNGJSlJInlHRixGJkYlRiUqKCIjOUYlRitGJUYuRiVGJSooIiNJRiVGKkYlKUYvIiIkRiVGJSooIiNARiVGK0YlKUYvIiIlRiUhIiIqKCIjOkYlRipGJSlGL0YtRiVGOiooIiM4RiVGK0YlRjRGJUYlKiYiIihGJUY4RiVGOiooRjxGJUYqRiVGL0YlRjoqKEYtRiVGK0YlRj1GJUY6KiYiIipGJUYqRiVGOiooRjlGJUYrRiVGL0YlRiUqJkYtRiVGPUYlRjpGJUY9RjopLCZGL0YlRiVGJUYtRjosJiomRjVGJUYrRiVGJUYlRiVGOkYrRjpGOg==

factor(%-Gy);

IiIh

convert(Gy,parfrac,y);convert(Iy,parfrac,y);

LC4qJiMiIzkiIiYiIiJJInlHNiJGJyEiIiooIyIiKEYmRicsJiomIiIlRidJInRHRilGJ0YnRidGJ0YnRjFGKkYnKiojRidGJkYnLCYqJiIjXkYnRjFGJ0YqRiVGKkYnLCZGKEYnRidGJ0YqRjFGKkYnKioiIipGJ0YxRicpRigiIiNGKiwmKiYiIzpGJ0YxRidGJ0YmRidGKkYnKigsJiomIiIkRidGMUYnRipGMEYqRidGKEYqRjxGKkYnKipGM0YnLCYqJiIjRkYnRjFGJ0YnRjlGJ0YnKUY3RjtGKkYxRipGJw==

LC4qJiIiJSIiIkkieUc2IkYlISIiKigiIiNGJSwmKiZGJEYlSSJ0R0YnRiVGJUYlRiVGJUYtRihGJSooLCYqJiIjO0YlRi1GJUYoRiRGKEYlLCZGJkYlRiVGJUYoRi1GKEYlKipGJEYlRi1GJSlGJkYqRigsJiomIiIkRiVGLUYlRiVGJUYlRihGJSooLCZGJEYoKiYiIilGJUYtRiVGKEYlRiZGKEY1RihGJSooLCYqJiIjN0YlRi1GJUYlRiRGJUYlKUYyRipGKEYtRihGJQ==

factor(Gy-9/5/4*Iy);

LCQqLCMiIiIiIiNGJUkieUc2IiEiIiwsKihGJkYlSSJ0R0YoRiUpRiciIiRGJUYlKihGJkYlRixGJSlGJ0YmRiVGKSokRjBGJUYpKiZGJkYlRixGJUYpRidGJUYlRixGKSwmRidGJUYlRiVGKUYp

convert(%,parfrac,y);

LCpJInlHNiIhIiIqKCMiIiIiIiNGKCwmKiYiIiVGKEkidEdGJEYoRihGKEYoRihGLUYlRigqKCwmKiYiIiRGKEYtRihGJUYoRiVGKCwmRiNGKEYoRihGJUYtRiVGKCokRiNGJUYo

factor(Fx+9/5/4*Ix);

LCQqLCMiIiIiIiNGJUkieEc2IiEiIiwsKihGJkYlSSJ0R0YoRiUpRiciIiRGJUYlKihGJkYlRixGJSlGJ0YmRiVGKSokRjBGJUYpKiZGJkYlRixGJUYpRidGJUYlLCZGJ0YlRiVGJUYpRixGKUYp

convert(%,parfrac,x);

LCpJInhHNiIhIiIqKCMiIiIiIiNGKCwmKiYiIiVGKEkidEdGJEYoRihGKEYoRihGLUYlRigqJEYjRiVGKCooLCYqJiIiJEYoRi1GKEYlRihGJUYoLCZGI0YoRihGKEYlRi1GJUYo

factor(Gy-9/5/4*Iy-(1/2)*(4*t+1)/t);convert(%,parfrac,y);

LCQqKkkieUc2IiEiIiwsKiZJInRHRiUiIiIpRiQiIiRGKkYqKiZGKUYqKUYkIiIjRipGKiooRi9GKkYpRipGJEYqRipGKUYmRiRGKkYqRilGJiwmRiRGKkYqRipGJkYm

LChJInlHNiIhIiIqKCwmKiYiIiQiIiJJInRHRiRGKkYlRipGJUYqLCZGI0YqRipGKkYlRitGJUYqKiRGI0YlRio=

factor(Fx+9/5/4*Ix+(1/2)*(4*t+1)/t);convert(%,parfrac,x);

LCQqKkkieEc2IiEiIiwsKiZJInRHRiUiIiIpRiQiIiRGKkYqKihGLEYqRilGKilGJCIiI0YqRiYqKEYvRipGKUYqRiRGKkYmKiRGLkYqRiZGKUYmRiosJkYkRipGKkYqRiZGKUYmRiY=

LCpJInhHNiIhIiIqJiwmKiYiIiUiIiJJInRHRiRGKkYqRipGKkYqRitGJUYqKiRGI0YlRioqKCwmKiYiIiRGKkYrRipGJUYqRiVGKiwmRiNGKkYqRipGJUYrRiVGKg==

Decoupling for other starting points (Section 7.3) r[1]:=[x,x^2]; r[2]:=phi(%); r[3]:=psi(%);r[4]:=phi(%); r[5]:=psi(%);r[6]:=phi(%); r[7]:=psi(%);r[8]:=phi(%); r[9]:=psi(%);r[10]:=phi(%); psi(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIiNyRJInhHRigqJClGMiIiI0YwNyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIjNyQqJkkieEdGKCIiIiwmKiQpRjNGMEY0RjRGNEY0ISIiRjY3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIkNyQqJkkieEdGKCIiIiwmKiQpRjMiIiNGNEY0RjRGNCEiIiomRjNGOSwoRjZGNEYzRjRGNEY0Rjk3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIlNyQqJkkieEdGKCEiIiwmRjMiIiJGNkY2RjQqJkYzRjQsKCokKUYzIiIjRjZGNkYzRjZGNkY2RjQ3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiImNyQqJkkieEdGKCEiIiwmRjMiIiJGNkY2RjRGMzcjRjE=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiInNyQqJClJInhHRigiIiMiIiJGNDcjRjE=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIoNyQqJClJInhHRigiIiMiIiIqJkY0RjYsJkYyRjZGNkY2ISIiNyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIpNyQqJkkieEdGKCEiIiwoKiQpRjMiIiMiIiJGOUYzRjlGOUY5RjQqJkYzRjksJkY2RjlGOUY5RjQ3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIqNyQqJkkieEdGKCEiIiwoKiQpRjMiIiMiIiJGOUYzRjlGOUY5RjQqJkYzRjQsJkYzRjlGOUY5RjQ3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiM1NyRJInhHRigqJkYyISIiLCZGMiIiIkY2RjZGNDcjRjE=

NyRJInhHNiIqJClGIyIiIyIiIg==

rseq:=[seq(r[i],i=1..10)];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVyc2VxR0YoNyw3JEkieEdGKCokKUYwIiIjIiIiNyQqJkYwRjQsJkYxRjRGNEY0ISIiRjE3JEY2KiZGMEY4LChGMUY0RjBGNEY0RjRGODckKiZGMEY4LCZGMEY0RjRGNEY4Rjo3JEY9RjA3JEYxRjA3JEYxRjY3JEY6RjY3JEY6Rj03JEYwRj03I0Yu

asy:=proc(p) local u, v: u:=p[1]: v:=p[2]: map(simplify,expand(asympt(u^(a+1)*v^(b+1), x,2) assuming a>0 assuming b>0),symbolic): end: map(asy,rseq);

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

forces a=b

liste:=2/x+y/x+y+1/x/y+x+x/y+2/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwwKiYiIiMiIiJJInhHRighIiJGMSomSSJ5R0YoRjFGMkYzRjFGNUYxKiZGNUYzRjJGM0YxRjJGMSomRjJGMUY1RjNGMSomRjBGMUY1RjNGMTcjRi4=

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLDIqKEkidEdGKCIiIilJInhHRigiIiNGMUkieUdGKEYxRjEqKEYwRjFGM0YxKUY1RjRGMUYxKiZGMEYxRjJGMUYxKiZGMEYxRjdGMUYxKihGNEYxRjBGMUYzRjFGMSooRjRGMUYwRjFGNUYxRjEqJkYzRjFGNUYxISIiRjBGMTcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YyLCoqJkkidEdGKEYyKUkieUdGKEYzRjIhIiIqJkYzRjJGNkYyRjlGOEYyKiQpKiZGOEYyLDAqJilGNkYzRjIpRjgiIiRGMkYyKigiIiVGMkZARjJGN0YyRjkqKCIiKUYyRkBGMkY4RjJGOSooRjNGMkY2RjJGN0YyRjkqJiIjN0YyRkBGMkY5KiZGREYyRjZGMkY5RjhGMkYyRjFGMkYyRjJGNkY5LCZGOEYyRjJGMkY5RjIsJCoqRjFGMiwqRjVGMkY7RjJGOkYyRjhGOUYyRjZGOUZLRjlGOTcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCoqIyIiIiIiI0YyLCoqJkkidEdGKEYyKUkieEdGKEYzRjIhIiIqJkYzRjJGNkYyRjlGOEYyKiQpKiZGOEYyLDAqJilGNkYzRjIpRjgiIiRGMkYyKigiIiVGMkZARjJGN0YyRjkqKCIiKUYyRkBGMkY4RjJGOSooRjNGMkY2RjJGN0YyRjkqJiIjN0YyRkBGMkY5KiZGREYyRjZGMkY5RjhGMkYyRjFGMkYyRjJGNkY5LCZGOEYyRjJGMkY5RjIsJCoqRjFGMiwqRjVGMkY7RjJGOkYyRjhGOUYyRjZGOUZLRjlGOTcjRi4=

qy:=coeff(K,x,0)/coeff(K,x,2); qx:=coeff(K,y,0)/coeff(K,y,2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeUdGKComLCgqJkkidEdGKCIiIilJInlHRigiIiNGMkYyKihGNUYyRjFGMkY0RjJGMkYxRjJGMiwmKiZGMUYyRjRGMkYyRjFGMiEiIjcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeEdGKComLCgqJkkidEdGKCIiIilJInhHRigiIiNGMkYyKihGNUYyRjFGMkY0RjJGMkYxRjJGMiwmKiZGMUYyRjRGMkYyRjFGMiEiIjcjRi4=

phi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([subs(y=v,qy)/u,v]): end: psi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([u,subs(x=u,qx)/v]): end: [x,y ]; phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);

NyRJInhHNiJJInlHRiQ=

NyQqJiwmSSJ5RzYiIiIiRidGJ0YnSSJ4R0YmISIiRiU=

NyQqJiwmSSJ5RzYiIiIiRidGJ0YnSSJ4R0YmISIiKihGKEYpLChGJUYnRidGJ0YoRidGJ0YlRik=

NyQqJiwmSSJ4RzYiIiIiRidGJ0YnSSJ5R0YmISIiKihGJUYpLChGKEYnRidGJ0YlRidGJ0YoRik=

NyQqJiwmSSJ4RzYiIiIiRidGJ0YnSSJ5R0YmISIiRiU=

NyRJInlHNiJJInhHRiQ=

NyRJInlHNiIqJiwmRiMiIiJGJ0YnRidJInhHRiQhIiI=

NyQqKEkieEc2IiEiIiwoSSJ5R0YlIiIiRilGKUYkRilGKUYoRiYqJiwmRihGKUYpRilGKUYkRiY=

NyQqKEkieEc2IiEiIiwoSSJ5R0YlIiIiRilGKUYkRilGKUYoRiYqJiwmRiRGKUYpRilGKUYoRiY=

NyRJInhHNiIqJiwmRiMiIiJGJ0YnRidJInlHRiQhIiI=

NyRJInhHNiJJInlHRiQ=

theta:=proc(p) psi(phi(p)): end: theta_itere:=proc(p,k) if k= 0 then p else theta_itere(theta(p),k-1): fi: end: n:=5: [seq(theta_itere([x,y],i),i=1..n-1)];

NyY3JComLCZJInlHNiIiIiJGKEYoRihJInhHRichIiIqKEYpRiosKEYmRihGKEYoRilGKEYoRiZGKjckKiYsJkYpRihGKEYoRihGJkYqRik3JEYmRiQ3JEYrRi4=

li:=map(p->p[1]*p[2],%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNsaUdGKDcmKiosJkkieUdGKCIiIkYyRjJGMilJInhHRigiIiMhIiIsKEYxRjJGMkYyRjRGMkYyRjFGNiooLCZGNEYyRjJGMkYyRjFGNkY0RjIqKEYxRjJGMEYyRjRGNioqRjRGNkY3RjIpRjFGNUY2RjlGMjcjRi4=

