<Text-field style="Heading 2" layout="Heading 2"><Font size="18">Solving the equation in one catalytic variable [Section 6.1]</Font></Text-field>

For this section to work, we need to have at hand a number of equations and series expansions. So we excute the session until the first section.

The series C and Q

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

eqC:=-K*C(x,y)+1-t/y*(x+1/x+1)*Cm(1/x)-t/x*(y+1/y+1)*Cm(1/y)-t/x/y*C00+t*(1/x+1/y)*Cm10;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcUNHRigsLiomLCYiIiJGMSomSSJ0R0YoRjEsLEkieEdGKEYxKiRGNSEiIkYxSSJ5R0YoRjEqJEY4RjdGMSomLCZGNUYxRjZGMUYxLCZGOEYxRjlGMUYxRjFGMUY3RjEtSSJDR0YoNiRGNUY4RjFGN0YxRjEqKkYzRjFGOEY3LChGNUYxRjZGMUYxRjFGMS1JI0NtR0YoNiNGNkYxRjcqKkYzRjFGNUY3LChGOEYxRjlGMUYxRjFGMS1GQzYjRjlGMUY3KipGM0YxRjVGN0Y4RjdJJEMwMEdGKEYxRjcqKEYzRjEsJkY2RjFGOUYxRjFJJUNtMTBHRihGMUYxNyNGLg==

Cmex:=proc(n) local res,i: res:=coeff(convert(Cex(n),polynom),y,0): add(coeff(res,x,-i)/x^i,i=1..n): end:

C00ex:=proc(n) local res: res:=coeff(convert(Cex(n),polynom),y,0): coeff(coeff(res,x,0),y,0): end:

Cex:=proc(n) option remember: if n=0 then 1 else normal(series(subs(C00=C00ex(n-1),C(x,y)=Cex(n-1),Cm(1/x)=Cmex(n-1),Cm(1/y)=subs(x=y,Cmex(n-1)),Cm10=coeff(Cmex(n-1),x,-1),eqC+C(x,y)),t,n+1)): fi: end:

map(expand,Cex(2));

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eqQ:=-K*Q(x,y)+1-t/y*(x+1/x+1)*Qp(x)-t/x*(y+1/y+1)*Qp(y)+t/x/y*Q00;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcVFHRigsLComLCYiIiJGMSomSSJ0R0YoRjEsLEkieEdGKEYxKiRGNSEiIkYxSSJ5R0YoRjEqJEY4RjdGMSomLCZGNUYxRjZGMUYxLCZGOEYxRjlGMUYxRjFGMUY3RjEtSSJRR0YoNiRGNUY4RjFGN0YxRjEqKkYzRjFGOEY3LChGNUYxRjZGMUYxRjFGMS1JI1FwR0YoNiNGNUYxRjcqKkYzRjFGNUY3LChGOEYxRjlGMUYxRjFGMS1GQzYjRjhGMUY3KipGM0YxRjVGN0Y4RjdJJFEwMEdGKEYxRjE3I0Yu

Qpex:=proc(n) subs(y=0,Qex(n)): end:

Q00ex:=proc(n) subs(x=0,y=0,Qex(n)): end:

Qex:=proc(n) option remember: if n=0 then 1 else normal(series(subs(Q00=Q00ex(n-1),Q(x,y)=Qex(n-1),Qp(x)=Qpex(n-1),Qp(y)=subs(x=y,Qpex(n-1)),eqQ+Q(x,y)),t,n+1)): fi: end:

Qex(4);

Ky1JInRHNiIiIiIiIiEsKComSSJ4R0YkRiVJInlHRiRGJUYlRilGJUYqRiVGJSw0KiYpRikiIiNGJSlGKkYuRiVGJSooRi5GJUYtRiVGKkYlRiUqKEYuRiVGKUYlRi9GJUYlKiZGLkYlRi1GJUYlKihGLkYlRilGJUYqRiVGJSomRi5GJUYvRiVGJSomRi5GJUYpRiVGJSomRi5GJUYqRiVGJSIiJEYlRi4sQiomKUYpRjdGJSlGKkY3RiVGJSooRjdGJUY6RiVGL0YlRiUqKEY3RiVGLUYlRjtGJUYlKigiIiZGJUY6RiVGKkYlRiUqKCIiJ0YlRi1GJUYvRiVGJSooRj9GJUYpRiVGO0YlRiUqJiIiJUYlRjpGJUYlKigiIipGJUYtRiVGKkYlRiUqKEZGRiVGKUYlRi9GJUYlKiZGREYlRjtGJUYlKiZGQUYlRi1GJUYlKigiIztGJUYpRiVGKkYlRiUqJkZBRiVGL0YlRiUqJiIjNkYlRilGJUYlKiZGTkYlRipGJUYlRkFGJUY3LFQqJilGKUZERiUpRipGREYlRiUqKEZERiVGUkYlRjtGJUYlKihGREYlRjpGJUZTRiVGJSooRkZGJUZSRiVGL0YlRiUqKCIjN0YlRjpGJUY7RiVGJSooRkZGJUYtRiVGU0YlRiUqKEZYRiVGUkYlRipGJUYlKigiI0NGJUY6RiVGL0YlRiUqKEZmbkYlRi1GJUY7RiVGJSooRlhGJUYpRiVGU0YlRiUqJkZGRiVGUkYlRiUqKCIjR0YlRjpGJUYqRiVGJSooIiNeRiVGLUYlRi9GJUYlKihGW29GJUYpRiVGO0YlRiUqJkZGRiVGU0YlRiUqJiIjP0YlRjpGJUYlKigiI2dGJUYtRiVGKkYlRiUqKEZjb0YlRilGJUYvRiVGJSomRmFvRiVGO0YlRiUqJiIjWEYlRi1GJUYlKigiI2tGJUYpRiVGKkYlRiUqJkZnb0YlRi9GJUYlKiYiI1tGJUYpRiVGJSomRlxwRiVGKkYlRiUiI1FGJUZE