Htau:=-1/n*add(i*li[i],i=1..n-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIdGF1R0YoLCoqLCMiIiIiIiZGMSwmSSJ5R0YoRjFGMUYxRjEpSSJ4R0YoIiIjISIiLChGNEYxRjFGMUY2RjFGMUY0RjhGOCoqI0Y3RjJGMSwmRjZGMUYxRjFGMUY0RjhGNkYxRjgqKiMiIiRGMkYxRjRGMUYzRjFGNkY4RjgqLCMiIiVGMkYxRjZGOEY5RjEpRjRGN0Y4RjxGMUY4NyNGLg==

The function F(x) Fx:=factor(rationalize(subs(y=Ysol[1],Htau)+ subs(y=Ysol[2],Htau)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiwjIiIiIiImRjEsOiooIiIjRjEpSSJ0R0YoRjVGMSlJInhHRihGMkYxRjEqKEYyRjFGNkYxKUY5IiIlRjEhIiIqKCIjREYxRjZGMSlGOSIiJEYxRjEqKEY1RjFGN0YxRjtGMUY9KihGP0YxRjZGMSlGOUY1RjFGMSooIiM3RjFGN0YxRkBGMUYxKigiI11GMUY2RjFGOUYxRjEqKCIiKEYxRjdGMUZERjFGPSomIiInRjFGNkYxRj0qKCIjPkYxRjdGMUY5RjFGMSomRkpGMUZERjFGPSomRjVGMUY3RjFGPUYxRjZGPSksJkY5RjFGMUYxRjVGPUY5Rj1GMTcjRi4=

Computing G(y) Gy:=factor(simplify(expand(rationalize(subs(x=Xsol[1],x*y-Fx))),symbolic));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiwjIiIiIiImRjEsOiooIiIoRjEpSSJ0R0YoIiIjRjEpSSJ5R0YoRjJGMUYxKihGMkYxRjZGMSlGOiIiJUYxRjEqKCIjP0YxRjZGMSlGOiIiJEYxRjEqKEY1RjFGN0YxRjxGMSEiIiooRj9GMUY2RjEpRjpGOEYxRjEqKEZBRjFGN0YxRkBGMUZDKihGMkYxRjZGMUY6RjFGQyooRjhGMUY3RjFGRUYxRkMqJiIjQEYxRjZGMUZDKiZGN0YxRjpGMUZDKiZGOEYxRkVGMUZDKiZGNUYxRjdGMUZDRjFGNkZDKSwmRjpGMUYxRjFGOEZDRjpGQ0ZDNyNGLg==

let us check the formula given in the paper for G(y) thetat:=proc(p) phi(psi(p)): end: thetat_itere:=proc(p,k) if k= 0 then p else thetat_itere(thetat(p),k-1): fi: end: [seq(thetat_itere([x,y],i),i=1..n-1)];

NyY3JCooSSJ4RzYiISIiLChJInlHRiYiIiJGKkYqRiVGKkYqRilGJyomLCZGJUYqRipGKkYqRilGJzckRikqJiwmRilGKkYqRipGKkYlRic3JEYrRiU3JEYuRiQ=

lit:=map(p->p[1]*p[2],%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRsaXRHRig3JioqSSJ4R0YoISIiLChJInlHRigiIiJGNEY0RjBGNEY0KUYzIiIjRjEsJkYwRjRGNEY0RjQqKEYzRjQsJkYzRjRGNEY0RjRGMEYxKihGN0Y0RjNGMUYwRjQqKkY5RjQpRjBGNkYxRjJGNEYzRjE3I0Yu

Htaut:=-1/n*add(i*lit[i],i=1..n-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIdGF1dEdGKCwqKiwjIiIiIiImRjFJInhHRighIiIsKEkieUdGKEYxRjFGMUYzRjFGMSlGNiIiI0Y0LCZGM0YxRjFGMUYxRjQqKiNGOEYyRjFGNkYxLCZGNkYxRjFGMUYxRjNGNEY0KiojIiIkRjJGMUY5RjFGNkY0RjNGMUY0KiwjIiIlRjJGMUY8RjEpRjNGOEY0RjVGMUY2RjRGNDcjRi4=

The invariant associated with H(x,y)=xy L:=[[x,y ]]: for i from 1 to 2*n-1 do if i mod 2 =1 then L:=[op(L), phi(L[i])]; else L:=[op(L), psi(L[i])]; fi: od: L;

Nyw3JEkieEc2IkkieUdGJTckKiYsJkYmIiIiRipGKkYqRiQhIiJGJjckRigqKEYkRissKEYmRipGKkYqRiRGKkYqRiZGKzckKiYsJkYkRipGKkYqRipGJkYrRi03JEYwRiQ3JEYmRiQ3JEYmRig3JEYtRig3JEYtRjA3JEYkRjA=

Hsig:=convert(map(p->p[1]*p[2], L),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIc2lnR0YoLCwqKCIiIyIiIkkieEdGKEYxSSJ5R0YoRjFGMSoqRjBGMUYzRjEsJkYzRjFGMUYxRjFGMiEiIkYxKixGMEYxRjVGMSlGMkYwRjYsKEYzRjFGMUYxRjJGMUYxRjNGNkYxKixGMEYxRjJGNkY5RjEpRjNGMEY2LCZGMkYxRjFGMUYxRjEqKkYwRjFGPEYxRjNGNkYyRjFGMTcjRi4=

Ix:=factor(simplify(expand(rationalize(subs(y=Ysol[1],Hsig))),symbolic));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeEdGKCwkKiwiIiMiIiIsOComKUkidEdGKEYwRjEpSSJ4R0YoIiImRjFGMSooRjhGMUY0RjEpRjciIiRGMUYxKiZGNUYxKUY3IiIlRjEhIiIqKEY4RjFGNEYxKUY3RjBGMUYxKiZGNUYxRjpGMUYxKihGOEYxRjRGMUY3RjFGMSomRjVGMUZBRjFGPyomRjtGMUY0RjFGPyooRjBGMUY1RjFGN0YxRjEqJEZBRjFGP0Y1Rj9GMUY0Rj8pLCZGN0YxRjFGMUYwRj9GN0Y/Rj83I0Yu

Iy:=factor(simplify(expand(rationalize(subs(x=Xsol[1],Hsig))),symbolic));;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeUdGKCwkKiwiIiMiIiIsOComKUkidEdGKEYwRjEpSSJ5R0YoIiImRjFGMSooRjhGMUY0RjEpRjciIiRGMUYxKiZGNUYxKUY3IiIlRjEhIiIqKEY4RjFGNEYxKUY3RjBGMUYxKiZGNUYxRjpGMUYxKihGOEYxRjRGMUY3RjFGMSomRjVGMUZBRjFGPyomRjtGMUY0RjFGPyooRjBGMUY1RjFGN0YxRjEqJEZBRjFGP0Y1Rj9GMUY0Rj8pLCZGN0YxRjFGMUYwRj9GN0Y/Rj83I0Yu

Expected value for G(y) factor(rationalize(subs(x=Xsol[1],Htaut)+subs(x=Xsol[2],Htaut)+(1-1/2/n)*Iy));

LCQqLCMiIiIiIiZGJSw6KigiIihGJSlJInRHNiIiIiNGJSlJInlHRixGJkYlRiUqKEYmRiVGKkYlKUYvIiIlRiVGJSooIiM/RiVGKkYlKUYvIiIkRiVGJSooRilGJUYrRiVGMUYlISIiKihGNEYlRipGJSlGL0YtRiVGJSooRjZGJUYrRiVGNUYlRjgqKEYmRiVGKkYlRi9GJUY4KihGLUYlRitGJUY6RiVGOComIiNARiVGKkYlRjgqJkYrRiVGL0YlRjgqJkYtRiVGOkYlRjgqJkYpRiVGK0YlRjhGJUYqRjgpLCZGL0YlRiVGJUYtRjhGL0Y4Rjg=

factor(%-Gy);

IiIh

convert(Gy,parfrac,y);convert(Iy,parfrac,y);

LCoqKCMiIiIiIiZGJSwsKigiIihGJUkidEc2IkYlKUkieUdGKyIiI0YlRiUqKCIiKkYlRipGJUYtRiUhIiIqJiIjSkYlRipGJUYlKiZGKUYlRi1GJUYxIiM2RiVGJUYqRjFGMSoqRiRGJSwoKiYiIzdGJSlGKkYuRiVGJSomIiM1RiVGKkYlRiVGLkYlRiUsJkYtRiVGJUYlRjFGOkYxRiUqKkYkRiUsJiomIiNARiVGKkYlRiVGKUYlRiVGLUYxRipGMUYlKipGJEYlLCgqJiIjPUYlRjpGJUYxKiZGOUYlRipGJUYxRi5GMUYlKUY9Ri5GMUY6RjFGJQ==

LCoqKCIiIyIiIiwsKiZJInRHNiJGJSlJInlHRilGJEYlRiUqKEYkRiVGKEYlRitGJSEiIiomIiIpRiVGKEYlRiVGK0YtIiIkRiVGJUYoRi1GLSooLCgqJiIjN0YlKUYoRiRGJUYlKiYiIzVGJUYoRiVGJUYkRiVGJSwmRitGJUYlRiVGLUY1Ri1GJSooLCYqJiIiJ0YlRihGJUYlRiRGJUYlRitGLUYoRi1GJSooLCgqJiIjPUYlRjVGJUYtKiZGNEYlRihGJUYtRiRGLUYlKUY4RiRGLUY1Ri1GJQ==

factor(Gy-1/5*Iy);

LCQqKCwwKiZJInRHNiIiIiIpSSJ5R0YnIiIkRihGKComRiZGKClGKiIiI0YoISIiKihGK0YoRiZGKEYqRihGKCokRi1GKEYvKiZGK0YoRiZGKEYvRipGKEYoRi9GKEYmRi9GKkYvRi8=

collect(%,y);

LCoqJClJInlHNiIiIiMiIiIhIiIqKCwmRihGKUkidEdGJkYpRihGLEYpRiVGKEYpKiZGLEYpLCYqJiIiJEYoRixGKEYoRihGKEYoRikqKCwmRi9GKUYoRilGKEYsRilGJUYpRik=

factor(Fx+1/5*Ix);

LCQqKCwyKiYpSSJ0RzYiIiIjIiIiKUkieEdGKCIiJEYqRioqKEYtRipGJkYqKUYsRilGKiEiIiooRi1GKkYmRipGLEYqRjAqKEYpRipGJ0YqRi9GKkYwKiYiIilGKkYmRipGMComRidGKkYsRipGKiomRi1GKkYnRipGMEYsRipGKkYmRjApLCZGLEYqRipGKkYpRjBGMA==

convert(%,parfrac,x);

LCpJInhHNiIhIiIqJkkidEdGJEYlLCYqJiIiJiIiIkYnRitGKyIiI0YrRitGKyooLCgqJiIiJ0YrKUYnRixGK0YlRilGJUYrRiVGKywmRiNGK0YrRitGJUYxRiVGKyooLCgqJiIiKkYrRjFGK0YrKiZGMEYrRidGK0YrRitGK0YrKUYyRixGJUYxRiVGKw==

factor(Gy-1/5*Iy+(3*t+1)/t);convert(%,parfrac,y);

LCQqKCwsKiZJInRHNiIiIiIpSSJ5R0YnIiIkRihGKComRiZGKClGKiIiI0YoISIiKiRGLUYoRi8qJkYrRihGJkYoRi9GKEYvRihGJkYvRipGL0Yv

LCgqJClJInlHNiIiIiMiIiIhIiIqKCwmRihGKEkidEdGJkYoRihGLEYpRiVGKEYoKigsJiomIiIkRihGLEYoRihGKEYoRihGJUYpRixGKUYo

factor(Fx+1/5*Ix-(3*t+1)/t);map(factor,convert(%,parfrac,x));

LCQqKCwwKiYpSSJ0RzYiIiIjIiIiKUkieEdGKCIiJEYqRioqKEYtRipGJkYqRixGKkYqKiZGJ0YqKUYsRilGKiEiIiomIiImRipGJkYqRjEqKEYtRipGJ0YqRixGKkYqKiZGKUYqRidGKkYxRixGKkYqRiZGMSksJkYsRipGKkYqRilGMUYx

LCpJInhHNiIhIiIqJkkidEdGJEYlLCYqJiIiIyIiIkYnRitGK0YrRitGK0YrKipGKEYrLCYqJiIiJEYrRidGK0YrRitGK0YrLCZGI0YrRitGK0YlKUYnRipGJUYlKigpRi1GKkYrKUYwRipGJUYxRiVGKw==

Decoupling for other starting points (Section 7.3) r[1]:=[x,x^2]; r[2]:=phi(%); r[3]:=psi(%);r[4]:=phi(%); r[5]:=psi(%);r[6]:=phi(%); r[7]:=psi(%);r[8]:=phi(%); r[9]:=psi(%);r[10]:=phi(%); psi(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIiNyRJInhHRigqJClGMiIiI0YwNyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIjNyQqJiwmKiQpSSJ4R0YoRjAiIiJGN0Y3RjdGN0Y2ISIiRjQ3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIkNyQqJiwmKiQpSSJ4R0YoIiIjIiIiRjhGOEY4RjhGNiEiIiomKUY2RjBGOSwoRjRGOEY2RjhGOEY4Rjg3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIlNyQqJiwmSSJ4R0YoIiIiRjVGNUY1KUY0IiIjISIiKiYpRjQiIiRGOCwoKiRGNkY1RjVGNEY1RjVGNUY1NyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiImNyQqJiwmSSJ4R0YoIiIiRjVGNUY1KUY0IiIjISIiRjQ3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiInNyQqJClJInhHRigiIiMiIiJGNDcjRjE=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIoNyQqJClJInhHRigiIiMiIiIqJiwmRjJGNkY2RjZGNkY0ISIiNyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIpNyQqJilJInhHRigiIiQhIiIsKCokKUY0IiIjIiIiRjtGNEY7RjtGO0Y7KiYsJkY4RjtGO0Y7RjtGNEY2NyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIqNyQqJilJInhHRigiIiQhIiIsKCokKUY0IiIjIiIiRjtGNEY7RjtGO0Y7KiYsJkY0RjtGO0Y7RjtGOUY2NyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiM1NyRJInhHRigqJiwmRjIiIiJGNUY1RjUpRjIiIiMhIiI3I0Yx

NyRJInhHNiIqJClGIyIiIyIiIg==

rseq:=[seq(r[i],i=1..10)];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVyc2VxR0YoNyw3JEkieEdGKCokKUYwIiIjIiIiNyQqJiwmRjFGNEY0RjRGNEYwISIiRjE3JEY2KiYpRjAiIiRGOCwoRjFGNEYwRjRGNEY0RjQ3JComLCZGMEY0RjRGNEY0RjJGOEY6NyRGP0YwNyRGMUYwNyRGMUY2NyRGOkY2NyRGOkY/NyRGMEY/NyNGLg==

asy:=proc(p) local u, v: u:=p[1]: v:=p[2]: map(simplify,expand(asympt(u^(a+1)*v^(b+1), x,2) assuming a>0 assuming b>0),symbolic): end: map(asy,rseq);

NywqKClJInhHNiJJImFHRiYiIiIpRiUsJComIiIjRihJImJHRiZGKEYoRigpRiUiIiRGKCwmKUYlLChGJ0YoRitGKEYvRihGKC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0YoRigsKilGJSwmRidGKEYtISIiRigqJilGJSwoRidGKEYtRjxGKEY8RihGLUYoRihGPkYoLUY0NiMqJClGJUYsRjxGKCwsKUYlLChGJ0Y8Ri1GPEYsRjxGKComKUYlLChGJ0Y8Ri1GPEYvRjxGKEYnRihGKComRkhGKEYtRihGKComRixGKEZIRihGKC1GNDYjKiQpRiUiIiVGPEYoLCopRiUsJkYtRihGJ0Y8RigqJilGJSwoRi1GKEYnRjxGKEY8RihGJ0YoRihGVUYoRkBGKCooKUYlLCQqJkYsRihGJ0YoRihGKClGJUYtRihGLkYoLCYpRiUsKEZaRihGLUYoRi9GKEYoRjNGKEZRRkRGOQ==

forces a=b

liste:=2/x+y/x+2*y+1/x/y+x+x*y+1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwwKiYiIiMiIiJJInhHRighIiJGMSomSSJ5R0YoRjFGMkYzRjEqJkYwRjFGNUYxRjEqJkY1RjNGMkYzRjFGMkYxKiZGMkYxRjVGMUYxKiRGNUYzRjE3I0Yu

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLDIqKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGMkYxRjZGMUYxKipGNEYxRjBGMUYzRjFGNUYxRjEqJkYwRjFGNUYxRjEqJkYwRjFGM0YxRjEqKEY0RjFGMEYxRjZGMUYxKiZGM0YxRjZGMSEiIkYwRjE3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCosIyIiIiIiI0YyLCoqKEYzRjJJInRHRihGMilJInlHRihGM0YyISIiRjZGOUY4RjIqJCksMCooIiM3RjIpRjZGM0YyKUY4IiIkRjJGOSooIiIpRjJGP0YyRjdGMkY5KigiIiVGMkY2RjJGQEYyRjkqKEZFRjJGP0YyRjhGMkY5KiRGP0YyRjIqKEYzRjJGNkYyRjhGMkY5KiRGN0YyRjJGMUYyRjJGMkY4RjlGNkY5LCZGOEYyRjJGMkY5RjIsJCosRjFGMiwqRjVGMkY6RjJGNkYyRjhGOUYyRjhGOUY2RjlGSkY5Rjk3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCoqIyIiIiIiI0YyLCoqJkkidEdGKEYyKUkieEdGKEYzRjIhIiIqJkYzRjJGNkYyRjlGOEYyKiQpKiZGOEYyLDAqJilGNkYzRjIpRjgiIiRGMkYyKigiIiVGMkZARjJGN0YyRjkqKCIiKUYyRkBGMkY4RjJGOSooRjNGMkY2RjJGN0YyRjkqJiIjN0YyRkBGMkY5KiZGREYyRjZGMkY5RjhGMkYyRjFGMkYyRjIpLCZGOEYyRjJGMkYzRjlGNkY5RjIsJCoqRjFGMiwqRjVGMkY7RjJGOkYyRjhGOUYyRktGOUY2RjlGOTcjRi4=

qy:=coeff(K,x,0)/coeff(K,x,2); qx:=coeff(K,y,0)/coeff(K,y,2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeUdGKComLCgqJkkidEdGKCIiIilJInlHRigiIiNGMkYyKihGNUYyRjFGMkY0RjJGMkYxRjJGMiwmRjBGMiomRjFGMkY0RjJGMiEiIjcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeEdGKComLCYqJkkidEdGKCIiIkkieEdGKEYyRjJGMUYyRjIsKComRjFGMilGMyIiI0YyRjIqKEY3RjJGMUYyRjNGMkYyRjFGMiEiIjcjRi4=

phi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([subs(y=v,qy)/u,v]): end: psi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([u,subs(x=u,qx)/v]): end: [x,y ]; phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);

NyRJInhHNiJJInlHRiQ=

NyQqKCwmSSJ5RzYiIiIiRidGJ0YnSSJ4R0YmISIiRiVGKUYl

NyQqKCwmSSJ5RzYiIiIiRidGJ0YnSSJ4R0YmISIiRiVGKSomRihGJywoKiZGKEYnRiVGJ0YnRiVGJ0YnRidGKQ==

NyQqJkkieUc2IiIiIiwmSSJ4R0YlRiZGJkYmRiYqJkYoRiYsKComRihGJkYkRiZGJkYkRiZGJkYmISIi

NyQqJkkieUc2IiIiIiwmSSJ4R0YlRiZGJkYmRiYqJEYoISIi

NyQqJEkieUc2IiEiIiokSSJ4R0YlRiY=

NyQqJEkieUc2IiEiIiooSSJ4R0YlIiIiRiRGKSwmRiRGKUYpRilGJg==

NyQqJiwoKiZJInhHNiIiIiJJInlHRidGKEYoRilGKEYoRihGKEYmISIiKihGJkYoRilGKCwmRilGKEYoRihGKg==

NyQqJiwoKiZJInhHNiIiIiJJInlHRidGKEYoRilGKEYoRihGKEYmISIiKiYsJkYmRihGKEYoRipGKUYq

NyRJInhHNiIqJiwmRiMiIiJGJ0YnISIiSSJ5R0YkRig=

NyRJInhHNiJJInlHRiQ=

theta:=proc(p) psi(phi(p)): end: theta_itere:=proc(p,k) if k= 0 then p else theta_itere(theta(p),k-1): fi: end: n:=5: [seq(theta_itere([x,y],i),i=1..n-1)];

NyY3JCooLCZJInlHNiIiIiJGKEYoRihJInhHRichIiJGJkYqKiZGKUYoLCgqJkYpRihGJkYoRihGJkYoRihGKEYqNyQqJkYmRigsJkYpRihGKEYoRigqJEYpRio3JCokRiZGKiooRilGKEYmRihGJUYqNyQqJkYsRihGKUYqKiZGMEYqRiZGKg==

li:=map(p->p[1]*p[2],%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNsaUdGKDcmKigsJkkieUdGKCIiIkYyRjJGMkYxISIiLCgqJkkieEdGKEYyRjFGMkYyRjFGMkYyRjJGMyooRjFGMiwmRjZGMkYyRjJGMkY2RjMqJkY2RjJGMEYzKipGNEYyRjZGM0Y4RjNGMUYzNyNGLg==

Htau:=-1/n*add(i*li[i],i=1..n-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIdGF1R0YoLCoqKiMiIiIiIiZGMSwmSSJ5R0YoRjFGMUYxRjFGNCEiIiwoKiZJInhHRihGMUY0RjFGMUY0RjFGMUYxRjVGNSoqIyIiI0YyRjFGNEYxLCZGOEYxRjFGMUYxRjhGNUY1KigjIiIkRjJGMUY4RjFGM0Y1RjUqLCMiIiVGMkYxRjZGMUY4RjVGPEY1RjRGNUY1NyNGLg==

The function F(x) Fx:=factor(rationalize(subs(y=Ysol[1],Htau)+ subs(y=Ysol[2],Htau)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKi4jIiIiIiImRjEsPiooIiIlRjEpSSJ0R0YoIiIjRjEpSSJ4R0YoRjJGMUYxKigiIzpGMUY2RjEpRjpGNUYxRjEqKEY8RjFGNkYxKUY6IiIkRjFGMSomRjdGMUY9RjFGMSooIiM1RjFGNkYxKUY6RjhGMUYxKihGNUYxRjdGMUY/RjFGMSooIiNJRjFGNkYxRjpGMUYxKigiIzZGMUY3RjFGREYxRjEqJEY/RjFGMSomIiM3RjFGNkYxISIiKigiI0dGMUY3RjFGOkYxRjEqJkZARjFGREYxRjEqJkY1RjFGN0YxRk0qJiIiJ0YxRjpGMUYxRjEsJiomRkBGMUY3RjFGMUYxRjFGTUY3Rk0pLCZGOkYxRjFGMUY4Rk1GOkZNRk03I0Yu

Computing G(y) Gy:=factor(simplify(expand(rationalize(subs(x=Xsol[1],x*y-Fx))),symbolic));

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

let us check the formula given in the paper for G(y) thetat:=proc(p) phi(psi(p)): end: thetat_itere:=proc(p,k) if k= 0 then p else thetat_itere(thetat(p),k-1): fi: end: [seq(thetat_itere([x,y],i),i=1..n-1)];

NyY3JComLCgqJkkieEc2IiIiIkkieUdGKEYpRilGKkYpRilGKUYpRichIiIqJiwmRidGKUYpRilGK0YqRis3JCokRipGKyooRidGKUYqRiksJkYqRilGKUYpRis3JComRipGKUYtRikqJEYnRis3JCooRjFGKUYnRitGKkYrKiZGJ0YpRiVGKw==

lit:=map(p->p[1]*p[2],%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRsaXRHRig3JioqLCgqJkkieEdGKCIiIkkieUdGKEYzRjNGNEYzRjNGM0YzRjIhIiIsJkYyRjNGM0YzRjVGNEY1KiZGMkYzLCZGNEYzRjNGM0Y1KihGNEYzRjZGM0YyRjUqKEY4RjNGNEY1RjBGNTcjRi4=

Htaut:=-1/n*add(i*lit[i],i=1..n-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZIdGF1dEdGKCwqKiwjIiIiIiImRjEsKComSSJ4R0YoRjFJInlHRihGMUYxRjZGMUYxRjFGMUY1ISIiLCZGNUYxRjFGMUY3RjZGN0Y3KigjIiIjRjJGMUY1RjEsJkY2RjFGMUYxRjdGNyoqIyIiJEYyRjFGNkYxRjhGMUY1RjdGNyoqIyIiJUYyRjFGPEYxRjZGN0YzRjdGNzcjRi4=

The invariant associated with H(x,y)=xy L:=[[x,y ]]: for i from 1 to 2*n-1 do if i mod 2 =1 then L:=[op(L), phi(L[i])]; else L:=[op(L), psi(L[i])]; fi: od: L;

Nyw3JEkieEc2IkkieUdGJTckKigsJkYmIiIiRipGKkYqRiQhIiJGJkYrRiY3JEYoKiZGJEYqLCgqJkYkRipGJkYqRipGJkYqRipGKkYrNyQqJkYmRiosJkYkRipGKkYqRipGLTckRjEqJEYkRis3JCokRiZGK0Y0NyRGNiooRiRGKkYmRipGKUYrNyQqJkYuRipGJEYrRjg3JEY6KiZGMkYrRiZGKzckRiRGPA==