The series A, P, M

Aex:=proc(n) expand(series(Cex(n)-1/3*(Qex(n)-1/x^2*subs(x=1/x,Qex(n))-1/y^2*subs(y=1/y,Qex(n))),t,n+1)): end:

Mex:=proc(n) local i: collect(add(1/x*coeff(convert(Aex(n),polynom),x,-i)*x^i,i=1..n+2),t,normal): end:

Mex(2);

LCgqJiw6KiYiIiMiIiIpSSJ5RzYiRiZGJ0YnKiZGJkYnRilGJ0YnIiIkRicqKCMiIiZGLEYnSSJ4R0YqRidGKEYnRicqKCMiIilGLEYnRjBGJ0YpRidGJyomRixGJ0YwRidGJyomI0YmRixGJylGMEYmRidGJyooRjZGJ0Y3RidGKUYnRicqKEY2RidGN0YnRihGJ0YnKiZGNkYnKUYwRixGJ0YnKigjRidGLEYnRjtGJ0YoRidGJyooRjZGJ0Y7RidGKUYnRidGJylJInRHRipGJkYnRicqJiwsRilGJ0YnRicqKEY9RidGMEYnRilGJ0YnKihGPUYnRjdGJ0YpRidGJyomRj1GJ0Y3RidGJ0YnRkBGJ0YnKiZGPUYnRjBGJ0Yn

Pex:=proc(n) local res,i,j: res:=convert(Aex(n),polynom): collect(add(add(coeff(coeff(res,x,i),y,j)*x^i*y^j,i=0..n),j=0..n),t): end:

Pex(3);

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

The discriminant

Delta:=(1-t*(x+1/x))^2-4*t^2*(1+x+1/x)^2;

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The series R(x) and S(x)

Rex:=proc(n) t^2*subs(y=0,Mex(n)): end:

series(Rex(5),t,5);

KytJInRHNiIsJComIyIiIiIiJEYoSSJ4R0YkRihGKCIiIywmRihGKComRidGKClGKkYrRihGKEYpLCpGKUYoKiZGKUYoRipGKEYoKiYjRitGKUYoRi5GKEYoKiZGMkYoKUYqRilGKEYoIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIiY=

Sex:=proc(n) t*x*subs(x=0,y=x,Mex(n)): end:

series(Sex(5),t,5);

KytJInRHNiIqJkkieEdGJCIiIiwmRiZGJ0YnRidGJyIiIyomRiZGJywoKiZGKUYnKUYmRilGJ0YnKiZGKUYnRiZGJ0YnIiIkRidGJ0YvKiZGJkYnLCoqJiIiJ0YnKUYmRi9GJ0YnKiYiIzdGJ0YtRidGJyomIiNDRidGJkYnRiciIz9GJ0YnIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiciIiY=

The equation in one catalytic variable satisfied by S(x)