Hsig:=convert(map(p->p[1]*p[2], L),`+`);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVIc2lnR0YoLDYqJkkieEdGKCIiIkkieUdGKEYxRjEqJiwmRjJGMUYxRjFGMUYwISIiRjEqKEY0RjFGMkY1LChGL0YxRjJGMUYxRjFGNUYxKipGMkYxLCZGMEYxRjFGMUYxRjBGMUY3RjVGMSooRjJGMUY5RjFGMEY1RjEqJkYyRjVGMEY1RjEqJkYwRjFGNEY1RjEqKEY3RjFGMkYxRjRGNUYxKipGN0YxRjBGNUY5RjVGMkY1RjEqKEYwRjFGMkY1RjlGNUYxNyNGLg==

Ix:=factor(simplify(expand(rationalize(subs(y=Ysol[1],Hsig))),symbolic));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeEdGKCwkKi4iIiMiIiIsOComKUkidEdGKEYwRjEpSSJ4R0YoIiImRjFGMSomRjVGMSlGNyIiJUYxISIiKihGOEYxRjRGMSlGN0YwRjFGPCooRjBGMUY1RjEpRjciIiRGMUYxKiZGNUYxRj5GMUYxKiRGQEYxRjEqJkZBRjFGNEYxRjwqKEZBRjFGNUYxRjdGMUYxKiRGPkYxRjFGNUY8RjdGMUYxLCYqJkZBRjFGNUYxRjFGMUYxRjxGNUY8KSwmRjdGMUYxRjFGMEY8RjdGPEYxNyNGLg==

Iy:=factor(simplify(expand(rationalize(subs(x=Xsol[1],Hsig))),symbolic));;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJeUdGKCwkKi4iIiMiIiIsOCooIiIkRjEpSSJ0R0YoRjBGMSlJInlHRigiIiZGMUYxKiZGNkYxRjdGMUYxKihGOUYxRjVGMSlGOEY0RjFGMSooRjRGMUY2RjEpRjgiIiVGMSEiIiomRjZGMUY8RjFGQCokRj5GMUZAKihGMEYxRjZGMSlGOEYwRjFGQCokRjxGMUZAKiRGNUYxRkAqJkY2RjFGOEYxRjEqJEZERjFGQEYxRkRGQCksJkY4RjFGMUYxRjBGQCwmKiZGNEYxRjZGMUYxRjFGMUZARjZGQEZANyNGLg==

Expected value for G(y) factor(rationalize(subs(x=Xsol[1],Htaut)+subs(x=Xsol[2],Htaut)+(1-1/2/n)*Iy));

LCQqLiMiIiIiIiZGJSw+KigiI0ZGJSlJInRHNiIiIiNGJSlJInlHRixGJkYlRiUqKCIjSUYlRipGJSlGLyIiJUYlRiUqKCIiKkYlRitGJUYuRiVGJSooIiNdRiVGKkYlKUYvIiIkRiVGJSooIiIpRiVGK0YlRjJGJSEiIiooRjFGJUYqRiUpRi9GLUYlRiUqKCIiJ0YlRitGJUY4RiVGPComRkBGJUYyRiVGPCooIiM6RiVGKkYlRi9GJUYlKihGM0YlRitGJUY+RiVGPComRjtGJUY4RiVGPComRjNGJUYqRiVGPCooRjVGJUYrRiVGL0YlRiUqJkZARiVGPkYlRjxGJUY+RjwpLCZGL0YlRiVGJUYtRjwsJiomRjlGJUYrRiVGJUYlRiVGPEYrRjxGPA==

factor(%-Gy);

IiIh

convert(Gy,parfrac,y);convert(Iy,parfrac,y);

LC4qJiMiIioiIiYiIiJJInlHNiJGJyEiIiooIyIiI0YmRicsJiomIiIlRidJInRHRilGJ0YnIiIkRidGJ0YxRipGJyoqI0YnRiZGJywmKiYiIztGJ0YxRidGKkYwRipGJywmRihGJ0YnRidGKkYxRipGJyoqRjBGJ0YxRicpRihGLUYqLCYqJiIjOkYnRjFGJ0YnRiZGJ0YqRicqKCwmKiYiI0JGJ0YxRidGKkYlRipGJ0YoRipGO0YqRicqKkY0RicsJiomIiM3RidGMUYnRidGMEYnRicpRjhGLUYqRjFGKkYn

LC4qJiIiIyIiIkkieUc2IkYlISIiKihGJEYlSSJ0R0YnRigsJiomRiRGJUYqRiVGJUYlRiVGJUYlKigsJiomIiIpRiVGKkYlRihGJEYoRiUsJkYmRiVGJUYlRihGKkYoRiUqKkYkRiVGKkYlKUYmRiRGKCwmKiYiIiRGJUYqRiVGJUYlRiVGKEYlKigsJiomIiIlRiVGKkYlRihGJEYoRiVGJkYoRjRGKEYlKigsJiomIiInRiVGKkYlRiVGJEYlRiUpRjFGJEYoRipGKEYl

factor(Gy-2/5*Iy);

LCQqKiMiIiIiIiZGJUkieUc2IiEiIiwoKihGJkYlSSJ0R0YoRiUpRiciIiNGJUYlKiZGJkYlRixGJUYlKiZGLkYlRidGJUYpRiVGLEYpRik=

collect(%,y);

LChJInlHNiIhIiIqJiMiIiMiIiYiIiJJInRHRiRGJUYqKiRGI0YlRiU=

factor(Fx+2/5*Ix);

LCQqKiMiIiIiIiZGJSwqKihGJkYlSSJ0RzYiRiUpSSJ4R0YqIiIjRiVGJSomIiM1RiVGKUYlRiUqJiIiJEYlRixGJSEiIkYtRiVGJUYpRjIsJkYsRiVGJUYlRjJGMg==

convert(%,parfrac,x);

LChJInhHNiIhIiIqKCMiIiIiIiZGKCwmKiZGKUYoSSJ0R0YkRihGKCIiJEYoRihGLEYlRigqKCwmKiZGLUYoRixGKEYlRihGJUYoLCZGI0YoRihGKEYlRixGJUYo

factor(Gy-2/5*Iy-2/5/t);convert(%,parfrac,y);

LCQqJkkieUc2IiEiIiwmKiQpRiQiIiMiIiJGK0YrRitGK0Ym

LCZJInlHNiIhIiIqJEYjRiVGJQ==

factor(Fx+2/5*Ix+2/5/t);map(factor,convert(%,parfrac,x));

LCQqKCwoKiZJInRHNiIiIiIpSSJ4R0YnIiIjRihGKComRitGKEYmRihGKEYqISIiRihGJkYtLCZGKkYoRihGKEYtRi0=

LChJInhHNiIhIiIqJiwmIiIiRihJInRHRiRGKEYoRilGJUYoKigsJiomIiIkRihGKUYoRihGKEYoRihGKUYlLCZGI0YoRihGKEYlRiU=

Decoupling for other starting points (Section 7.3) r[1]:=[x,x^2]; r[2]:=phi(%); r[3]:=psi(%);r[4]:=phi(%); r[5]:=psi(%);r[6]:=phi(%); r[7]:=psi(%);r[8]:=phi(%); r[9]:=psi(%);r[10]:=phi(%); psi(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIiNyRJInhHRigqJClGMiIiI0YwNyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIjNyQqJiwmKiQpSSJ4R0YoRjAiIiJGN0Y3RjdGNylGNiIiJCEiIkY0NyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIkNyQqJiwmKiQpSSJ4R0YoIiIjIiIiRjhGOEY4RjgpRjZGMCEiIiomRjZGOCwoKiRGOUY4RjhGNEY4RjhGOEY6NyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIlNyQqJiwmSSJ4R0YoIiIiRjVGNUY1KUY0IiIjRjUqJkY0RjUsKCokKUY0IiIkRjVGNSokRjZGNUY1RjVGNSEiIjcjRjE=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiImNyQqJiwmSSJ4R0YoIiIiRjVGNUY1KUY0IiIjRjUqJEY0ISIiNyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiInNyQqJClJInhHRigiIiMhIiIqJEY0RjY3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIoNyQqJClJInhHRigiIiMhIiIqJilGNCIiJCIiIiwmKiRGM0Y6RjpGOkY6RjY3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIpNyQqJiwoKiQpSSJ4R0YoIiIkIiIiRjgqJClGNiIiI0Y4RjhGOEY4RjhGNiEiIiomRjVGOCwmRjlGOEY4RjhGPDcjRjE=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIqNyQqJiwoKiQpSSJ4R0YoIiIkIiIiRjgqJClGNiIiI0Y4RjhGOEY4RjhGNiEiIiomRjpGPCwmRjZGOEY4RjhGPDcjRjE=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiM1NyRJInhHRigqJilGMiIiIyEiIiwmRjIiIiJGOEY4RjY3I0Yx

NyRJInhHNiIqJClGIyIiIyIiIg==

rseq:=[seq(r[i],i=1..10)];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVyc2VxR0YoNyw3JEkieEdGKCokKUYwIiIjIiIiNyQqJiwmRjFGNEY0RjRGNClGMCIiJCEiIkYxNyRGNiomRjBGNCwoKiRGOEY0RjRGMUY0RjRGNEY6NyQqJiwmRjBGNEY0RjRGNEYyRjRGPDckRkAqJEYwRjo3JCokRjJGOkZDNyRGRSomRjhGNEY3Rjo3JComRj1GNEYwRjpGRzckRkkqJkYyRjpGQUY6NyRGMEZLNyNGLg==

asy:=proc(p) local u, v: u:=p[1]: v:=p[2]: map(simplify,expand(asympt(u^(a+1)*v^(b+1),x,2) assuming a>0 assuming b>0),symbolic): end: map(asy,rseq);

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

case a=b asy:=proc(p) local u, v: u:=p[1]: v:=p[2]: map(simplify,expand(asympt(u^(a+1)*v^(a+1), x,2) assuming a>0 assuming b>0),symbolic): end: map(asy,rseq);

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

case a=2b+1 asy:=proc(p) local u, v: u:=p[1]: v:=p[2]: map(simplify,expand(asympt(u^(2*b+2)*v^(b+1), x,3) assuming a>0 assuming b>0),symbolic): end: map(asy,rseq);

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

map(simplify,%[1]-%[2]+%[3]-%[4]+%[5]-%[6]+%[7]-%[8]+%[9]-%[10],symbolic);

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

forces b=0. Let us chech that x^2y is indeed decoupled map(Z-> Z[1]^2*Z[2], rseq);

NywqJClJInhHNiIiIiUiIiIqJiksJiokKUYlIiIjRihGKEYoRihGLkYoRiQhIiIqKEYqRigpRiUiIiZGLywoKiQpRiUiIiRGKEYoRixGKEYoRihGLyooKSwmRiVGKEYoRihGLkYoRjFGKEYzRi8qJkY4RihGNUYoKiRGMUYvKiZGJUYvRitGLyooKUYzRi5GKEYlRihGK0YvKihGPkYoRiRGL0Y5Ri8qJEY5Ri8=

factor(%[1]-%[2]+%[3]-%[4]+%[5]-%[6]+%[7]-%[8]+%[9]-%[10]);

IiIh

Now, the infinite groups: we check that the functions given in the paper are correct

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwqSSJ4R0YoIiIiSSJ5R0YoRjAqJEYvISIiRjAqJkYxRjNGL0YzRjA3I0Yu

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCwqKEkidEdGKCIiIilJInhHRigiIiNGMUkieUdGKEYxRjEqKEYwRjFGM0YxKUY1RjRGMUYxKiZGMEYxRjVGMUYxKiZGM0YxRjVGMSEiIkYwRjE3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ5R0YoISIiSSJ0R0YoRjUsKComRjZGMilGNEYzRjJGNUY0RjIqJCkqJkY0RjIsLComKUY2RjNGMilGNCIiJEYyRjIqKCIiJUYyRj9GMkY0RjJGNSooRjNGMkY2RjJGOUYyRjUqJkZDRjJGP0YyRjVGNEYyRjJGMUYyRjJGMkYyLCQqKkYxRjIsKEY4RjJGOkYyRjRGNUYyRjRGNUY2RjVGNTcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ4R0YoISIiSSJ0R0YoRjUsKiomRjZGMilGNEYzRjJGNUY2RjVGNEYyKiQpLDAqJilGNkYzRjIpRjQiIiVGMkYyKihGM0YyRj5GMkY5RjJGMiooRjNGMkY2RjIpRjQiIiRGMkY1KihGQEYyRj5GMkY0RjJGNSokRj5GMkYyKihGM0YyRjZGMkY0RjJGNSokRjlGMkYyRjFGMkYyRjJGMiwkKipGMUYyLCpGOEYyRjpGMkY2RjJGNEY1RjJGNEY1RjZGNUY1NyNGLg==

Gy:=-1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiRJInlHRighIiJGMTcjRi4=

Fx:=rationalize(subs(y=Ysol[1],x*y-Gy));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKComLCgqJkkidEdGKCIiIilJInhHRigiIiNGMiEiIkYxRjZGNEYyRjJGMUY2NyNGLg==

expand(Fx);

LCgqJClJInhHNiIiIiMiIiIhIiJGKEYpKiZJInRHRiZGKUYlRihGKA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwqSSJ4R0YoIiIiSSJ5R0YoRjAqJkYxRjBGLyEiIkYwKiZGMUYzRi9GM0YwNyNGLg==

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCwqKEkidEdGKCIiIilJInhHRigiIiNGMUkieUdGKEYxRjEqKEYwRjFGM0YxKUY1RjRGMUYxKiZGMEYxRjdGMUYxKiZGM0YxRjVGMSEiIkYwRjE3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ5R0YoISIiSSJ0R0YoRjUsKComRjZGMilGNEYzRjJGNUY0RjIqJCkqJkY0RjIsLComKUY2RjNGMilGNCIiJEYyRjIqKCIiJUYyRj9GMkY5RjJGNSooRjNGMkY2RjJGOUYyRjUqJkZDRjJGP0YyRjVGNEYyRjJGMUYyRjJGMkYyLCQqKkYxRjIsKEY4RjJGOkYyRjRGNUYyRjRGNUY2RjVGNTcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCoqIyIiIiIiI0YyLCgqJkkidEdGKEYyKUkieEdGKEYzRjIhIiJGOEYyKiQpLCwqJilGNkYzRjIpRjgiIiVGMkYyKihGM0YyRjZGMilGOCIiJEYyRjkqKEZARjJGPkYyRjhGMkY5KiZGQEYyRj5GMkY5KiRGN0YyRjJGMUYyRjJGMkY2RjksJkY4RjJGMkYyRjlGMiwkKipGMUYyLChGNUYyRjpGMkY4RjlGMkY2RjlGR0Y5Rjk3I0Yu

Gy:=-y-1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwmSSJ5R0YoISIiKiRGL0YwRjA3I0Yu

Fx:=rationalize(subs(y=Ysol[1],x*y-Gy));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKihJInhHRigiIiIsJiomSSJ0R0YoRjFGMEYxRjFGMSEiIkYxRjRGNUY1NyNGLg==

expand(Fx);

LCYqJClJInhHNiIiIiMiIiIhIiIqJkkidEdGJkYpRiVGKEYo

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwqSSJ5R0YoIiIiKiRJInhHRighIiJGMCokRi9GM0YwKiZGMkYwRi9GMEYwNyNGLg==

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCwqKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGM0YxRjVGMUYxKiZGMEYxRjNGMUYxKiZGMEYxRjZGMUYxKiZGM0YxRjZGMSEiIjcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ0R0YoISIiKUkieUdGKEYzRjUsKiomRjRGMkY2RjJGNUY0RjVGN0YyKiQpLDAqJilGNEYzRjIpRjciIiVGMkYyKihGQEYyRj5GMilGNyIiJEYyRjUqKEYzRjJGPkYyRjZGMkYyKihGM0YyRjRGMkZCRjJGNSokRj5GMkYyKihGM0YyRjRGMkY3RjJGNSokRjZGMkYyRjFGMkYyRjJGMiwkKipGMUYyLCpGOUYyRjpGMkY0RjJGN0Y1RjJGNEY1RjZGNUY1NyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCosIyIiIiIiI0YyLChJInRHRighIiJJInhHRihGMiokKSwsKigiIiVGMilGNUYzRjIpRjciIiRGMkY2KihGPEYyRj1GMilGN0YzRjJGNiokRj1GMkYyKihGM0YyRjVGMkY3RjJGNiokRkFGMkYyRjFGMkYyRjJGN0Y2RjVGNiwmRjdGMkYyRjJGNkYyLCQqLEYxRjIsKEY1RjJGN0Y2RjhGMkYyRjdGNkY1RjZGRUY2RjY3I0Yu

Gy:=-y-1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwmSSJ5R0YoISIiKiRGL0YwRjA3I0Yu

Fx:=rationalize(subs(y=Ysol[1],x*y-Gy));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCooLCZJInRHRighIiJJInhHRigiIiJGM0YyRjFGMEYxNyNGLg==

expand(Fx);

LCYqJEkieEc2IiEiIkYmKiRJInRHRiVGJiIiIg==

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwqSSJ4R0YoIiIiSSJ5R0YoRjAqJEYvISIiRjAqJkYvRjBGMUYzRjA3I0Yu

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCwqKEkidEdGKCIiIilJInhHRigiIiNGMUkieUdGKEYxRjEqKEYwRjFGM0YxKUY1RjRGMUYxKiZGMEYxRjJGMUYxKiZGMEYxRjVGMUYxKiZGM0YxRjVGMSEiIjcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YyLCgqJkkidEdGKEYyKUkieUdGKEYzRjIhIiJGOEYyKiQpKiZGOEYyLCwqJilGNkYzRjIpRjgiIiRGMkYyKigiIiVGMkY/RjJGOEYyRjkqKEYzRjJGNkYyRjdGMkY5KiZGQ0YyRj9GMkY5RjhGMkYyRjFGMkYyRjJGNkY5LCZGOEYyRjJGMkY5RjIsJCoqRjFGMiwoRjVGMkY6RjJGOEY5RjJGNkY5RkZGOUY5NyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ4R0YoISIiSSJ0R0YoRjUsKiomRjZGMilGNEYzRjJGNUY2RjVGNEYyKiQpLDAqJilGNkYzRjIpRjQiIiVGMkYyKihGQEYyRj5GMilGNCIiJEYyRjUqKEYzRjJGPkYyRjlGMkYyKihGM0YyRjZGMkZCRjJGNSokRj5GMkYyKihGM0YyRjZGMkY0RjJGNSokRjlGMkYyRjFGMkYyRjJGMiwkKipGMUYyLCpGOEYyRjpGMkY2RjJGNEY1RjJGNEY1RjZGNUY1NyNGLg==

Gy:=-y^2+y/t+1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwoKiQpSSJ5R0YoIiIjIiIiISIiKiZGMUYzSSJ0R0YoRjRGMyokRjFGNEYzNyNGLg==

Fx:=factor(simplify(expand(rationalize(subs(y=Ysol[1],x*y-Gy))),symbolic));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKigpSSJ4R0YoIiIjISIiSSJ0R0YoRjMsKiomRjQiIiIpRjEiIiRGN0Y3KiZGNEY3RjBGN0YzRjRGM0YxRjdGN0YzNyNGLg==

expand(Fx);

LCpJInhHNiIhIiIiIiJGJiokKUYjIiIjRiVGJiomRiNGJUkidEdGJEYlRiU=

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwsSSJ4R0YoIiIiSSJ5R0YoRjAqJEYvISIiRjAqJkYxRjNGL0YzRjAqJkYvRjBGMUYwRjA3I0Yu

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGMkYxRjZGMUYxKihGMEYxRjNGMUY1RjFGMSomRjBGMUY2RjFGMSomRjNGMUY2RjEhIiJGMEYxNyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCosIyIiIiIiI0YyLCgqJkkidEdGKEYyKUkieUdGKEYzRjIhIiJGOEYyKiQpKiZGOEYyLC4qJilGNkYzRjIpRjgiIiRGMkYyKigiIiVGMkY/RjJGN0YyRjkqKCIiKUYyRj9GMkY4RjJGOSooRjNGMkY2RjJGN0YyRjkqJkZDRjJGP0YyRjlGOEYyRjJGMUYyRjJGMkY4RjlGNkY5LCZGOEYyRjJGMkY5RjIsJCosRjFGMiwoRjVGMkY6RjJGOEY5RjJGOEY5RjZGOUZIRjlGOTcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCosIyIiIiIiI0YyLCoqJkkidEdGKEYyKUkieEdGKEYzRjIhIiJGNkY5RjhGMiokKSwwKiYpRjZGM0YyKUY4IiIlRjJGMiooRjNGMkY+RjJGN0YyRjkqKEYzRjJGNkYyKUY4IiIkRjJGOSooRkBGMkY+RjJGOEYyRjkqJEY+RjJGMiooRjNGMkY2RjJGOEYyRjkqJEY3RjJGMkYxRjJGMkYyRjhGOUY2RjksJkY4RjJGMkYyRjlGMiwkKixGMUYyLCpGNUYyRjpGMkY2RjJGOEY5RjJGOEY5RjZGOUZJRjlGOTcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwmSSJ5R0YoISIiKigsJiIiIkYzSSJ0R0YoRjNGM0Y0RjAsJkYvRjNGM0YzRjBGMDcjRi4=

Fx:=factor(simplify(expand(rationalize(subs(y=Ysol[1],x*y-Gy))),symbolic));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCooLCgqJkkidEdGKCIiIkkieEdGKEYyRjJGMSEiIkYzRjJGMkYzRjRGMUY0NyNGLg==

expand(Fx);

LCgiIiJGIyokSSJ4RzYiISIiRicqJEkidEdGJkYnRiM=

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwsKiRJInlHRighIiIiIiJGMEYyKiRJInhHRihGMUYyKiZGMEYxRjRGMUYyKiZGNEYyRjBGMkYyNyNGLg==

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGM0YxRjVGMUYxKiZGMEYxRjNGMUYxKiZGMEYxRjZGMUYxKiZGM0YxRjZGMSEiIkYwRjE3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ0R0YoISIiKUkieUdGKEYzRjUsKiomRjRGMkY2RjJGNUY0RjVGN0YyKiQpLDAqJilGNEYzRjIpRjciIiVGMkYyKihGQEYyRj5GMilGNyIiJEYyRjUqKEYzRjJGPkYyRjZGMkY1KihGM0YyRjRGMkZCRjJGNSokRj5GMkYyKihGM0YyRjRGMkY3RjJGNSokRjZGMkYyRjFGMkYyRjJGMiwkKipGMUYyLCpGOUYyRjpGMkY0RjJGN0Y1RjJGNEY1RjZGNUY1NyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCosIyIiIiIiI0YyLChJInRHRighIiJJInhHRihGMiokKSwuKigiIiVGMilGNUYzRjIpRjciIiRGMkY2KigiIilGMkY9RjIpRjdGM0YyRjYqKEY8RjJGPUYyRjdGMkY2KiRGPUYyRjIqKEYzRjJGNUYyRjdGMkY2KiRGQkYyRjJGMUYyRjJGMkY3RjZGNUY2LCZGN0YyRjJGMkY2RjIsJCosRjFGMiwoRjVGMkY3RjZGOEYyRjJGN0Y2RjVGNkZHRjZGNjcjRi4=

Gy:=-1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiRJInlHRighIiJGMTcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKigsJkkidEdGKCIiIkkieEdGKCEiIkYyRjFGNCwmRjNGMkYyRjJGNEY0NyNGLg==

convert(Fx,parfrac,x);

LCYqJEkidEc2IiEiIiIiIiooLCZGJ0YmRiRGJkYnRiRGJiwmSSJ4R0YlRidGJ0YnRiZGJw==

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwsSSJ4R0YoIiIiSSJ5R0YoRjAqJEYvISIiRjAqJkYxRjNGL0YzRjAqJkYxRjBGL0YzRjA3I0Yu

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMUkieUdGKEYxRjEqKEYwRjFGM0YxKUY1RjRGMUYxKiZGMEYxRjdGMUYxKiZGMEYxRjVGMUYxKiZGM0YxRjVGMSEiIkYwRjE3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ5R0YoISIiSSJ0R0YoRjUsKComRjZGMilGNEYzRjJGNUY0RjIqJCkqJkY0RjIsLiomKUY2RjNGMilGNCIiJEYyRjIqKCIiJUYyRj9GMkY5RjJGNSooRkNGMkY/RjJGNEYyRjUqKEYzRjJGNkYyRjlGMkY1KiZGQ0YyRj9GMkY1RjRGMkYyRjFGMkYyRjJGMiwkKipGMUYyLChGOEYyRjpGMkY0RjVGMkY0RjVGNkY1RjU3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCoqIyIiIiIiI0YyLCoqJkkidEdGKEYyKUkieEdGKEYzRjIhIiJGNkY5RjhGMiokKSwwKiYpRjZGM0YyKUY4IiIlRjJGMiooRjNGMkY+RjJGN0YyRjIqKEYzRjJGNkYyKUY4IiIkRjJGOSooRkBGMkY+RjJGOEYyRjkqJkZERjJGPkYyRjkqKEYzRjJGNkYyRjhGMkY5KiRGN0YyRjJGMUYyRjJGMkY2RjksJkY4RjJGMkYyRjlGMiwkKipGMUYyLCpGNUYyRjpGMkY2RjJGOEY5RjJGNkY5RklGOUY5NyNGLg==