eqScat:=-(t*x^2+2*t*x+t+x)*(3*t*x^2+2*t*x+3*t-x)*S(x)^3/x^2-(t*x^2+2*t*x+t+x)*(3*t*x^2+2*t*x+3*t-x)*(2*x^2+1)*S(x)^2/(x^2*(x^2+x+1))+(-B1+B2-9*R1^2-(3*(t*x^4+3*R0*x^3-t*x^3+3*R0*x^2+t*x^2-x^3+3*R0*x-t*x+t-x))*R1/(x*(x^2+x+1))+(-4*t^2*x^8-10*t^2*x^7+2*R0*t*x^6-17*t^2*x^6-2*t*x^7+4*R0*t*x^5+2*R0*x^6-20*t^2*x^5+6*R0*t*x^4+4*R0*x^5-18*t^2*x^4-4*t*x^5+x^6+4*R0*t*x^3+6*R0*x^4-12*t^2*x^3+2*R0*t*x^2+4*R0*x^3-7*t^2*x^2-2*t*x^3+x^4+2*R0*x^2-2*t^2*x-t^2)/((x^2+x+1)^2*x^2))*S(x)-(x^2+2)*B1/(3*(x^2+x+1))+x^2*B2/(x^2+x+1)-9*x^2*R1^2/(x^2+x+1)-(3*(t*x^5+4*R0*x^4-t*x^4+4*R0*x^3+t*x^3-x^4+3*R0*x^2-t*x^2-R0*x+t*x-x^2-R0))*R1/(x^2+x+1)^2-(3*t^2*x^7+3*R0*t*x^6+3*t^2*x^6+9*R0^2*x^5-7*R0*t*x^5+6*t^2*x^5+9*R0^2*x^4-4*R0*t*x^4-7*R0*x^5+3*t^2*x^4-6*R0*t*x^3-4*R0*x^4+6*t^2*x^3-9*R0^2*x^2-2*R0*t*x^2-6*R0*x^3+3*t^2*x^2-9*R0^2*x+R0*t*x-2*R0*x^2+3*t^2*x-3*R0*t+R0*x)/(3*(x^2+x+1)^2*x)

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In compact form: equation for S-tilde

eqStcatz:=(27*(2*t+z+1))*(10*t-3*z+1)*St^3+((54-243*R1+54*t)*R0-243*R1^2+243*R1*t-81*z*R1-45*t^2+18*z*t-27*B1+27*B2+81*R1-54*t+18*z-9)*St-81*R0^2+(81*t+27-27*z-162*R1)*R0-81*R1^2+81*R1*t-27*z*R1+5*t^2+10*z*t-3*z^2+9*B1+9*B2+27*R1-6*t+4*z-2;

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

zx:=t*(x+1+1/x)+(t+1)/(x+1+1/x); StS:=1/(x-1/x)*((x+1/x+1)*S(x)+2*x/3+1/x/3)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN6eEdGKCwmKiZJInRHRigiIiIsKEkieEdGKEYxKiRGMyEiIkYxRjFGMUYxRjEqJiwmRjBGMUYxRjFGMUYyRjVGMTcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRTdFNHRigqJiwmSSJ4R0YoIiIiKiRGMCEiIkYzRjMsKComLChGMEYxRjJGMUYxRjFGMS1JIlNHRig2I0YwRjFGMSomIyIiIyIiJEYxRjBGMUYxKiYjRjFGPUYxRjBGM0YxRjE3I0Yu

Let us check

jj:=exp(2*I*Pi/3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNqakdGKCwmIyIiIiIiIyEiIiomKiZGL0YwXiNGMEYwRjApIiIkRi9GMEYwNyNGLg==

B1ex:=proc(n) map(expand,map(rationalize,1/2*convert(normal(series((1+t)^2*jj*subs(x=jj,diff(Sex(n),x))+(1+t)^2/jj*subs(x=1/jj,diff(Sex(n),x)),t,n+1)),polynom))): end:

B1ex(4);

LCgqJiMiIiQiIiMiIiIpSSJ0RzYiRiZGJyEiIiomI0YnRiZGJylGKUYlRidGKyomIyIjOEYmRicpRikiIiVGJ0Yr

B2ex:=proc(n) collect(1/2/I/sqrt(3)*map(expand,map(rationalize,convert(normal(series((1+t)^2*jj*subs(x=jj,diff(Sex(n),x))-(1+t)^2/jj*subs(x=1/jj,diff(Sex(n),x)),t,n+1)),polynom))),t): end:

B2ex(4);

LCgqJiMiIigiIiMiIiIpSSJ0RzYiIiIlRichIiIqJiMiIiRGJkYnKUYpRi9GJ0YsKiYjRidGJkYnKUYpRiZGJ0Ys

R0ex:=proc(n) coeff(Rex(n),x,0): end:

R1ex:=proc(n) coeff(Rex(n),x,1): end:

n:=10: normal(series(subs(R0=R0ex(n),R1=R1ex(n),B1=B1ex(n),B2=B2ex(n),St=StS,S(x)=Sex(n), z=zx,eqStcatz),t,n))

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

Let us check

n:=22: normal(series(subs(S(x)=Sex(n), R0=R0ex(n),R1=R1ex(n),B1=B1ex(n),B2=B2ex(n),eqScat),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0I=