Gy:=-y-1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwmSSJ5R0YoISIiKiRGL0YwRjA3I0Yu

Fx:=rationalize(subs(y=Ysol[1],x*y-Gy));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKComLCgqJkkidEdGKCIiIilJInhHRigiIiNGMiEiIkYxRjZGNEYyRjJGMUY2NyNGLg==

expand(Fx);

LCgqJClJInhHNiIiIiMiIiIhIiJGKEYpKiZJInRHRiZGKUYlRihGKA==

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwsSSJ4R0YoIiIiSSJ5R0YoRjAqJEYvISIiRjAqJEYxRjNGMComRi9GMEYxRjBGMDcjRi4=

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGMkYxRjZGMUYxKihGMEYxRjNGMUY1RjFGMSomRjBGMUYzRjFGMSomRjBGMUY2RjFGMSomRjNGMUY2RjEhIiI3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCosIyIiIiIiI0YyLCoqJkkidEdGKEYyKUkieUdGKEYzRjIhIiJGNkY5RjhGMiokKSwwKiYpRjZGM0YyKUY4IiIlRjJGMiooRkBGMkY+RjIpRjgiIiRGMkY5KihGM0YyRj5GMkY3RjJGOSooRjNGMkY2RjJGQkYyRjkqJEY+RjJGMiooRjNGMkY2RjJGOEYyRjkqJEY3RjJGMkYxRjJGMkYyRjhGOUY2RjksJkY4RjJGMkYyRjlGMiwkKixGMUYyLCpGNUYyRjpGMkY2RjJGOEY5RjJGOEY5RjZGOUZJRjlGOTcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCosIyIiIiIiI0YyLCoqJkkidEdGKEYyKUkieEdGKEYzRjIhIiJGNkY5RjhGMiokKSwwKiYpRjZGM0YyKUY4IiIlRjJGMiooRkBGMkY+RjIpRjgiIiRGMkY5KihGM0YyRj5GMkY3RjJGOSooRjNGMkY2RjJGQkYyRjkqJEY+RjJGMiooRjNGMkY2RjJGOEYyRjkqJEY3RjJGMkYxRjJGMkYyRjhGOUY2RjksJkY4RjJGMkYyRjlGMiwkKixGMUYyLCpGNUYyRjpGMkY2RjJGOEY5RjJGOEY5RjZGOUZJRjlGOTcjRi4=

Gy:=-y-1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwmSSJ5R0YoISIiKiRGL0YwRjA3I0Yu

Fx:=rationalize(subs(y=Ysol[1],x*y-Gy));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCooLCgqJkkidEdGKCIiIilJInhHRigiIiNGMiEiIkYxRjZGNEYyRjJGNEY2RjFGNjcjRi4=

expand(Fx);

LChJInhHNiIhIiIqJEYjRiVGJSokSSJ0R0YkRiUiIiI=

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwsSSJ4R0YoIiIiSSJ5R0YoRjAqJEYxISIiRjAqJkYvRjBGMUYzRjAqJkYxRjBGL0YzRjA3I0Yu

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMUkieUdGKEYxRjEqKEYwRjFGM0YxKUY1RjRGMUYxKiZGMEYxRjJGMUYxKiZGMEYxRjdGMUYxKiZGMEYxRjNGMUYxKiZGM0YxRjVGMSEiIjcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YyLCoqJkkidEdGKEYyKUkieUdGKEYzRjIhIiJGNkY5RjhGMiokKSwwKiYpRjZGM0YyKUY4IiIlRjJGMiooRkBGMkY+RjIpRjgiIiRGMkY5KihGM0YyRj5GMkY3RjJGOSooRjNGMkY2RjJGQkYyRjkqJEY+RjJGMiooRjNGMkY2RjJGOEYyRjkqJEY3RjJGMkYxRjJGMkYyRjZGOSwmRjhGMkYyRjJGOUYyLCQqKkYxRjIsKkY1RjJGOkYyRjZGMkY4RjlGMkY2RjlGSUY5Rjk3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCoqIyIiIiIiI0YyLCgqJkkidEdGKEYyKUkieEdGKEYzRjIhIiJGOEYyKiQpKiZGOEYyLC4qJilGNkYzRjIpRjgiIiRGMkYyKigiIiVGMkY/RjJGN0YyRjkqKCIiKUYyRj9GMkY4RjJGOSooRjNGMkY2RjJGN0YyRjkqJkZDRjJGP0YyRjlGOEYyRjJGMUYyRjJGMkY2RjksJkY4RjJGMkYyRjlGMiwkKipGMUYyLChGNUYyRjpGMkY4RjlGMkY2RjlGSEY5Rjk3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwqKiQpSSJ5R0YoIiIjISIiRjMqJkkidEdGKEYzRjFGMyIiIiooLCZGNkY2RjVGNkY2RjFGNkY1RjNGNiokRjBGNkYzNyNGLg==

Fx:=factor(simplify(expand(rationalize(subs(y=Ysol[1],x*y-Gy))),symbolic));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwkKiosMiomKUkidEdGKCIiIyIiIilJInhHRigiIiVGNUY1KiZGMkY1KUY3IiIkRjVGNSooRjRGNUYyRjUpRjdGNEY1RjUqJkYzRjVGOkY1ISIiKihGNEY1RjJGNUY3RjVGNSomRjNGNUY9RjVGNSokRjJGNUY1KiRGPUY1RjVGNUYyRj8pLCZGN0Y1RjVGNUY0Rj9GN0Y/Rj83I0Yu

map(factor,convert(Fx,parfrac,x));

LCxJInhHNiIhIiIqJiwmIiIiRihJInRHRiRGKEYoRilGJUYoKiRGI0YlRiUqKkYnRigsJiomIiIjRihGKUYoRihGKEYoRigsJkYjRihGKEYoRiUpRilGLkYlRiUqKClGJ0YuRigpRi9GLkYlRjBGJUYo

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH

Model 9, point (1,1) liste:=x+y+1/y+x/y+y/x;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwsSSJ4R0YoIiIiSSJ5R0YoRjAqJEYxISIiRjAqJkYvRjBGMUYzRjAqJkYxRjBGL0YzRjA3I0Yu

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMUkieUdGKEYxRjEqKEYwRjFGM0YxKUY1RjRGMUYxKiZGMEYxRjJGMUYxKiZGMEYxRjdGMUYxKiZGMEYxRjNGMUYxKiZGM0YxRjVGMSEiIjcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCoqIyIiIiIiI0YyLCgqJkkidEdGKEYyKUkieEdGKEYzRjIhIiJGOEYyKiQpKiZGOEYyLC4qJilGNkYzRjIpRjgiIiRGMkYyKigiIiVGMkY/RjJGN0YyRjkqKCIiKUYyRj9GMkY4RjJGOSooRjNGMkY2RjJGN0YyRjkqJkZDRjJGP0YyRjlGOEYyRjJGMUYyRjJGMkY2RjksJkY4RjJGMkYyRjlGMiwkKipGMUYyLChGNUYyRjpGMkY4RjlGMkY2RjlGSEY5Rjk3I0Yu

Gy:=map(factor,y^4-2*y^3*(1+2*t)/t+y^2/t^2*(5*t^2+4*t+1)-2*y/t^2*(1+t)*(1+2*t)-2/t^2/y*(1+t)^2+1/t^2/y^2*(1+2*t+2*t^2)-2/t/y^3+1/y^4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwyKiQpSSJ5R0YoIiIlIiIiRjMqKiIiI0YzKUYxIiIkRjMsJiomRjVGM0kidEdGKEYzRjNGM0YzRjNGOiEiIkY7KigpRjFGNUYzKUY6RjVGOywoKiYiIiZGM0Y+RjNGMyomRjJGM0Y6RjNGM0YzRjNGM0YzKixGNUYzRjFGM0Y+RjssJkYzRjNGOkYzRjNGOEYzRjsqKkY1RjNGPkY7RjFGOylGREY1RjNGOyooRj5GO0Y9RjssKComRjVGM0Y+RjNGM0Y5RjNGM0YzRjNGMyooRjVGM0Y6RjtGNkY7RjsqJEYwRjtGMzcjRi4=

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

IiIh

Fx:=factor(simplify(expand(rationalize(subs(y=Ysol[1],x^2*y^2-Gy))),symbolic));

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

map(factor,convert(Fx,parfrac,x));

LDAqKCwoKiZJInRHNiIiIiJJInhHRidGKEYoKiYiIiZGKEYmRihGKCIiJEYoRigsKEYlRihGJiEiIkYoRi5GKClGJiIiI0YuRigqKCksJkYoRihGJkYoIiIlRigpLCZGKUYoRihGKEY0Ri4pRiZGNEYuRi4qKkYwRihGM0YoRiZGLkYpRi5GKCosRjBGKEYzRigsKComIiIoRihGL0YoRigqJiIiKUYoRiZGKEYoRjBGKEYoKUYmRixGLkY2Ri5GKCosRjBGKCwmKiZGMEYoRiZGKEYoRihGKEYoKUYzRixGKEY3Ri4pRjZGLEYuRigqLEZBRigsJiomIiInRihGJkYoRihGKEYoRigpRjNGMEYoRjdGLilGNkYwRi5GLiokKUYpRjBGLkYu

Point (0,1) model 3 liste:=y+1/x+1/y+x*y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwqSSJ5R0YoIiIiKiRJInhHRighIiJGMCokRi9GM0YwKiZGMkYwRi9GMEYwNyNGLg==

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCwqKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGM0YxRjVGMUYxKiZGMEYxRjNGMUYxKiZGMEYxRjZGMUYxKiZGM0YxRjZGMSEiIjcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ0R0YoISIiKUkieUdGKEYzRjUsKiomRjRGMkY2RjJGNUY0RjVGN0YyKiQpLDAqJilGNEYzRjIpRjciIiVGMkYyKihGQEYyRj5GMilGNyIiJEYyRjUqKEYzRjJGPkYyRjZGMkYyKihGM0YyRjRGMkZCRjJGNSokRj5GMkYyKihGM0YyRjRGMkY3RjJGNSokRjZGMkYyRjFGMkYyRjJGMiwkKipGMUYyLCpGOUYyRjpGMkY0RjJGN0Y1RjJGNEY1RjZGNUY1NyNGLg==

Fx:=-x-1/x/t+1/x^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwoSSJ4R0YoISIiKiZGL0YwSSJ0R0YoRjBGMCokKUYvIiIjRjAiIiI3I0Yu

rationalize(subs(x=Xsol[1], x*y^2-Fx)-subs(x=Xsol[2], x*y^2-Fx));

IiIh

Gy:=factor(expand(simplify(rationalize(subs(x=Xsol[1],x*y^2-Fx)),symbolic)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKigsLiomSSJ0R0YoIiIiKUkieUdGKCIiJUYzRjMqJkYyRjMpRjUiIiRGMyEiIiooIiIjRjNGMkYzKUY1RjxGM0YzKiRGOEYzRjpGMkYzRjVGOkYzRjJGOkY9RjpGOjcjRi4=

expand(Gy);

LC4qJClJInlHNiIiIiMiIiIhIiJGJUYoRidGKSomRiVGKEkidEdGJkYpRigqJEYkRilGKSomRitGKUYlRilGKA==

model 5 liste:=x+y+1/x+1/x/y+x*y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwsSSJ4R0YoIiIiSSJ5R0YoRjAqJEYvISIiRjAqJkYxRjNGL0YzRjAqJkYvRjBGMUYwRjA3I0Yu

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGMkYxRjZGMUYxKihGMEYxRjNGMUY1RjFGMSomRjBGMUY2RjFGMSomRjNGMUY2RjEhIiJGMEYxNyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCosIyIiIiIiI0YyLCgqJkkidEdGKEYyKUkieUdGKEYzRjIhIiJGOEYyKiQpKiZGOEYyLC4qJilGNkYzRjIpRjgiIiRGMkYyKigiIiVGMkY/RjJGN0YyRjkqKCIiKUYyRj9GMkY4RjJGOSooRjNGMkY2RjJGN0YyRjkqJkZDRjJGP0YyRjlGOEYyRjJGMUYyRjJGMkY4RjlGNkY5LCZGOEYyRjJGMkY5RjIsJCosRjFGMiwoRjVGMkY6RjJGOEY5RjJGOEY5RjZGOUZIRjlGOTcjRi4=

Fx:=x^2-x/t-(1+t)/t/x+1/x^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwqKiQpSSJ4R0YoIiIjIiIiRjMqJkkidEdGKCEiIkYxRjNGNiooLCZGM0YzRjVGM0YzRjVGNkYxRjZGNiokRjBGNkYzNyNGLg==

rationalize(subs(x=Xsol[1], x*y^2-Fx)-subs(x=Xsol[2], x*y^2-Fx));

IiIh

Gy:=factor(expand(simplify(rationalize(subs(x=Xsol[1],x*y^2-Fx)),symbolic)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiosMiomKUkidEdGKCIiIyIiIilJInlHRigiIiZGNUY1KiZGMkY1KUY3IiIkRjUhIiIqJkYzRjUpRjciIiVGNUY8KihGNEY1RjJGNSlGN0Y0RjVGPCooRjRGNUYyRjVGN0Y1RjwqJkYzRjVGQUY1RjwqJEYyRjVGPCokRkFGNUY8RjVGMkY8KSwmRjdGNUY1RjVGNEY8RjdGPEY8NyNGLg==

expand(Gy);

LDIqJiksJkkieUc2IiIiIkYoRigiIiMhIiIpRiYiIiVGKEYqKiZGJEYqKUYmRilGKEYoKihJInRHRidGKkYkRiopRiYiIiRGKEYoKihGKUYoRiRGKkYmRihGKComRilGKEYkRipGKCooRjBGKkYkRipGJkYoRigqJkYkRipGJkYqRigqKClGMEYpRipGJEYqRiZGKEYo

model 6 liste:=1/y+y+1/x+1/x/y+x*y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwsKiRJInlHRighIiIiIiJGMEYyKiRJInhHRihGMUYyKiZGMEYxRjRGMUYyKiZGNEYyRjBGMkYyNyNGLg==

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGM0YxRjVGMUYxKiZGMEYxRjNGMUYxKiZGMEYxRjZGMUYxKiZGM0YxRjZGMSEiIkYwRjE3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YySSJ0R0YoISIiKUkieUdGKEYzRjUsKiomRjRGMkY2RjJGNUY0RjVGN0YyKiQpLDAqJilGNEYzRjIpRjciIiVGMkYyKihGQEYyRj5GMilGNyIiJEYyRjUqKEYzRjJGPkYyRjZGMkY1KihGM0YyRjRGMkZCRjJGNSokRj5GMkYyKihGM0YyRjRGMkY3RjJGNSokRjZGMkYyRjFGMkYyRjJGMiwkKipGMUYyLCpGOUYyRjpGMkY0RjJGN0Y1RjJGNEY1RjZGNUY1NyNGLg==

Fx:=x+(1+t)/((x+1)*t^2)-(t+1)^2/((x+1)^2*t^2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwoSSJ4R0YoIiIiKigsJkYwRjBJInRHRihGMEYwLCZGL0YwRjBGMCEiIilGMyIiI0Y1RjAqKClGMkY3RjApRjRGN0Y1RjZGNUY1NyNGLg==

rationalize(subs(x=Xsol[1], x*y^2-Fx)-subs(x=Xsol[2], x*y^2-Fx));

IiIh

Gy:=factor(expand(simplify(rationalize(subs(x=Xsol[1],x*y^2-Fx)),symbolic)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKigsKiomSSJ0R0YoIiIiKUkieUdGKCIiJEYzRjMqJkYyRjMpRjUiIiNGM0YzRjIhIiJGNUYzRjNGMkY6RjhGOkY6NyNGLg==

expand(Gy);

LCpJInlHNiIhIiIiIiJGJSokKUYjIiIjRiVGJiomSSJ0R0YkRiVGI0YlRiU=

model 10 liste:=y+x*y+x/y+1/x/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwqSSJ5R0YoIiIiKiZJInhHRihGMEYvRjBGMComRjJGMEYvISIiRjAqJkYvRjRGMkY0RjA3I0Yu

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCwqKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGM0YxRjVGMUYxKiZGMEYxRjJGMUYxKiZGM0YxRjZGMSEiIkYwRjE3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YyLCgqJkkidEdGKEYyKUkieUdGKEYzRjIhIiJGOEYyKiQpLCwqJilGNkYzRjIpRjgiIiVGMkYyKihGQEYyRj5GMkY3RjJGOSooRjNGMkY2RjIpRjgiIiRGMkY5KiZGQEYyRj5GMkY5KiRGN0YyRjJGMUYyRjJGMkY2RjksJkZGRjJGMkYyRjlGMiwkKipGMUYyLChGNUYyRjpGMkY4RjlGMkY2RjlGR0Y5Rjk3I0Yu

Fx:=-x-1/x;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwmSSJ4R0YoISIiKiRGL0YwRjA3I0Yu

rationalize(subs(x=Xsol[1], x*y^2-Fx)-subs(x=Xsol[2], x*y^2-Fx));

IiIh

Gy:=factor(expand(simplify(rationalize(subs(x=Xsol[1],x*y^2-Fx)),symbolic)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKihJInlHRigiIiIsJiomSSJ0R0YoRjFGMEYxRjFGMSEiIkYxRjRGNUY1NyNGLg==

convert(Gy,parfrac,y);

LCYqJClJInlHNiIiIiMiIiIhIiIqJkYlRihJInRHRiZGKUYo

model 12 liste:=y+x*y+x/y+1/x/y+1/y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwsSSJ5R0YoIiIiKiZJInhHRihGMEYvRjBGMComRjJGMEYvISIiRjAqJkYvRjRGMkY0RjAqJEYvRjRGMDcjRi4=

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGM0YxRjVGMUYxKiZGMEYxRjJGMUYxKiZGMEYxRjNGMUYxKiZGM0YxRjZGMSEiIkYwRjE3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YyLCoqJkkidEdGKEYyKUkieUdGKEYzRjIhIiJGNkY5RjhGMiokKSwuKiYpRjZGM0YyKUY4IiIlRjJGMiooRjNGMkY+RjJGN0YyRjkqKEYzRjJGNkYyKUY4IiIkRjJGOSomRkRGMkY+RjJGOSooRjNGMkY2RjJGOEYyRjkqJEY3RjJGMkYxRjJGMkYyRjZGOSwmRkdGMkYyRjJGOUYyLCQqKkYxRjIsKkY1RjJGOkYyRjZGMkY4RjlGMkY2RjlGSEY5Rjk3I0Yu

Fx:=-1/x-x;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwmSSJ4R0YoISIiKiRGL0YwRjA3I0Yu

rationalize(subs(x=Xsol[1], x*y^2-Fx)-subs(x=Xsol[2], x*y^2-Fx));

IiIh

Gy:=factor(expand(simplify(rationalize(subs(x=Xsol[1],x*y^2-Fx)),symbolic)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKiYsKComSSJ0R0YoIiIiKUkieUdGKCIiI0YzRjNGMkYzRjUhIiJGM0YyRjdGNzcjRi4=

convert(Gy,parfrac,y);

LCgqJClJInlHNiIiIiMiIiIhIiJGKEYpKiZGJUYoSSJ0R0YmRilGKA==

model 11 liste:=y+x*y+x/y+1/x;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwqSSJ5R0YoIiIiKiZJInhHRihGMEYvRjBGMComRjJGMEYvISIiRjAqJEYyRjRGMDcjRi4=

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLCwqKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGM0YxRjVGMUYxKiZGMEYxRjJGMUYxKiZGMEYxRjZGMUYxKiZGM0YxRjZGMSEiIjcjRi4=

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCoqIyIiIiIiI0YyLCgqJkkidEdGKEYyKUkieUdGKEYzRjIhIiJGOEYyKiQpKiZGOEYyLCwqJilGNkYzRjIpRjgiIiRGMkYyKigiIiVGMkY/RjJGN0YyRjkqKEYzRjJGNkYyRjdGMkY5KiZGQ0YyRj9GMkY5RjhGMkYyRjFGMkYyRjJGNkY5LCYqJEY3RjJGMkYyRjJGOUYyLCQqKkYxRjIsKEY1RjJGOkYyRjhGOUYyRjZGOUZGRjlGOTcjRi4=

Fx:=-x-1/t/x+1/x^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwoSSJ4R0YoISIiKiZGL0YwSSJ0R0YoRjBGMCokKUYvIiIjRjAiIiI3I0Yu

rationalize(subs(x=Xsol[1], x*y^2-Fx)-subs(x=Xsol[2], x*y^2-Fx));

IiIh

Gy:=factor(expand(simplify(rationalize(subs(x=Xsol[1],x*y^2-Fx)),symbolic)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCwkKigsKiomSSJ0R0YoIiIiKUkieUdGKCIiJEYzRjMqJkYyRjMpRjUiIiNGMyEiIiokRjhGM0Y6RjJGOkYzRjVGOkYyRjpGOjcjRi4=

convert(Gy,parfrac,y);

LCgqJClJInlHNiIiIiMiIiIhIiIqKCwmRihGKEkidEdGJkYoRihGJUYoRixGKUYoKiRGJUYpRig=

Point (1,0) model 5 liste:=x+y+1/x+1/x/y+x*y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwsSSJ4R0YoIiIiSSJ5R0YoRjAqJEYvISIiRjAqJkYxRjNGL0YzRjAqJkYvRjBGMUYwRjA3I0Yu

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGMkYxRjZGMUYxKihGMEYxRjNGMUY1RjFGMSomRjBGMUY2RjFGMSomRjNGMUY2RjEhIiJGMEYxNyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCosIyIiIiIiI0YyLCgqJkkidEdGKEYyKUkieUdGKEYzRjIhIiJGOEYyKiQpKiZGOEYyLC4qJilGNkYzRjIpRjgiIiRGMkYyKigiIiVGMkY/RjJGN0YyRjkqKCIiKUYyRj9GMkY4RjJGOSooRjNGMkY2RjJGN0YyRjkqJkZDRjJGP0YyRjlGOEYyRjJGMUYyRjJGMkY4RjlGNkY5LCZGOEYyRjJGMkY5RjIsJCosRjFGMiwoRjVGMkY6RjJGOEY5RjJGOEY5RjZGOUZIRjlGOTcjRi4=

Fx:=x/t-x^2+1/x;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwoKiZJInRHRighIiJJInhHRigiIiJGMyokKUYyIiIjRjNGMSokRjJGMUYzNyNGLg==

rationalize(subs(x=Xsol[1], x^2*y-Fx)-subs(x=Xsol[2], x^2*y-Fx));

IiIh

Gy:=factor(expand(simplify(rationalize(subs(x=Xsol[1],x^2*y-Fx)),symbolic)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCoqLCwqJkkidEdGKCIiIilJInlHRigiIiRGMkYyKiZGMUYyKUY0IiIjRjIhIiIqKEY4RjJGMUYyRjRGMkY5KiRGN0YyRjlGMUY5RjJGNEY5RjFGOSwmRjRGMkYyRjJGOTcjRi4=

convert(Gy,parfrac,y);

LCpJInlHNiIiIiIqJkkidEdGJCEiIiwmKiYiIiNGJUYnRiVGJUYlRiVGJUYoKigsJkYlRiVGJ0YlRiVGJ0YoLCZGI0YlRiVGJUYoRiUqJEYjRihGKA==

Point (1,3) model 5 liste:=x+y+1/x+1/x/y+x*y;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwsSSJ4R0YoIiIiSSJ5R0YoRjAqJEYvISIiRjAqJkYxRjNGL0YzRjAqJkYvRjBGMUYwRjA3I0Yu

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLC4qKEkidEdGKCIiIilJInhHRigiIiNGMSlJInlHRihGNEYxRjEqKEYwRjFGMkYxRjZGMUYxKihGMEYxRjNGMUY1RjFGMSomRjBGMUY2RjFGMSomRjNGMUY2RjEhIiJGMEYxNyNGLg==

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCosIyIiIiIiI0YyLCgqJkkidEdGKEYyKUkieUdGKEYzRjIhIiJGOEYyKiQpKiZGOEYyLC4qJilGNkYzRjIpRjgiIiRGMkYyKigiIiVGMkY/RjJGN0YyRjkqKCIiKUYyRj9GMkY4RjJGOSooRjNGMkY2RjJGN0YyRjkqJkZDRjJGP0YyRjlGOEYyRjJGMUYyRjJGMkY4RjlGNkY5LCZGOEYyRjJGMkY5RjIsJCosRjFGMiwoRjVGMkY6RjJGOEY5RjJGOEY5RjZGOUZIRjlGOTcjRi4=

Fx:=-x^4+2*x^3/t-x^2/t^2*(2*t^2+2*t+1)+2*x*(1+t)^2/t^2+2*(1+t)*(1+2*t)/x/t^2-(5*t^2+4*t+1)/t^2/x^2+2*(1+2*t)/t/x^3-1/x^4;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNGeEdGKCwyKiQpSSJ4R0YoIiIlIiIiISIiKigiIiNGMylGMSIiJEYzSSJ0R0YoRjRGMyooKUYxRjZGMylGOUY2RjQsKComRjZGM0Y8RjNGMyomRjZGM0Y5RjNGM0YzRjNGM0Y0KipGNkYzRjFGMyksJkYzRjNGOUYzRjZGM0Y8RjRGMyosRjZGM0ZCRjMsJkY/RjNGM0YzRjNGMUY0RjxGNEYzKigsKComIiImRjNGPEYzRjMqJkYyRjNGOUYzRjNGM0YzRjNGPEY0RjtGNEY0KipGNkYzRkRGM0Y5RjRGN0Y0RjMqJEYwRjRGNDcjRi4=

rationalize(subs(x=Xsol[1], x^2*y^4-Fx)-subs(x=Xsol[2], x^2*y^4-Fx));

IiIh

Gy:=factor(expand(simplify(rationalize(subs(x=Xsol[1],x^2*y^4-Fx)),symbolic)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHeUdGKCoqLFoqJilJInRHRigiIiUiIiIpSSJ5R0YoIiM1RjRGNCooIiIpRjRGMUY0KUY2RjlGNCEiIiooIiIjRjQpRjIiIiRGNClGNiIiKkY0RjsqKCIjP0Y0RjFGNClGNiIiKEY0RjsqKEYzRjRGPkY0RjpGNEY7KigiI0RGNEYxRjQpRjYiIidGNEY7KihGM0Y0Rj5GNEZERjRGOyomKUYyRj1GNEY6RjRGNCooIiNFRjRGMUY0KUY2IiImRjRGOyooRj1GNEZNRjRGREY0RjQqKCIjQ0Y0RjFGNClGNkYzRjRGOyooIiM3RjRGPkY0RlBGNEY7KihGM0Y0Rk1GNEZJRjRGNCooIiM5RjRGMUY0KUY2Rj9GNEY7KihGT0Y0Rj5GNEZVRjRGOyooRjlGNEZNRjRGUEY0RjsqKEY9RjRGMUY0KUY2Rj1GNEY7KigiI0FGNEY+RjRGZW5GNEY7KigiIzhGNEZNRjRGVUY0RjsqKEYzRjRGMkY0RlBGNEY7KihGPUY0RjFGNEY2RjRGNCooRjlGNEY+RjRGaW5GNEY7KihGOUY0Rk1GNEZlbkY0RjsqKEY9RjRGMkY0RlVGNEY7KiRGMUY0RjQqKEY9RjRGPkY0RjZGNEY7KihGPUY0RjJGNEZlbkY0RjsqJEZVRjRGNEY0KSwmRjZGNEY0RjRGM0Y7RjFGO0ZpbkY7NyNGLg==

convert(Gy,parfrac,y);

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

liste:=subs(x=1/x,y=1/y,1/x+lambda/y+x*y+2*x+1/x/y+x/y);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZsaXN0ZUdGKCwuSSJ4R0YoIiIiKiZJJ2xhbWJkYUdGKEYwSSJ5R0YoRjBGMComRjMhIiJGL0Y1RjAqJiIiI0YwRi9GNUYwKiZGL0YwRjNGMEYwKiZGL0Y1RjNGMEYwNyNGLg==

K:=expand(x*y*(t*liste-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJLR0YoLDAqKkknbGFtYmRhR0YoIiIiSSJ0R0YoRjFJInhHRihGMSlJInlHRigiIiNGMUYxKihGMkYxKUYzRjZGMUY0RjFGMSooRjJGMUY4RjFGNUYxRjEqJkYyRjFGNEYxRjEqKEY2RjFGMkYxRjVGMUYxKiZGM0YxRjVGMSEiIkYyRjE3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVYc29sR0YoNyQsJCosIyIiIiIiI0YyLCgqKEknbGFtYmRhR0YoRjJJInRHRihGMilJInlHRihGM0YyISIiRjlGMiokKSomRjlGMiwwKigpRjZGM0YyKUY3RjNGMilGOSIiJEYyRjIqKCIiJUYyRkFGMkZCRjJGOioqRjNGMkY2RjJGN0YyRjhGMkY6KigiIzdGMkZBRjJGOEYyRjoqKEZIRjJGQUYyRjlGMkY6KiZGRUYyRkFGMkY6RjlGMkYyRjFGMkYyRjJGOUY6RjdGOiwmRjlGMkYyRjJGOkYyLCQqLEYxRjIsKEY1RjJGO0YyRjlGOkYyRjlGOkY3RjpGS0Y6Rjo3I0Yu

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

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVZc29sR0YoNyQsJCoqIyIiIiIiI0YyLCoqJkkidEdGKEYyKUkieEdGKEYzRjIhIiIqJkYzRjJGNkYyRjlGOEYyKiQpLCQqJkY4RjIsLComKUY2RjNGMilGOCIiJEYyRjkqKCIiJUYySSdsYW1iZGFHRihGMkZBRjJGMiooRjNGMkY2RjJGN0YyRjIqJkZFRjJGNkYyRjJGOEY5RjJGOUYxRjJGMkYyRjZGOSwoKiZGRkYyRjhGMkYyKiRGN0YyRjJGMkYyRjlGMiwkKipGMUYyLCpGNUYyRjtGMkY6RjJGOEY5RjJGNkY5RklGOUY5NyNGLg==

qy:=coeff(K,x,0)/coeff(K,x,2); qx:=coeff(K,y,0)/coeff(K,y,2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeUdGKComLCgqJkkidEdGKCIiIilJInlHRigiIiNGMkYyKihGNUYyRjFGMkY0RjJGMkYxRjJGMiwmRjBGMiomRjFGMkY0RjJGMiEiIjcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxeEdGKComSSJ0R0YoIiIiLCgqKEknbGFtYmRhR0YoRjBGL0YwSSJ4R0YoRjBGMComRi9GMClGNCIiI0YwRjBGL0YwISIiNyNGLg==

phi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([subs(y=v,qy)/u,v]): end: psi:=proc(p) local u,v: u:=p[1]: v:=p[2]: factor([u,subs(x=u,qx)/v]): end: [x,y ]; phi(%); psi(%);phi(%); psi(%);phi(%); psi(%);

NyRJInhHNiJJInlHRiQ=

NyQqKCwmSSJ5RzYiIiIiRidGJ0YnSSJ4R0YmISIiRiVGKUYl

NyQqKCwmSSJ5RzYiIiIiRidGJ0YnSSJ4R0YmISIiRiVGKSooKUYoIiIjRidGJUYnLC4qKEknbGFtYmRhR0YmRidGKEYnKUYlRixGJ0YnKiZGK0YnRjBGJ0YnKihGL0YnRihGJ0YlRidGJyokRjBGJ0YnKiZGLEYnRiVGJ0YnRidGJ0Yp

NyQqJkkieEc2IiEiIiwqKihJJ2xhbWJkYUdGJSIiIkYkRipJInlHRiVGKkYqKiYpRiQiIiNGKkYrRipGKkYrRipGKkYqRioqKEYtRipGK0YqLC4qKEYpRipGJEYqKUYrRi5GKkYqKiZGLUYqRjJGKkYqRihGKiokRjJGKkYqKiZGLkYqRitGKkYqRipGKkYm

NyQqJkkieEc2IiEiIiwqKihJJ2xhbWJkYUdGJSIiIkYkRipJInlHRiVGKkYqKiYpRiQiIiNGKkYrRipGKkYrRipGKkYqRioqJkYrRiYsKComRilGKkYkRipGKiokRi1GKkYqRipGKkYm

NyRJInhHNiIqJkkieUdGJCEiIiwoKiZJJ2xhbWJkYUdGJCIiIkYjRitGKyokKUYjIiIjRitGK0YrRitGJw==

NyRJInhHNiJJInlHRiQ=

In Section 7.3 r[0]:=[x,y ]; r[1]:=phi(%); r[2]:=psi(%);r[3]:=phi(%); r[4]:=psi(%);r[5]:=phi(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIhNyRJInhHRihJInlHRig3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIiNyQqKCwmSSJ5R0YoRjBGMEYwRjBJInhHRighIiJGNEY2RjQ3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIjNyQqKCwmSSJ5R0YoIiIiRjVGNUY1SSJ4R0YoISIiRjRGNyooKUY2RjBGNUY0RjUsLiooSSdsYW1iZGFHRihGNUY2RjUpRjRGMEY1RjUqJkY5RjVGPUY1RjUqKEY8RjVGNkY1RjRGNUY1KiRGPUY1RjUqJkYwRjVGNEY1RjVGNUY1Rjc3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIkNyQqJkkieEdGKCEiIiwqKihJJ2xhbWJkYUdGKCIiIkYzRjhJInlHRihGOEY4KiYpRjMiIiNGOEY5RjhGOEY5RjhGOEY4RjgqKEY7RjhGOUY4LC4qKEY3RjhGM0Y4KUY5RjxGOEY4KiZGO0Y4RkBGOEY4RjZGOCokRkBGOEY4KiZGPEY4RjlGOEY4RjhGOEY0NyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiIlNyQqJkkieEdGKCEiIiwqKihJJ2xhbWJkYUdGKCIiIkYzRjhJInlHRihGOEY4KiYpRjMiIiNGOEY5RjhGOEY5RjhGOEY4RjgqJkY5RjQsKComRjdGOEYzRjhGOCokRjtGOEY4RjhGOEY0NyNGMQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkickdGKDYjIiImNyRJInhHRigqJkkieUdGKCEiIiwoKiZJJ2xhbWJkYUdGKCIiIkYyRjlGOSokKUYyIiIjRjlGOUY5RjlGNTcjRjE=

rseq:=[seq(r[i],i=0..5)];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVyc2VxR0YoNyg3JEkieEdGKEkieUdGKDckKigsJkYxIiIiRjVGNUY1RjAhIiJGMUY2RjE3JEYzKigpRjAiIiNGNUYxRjUsLiooSSdsYW1iZGFHRihGNUYwRjUpRjFGOkY1RjUqJkY5RjVGPkY1RjUqKEY9RjVGMEY1RjFGNUY1KiRGPkY1RjUqJkY6RjVGMUY1RjVGNUY1RjY3JComRjBGNiwqRkBGNSomRjlGNUYxRjVGNUYxRjVGNUY1RjVGODckRkQqJkYxRjYsKComRj1GNUYwRjVGNSokRjlGNUY1RjVGNUY2NyRGMEZINyNGLg==

a:=1: b:=0: asy:=proc(p) local u, v: u:=p[1]: v:=p[2]: expand(simplify(((u^(a+1)*v^(b+1)) ))): end: map(asy,rseq);

NygqJilJInhHNiIiIiMiIiJJInlHRiZGKCwoKiZGJCEiIkYpRihGKComRidGKEYkRixGKComRiRGLEYpRixGKCwoKiZGKUYoLC4qKEknbGFtYmRhR0YmRihGJUYoKUYpRidGKEYoKiZGJEYoRjRGKEYoKihGM0YoRiVGKEYpRihGKCokRjRGKEYoKiZGJ0YoRilGKEYoRihGKEYsRigqJkYnRihGMUYsRigqJkYpRixGMUYsRigsNioqKUYpIiIkRihGMUYsKUYzRidGKEYkRihGKCosRidGKEY9RihGMUYsRjNGKClGJUY+RihGKCooRj1GKEYxRiwpRiUiIiVGKEYoKixGJ0YoRj1GKEYxRixGM0YoRiVGKEYoKipGJ0YoRj1GKEYxRixGJEYoRigqLEYnRihGNEYoRjFGLEYzRihGJUYoRigqKkYnRihGNEYoRjFGLEYkRihGKComRj1GKEYxRixGKCooRidGKEY0RihGMUYsRihGMEYoLDYqKEYpRigsKComRjNGKEYlRihGKCokRiRGKEYoRihGKEYsRj9GKEYoKixGJ0YoRiVGKEYpRihGTUYsRjNGKEYoKihGJEYoRilGKEZNRixGKCosRidGKEYlRixGKUYoRk1GLEYzRihGKCooRidGKEYpRihGTUYsRigqKkYnRihGJUYsRk1GLEYzRihGKComRidGKEZNRixGKCooRiRGLEYpRihGTUYsRigqKEYnRihGJEYsRk1GLEYoKihGJEYsRilGLEZNRixGKCooRiRGKEYpRixGTUYs

factor(%[1]-%[2]+%[3]-%[4]+%[5]-%[6]);

LCQqMkknbGFtYmRhRzYiIiIiLCgqJilJInhHRiUiIiNGJkkieUdGJUYmRiZGLCEiIkYmRi1GJiwqKihGJEYmRipGJilGLEYrRiZGJiomRilGJkYwRiZGJiokRjBGJkYmRiZGLUYmLCwqKEYkRiZGKkYmRixGJkYmRihGJiokRilGJkYtRixGJkYmRiZGJkYqRi0sKComRiRGJkYqRiZGJkY1RiZGJkYmRi0sLkYvRiZGMUYmRjRGJkYyRiYqJkYrRiZGLEYmRiZGJkYmRi1GLEYtRi0=

This vanishes when lambda=0

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=