The four univariate series involved in the equation for S(x)

jj:=exp(2*I*Pi/3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNqakdGKCwmIyIiIiIiIyEiIiomKiZGL0YwXiNGMEYwRjApIiIkRi9GMEYwNyNGLg==

B1ex:=proc(n) map(expand,map(rationalize,1/2*convert(normal(series((1+t)^2*jj*subs(x=jj,diff(Sex(n),x))+(1+t)^2/jj*subs(x=1/jj,diff(Sex(n),x)),t,n+1)),polynom))): end:

B1ex(4);

LCgqJiMiIiQiIiMiIiIpSSJ0RzYiRiZGJyEiIiomI0YnRiZGJylGKUYlRidGKyomIyIjOEYmRicpRikiIiVGJ0Yr

B2ex:=proc(n) collect(1/2/I/sqrt(3)*map(expand,map(rationalize,convert(normal(series((1+t)^2*jj*subs(x=jj,diff(Sex(n),x))-(1+t)^2/jj*subs(x=1/jj,diff(Sex(n),x)),t,n+1)),polynom))),t): end:

B2ex(4);

LCgqJiMiIigiIiMiIiIpSSJ0RzYiIiIlRichIiIqJiMiIiRGJkYnKUYpRi9GJ0YsKiYjRidGJkYnKUYpRiZGJ0Ys

R0ex:=proc(n) coeff(Rex(n),x,0): end:

R1ex:=proc(n) coeff(Rex(n),x,1): end:

Let us check

n:=22: normal(series(subs(S(x)=Sex(n), R0=R0ex(n),R1=R1ex(n),B1=B1ex(n),B2=B2ex(n),eqScat),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiI0I=

Now we can start!

<Text-field style="Heading 2" layout="Heading 2"><Font size="18">Solving the equation in one catalytic variable [Section 6.1]</Font></Text-field>

eqcat:=numer(factor(subs(S(x)=Sx,eqScat))):nops(%);

IiRVIg==

deqS:=factor(diff(eqcat,Sx)):nops(%);indets(deqS);

IiMnKg==

PClJI0IxRzYiSSNCMkdGJEkjUjBHRiRJI1IxR0YkSSNTeEdGJEkidEdGJEkieEdGJA==

<Text-field style="Heading 3" layout="Heading 3">The series Xi</Text-field>

At t=0

n:=4: factor(series(subs(Sx=Sex(n),B1=B1ex(n),B2=B2ex(n),R0=R0ex(n),R1=R1ex(n),deqS),t,n));

Ky1JInRHNiIsJCooIiIkIiIiKUkieEdGJCIiJUYoLCYqJClGKiIiI0YoRihGKEYoRihGKCIiISwkKigiIidGKClGKkYnRigpRixGL0YoISIiRigsJCoqRidGKCwmRipGKEYoRihGKCwoKiZGL0YoRi5GKEYoRipGKEYoRihGKCwqKiRGKUYoRigqJEY0RihGKEY7RihGKEYoRihGNkYvLCQqKkYnRihGKkYoLChGLUYoRipGKEYoRihGKCwsKiZGJ0YoRilGKEYoKiZGM0YoRjRGKEY2RjtGKComRi9GKEYqRihGNkYoRjZGKEYoRictSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYr

Since the first coefficient has degree 6 in x, 6 series in t (or in power of t) will be solutions

A closer look

n:=6: factor(series(subs(Sx=Sex(n),B1=B1ex(n),B2=B2ex(n),R0=R0ex(n),R1=R1ex(n),deqS),t,n));

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

with(gfun):map(allvalues,algeqtoseries(convert(%%,polynom),t,x,3,true));

NygrKUkidEc2Il4jIiIiIiIhRiciIiMtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJyIiJCspRiQsJEYmISIiRihGJ0YpRipGMCwqKiQpRiQjRidGKUYnRidGJEYnKiYjRjBGKUYnKUYkRjlGJ0YnLUYrNiMqJClGJEYpRidGJywoKiZGJkYnRjZGJ0YnKiYsJEYmRidGJ0Y6RidGM0Y7RicsKkY1RjNGJEYnRjhGM0Y7RicsKComRkJGJ0Y2RidGMyomRiZGJ0Y6RidGJ0Y7Ric=

This means that one solution has I and one -I as constant term, the others four vanish at t=0

We compute them inductively using the first few coefficients of S, R, .... We start with the series with non-zero constant terms

unassign('x1'):

X1ser := proc(N) add(x1[n]*t^n,n=0..N): end:

Warning, (in X1ser) `n` is implicitly declared local

n:=0:

factor(series(subs(Sx=Sex(n),B1=B1ex(n),B2=B2ex(n),R0=R0ex(n),R1=R1ex(n), x=X1ser(n),deqS),t,n+1));

KydJInRHNiIsJCooIiIkIiIiKSZJI3gxR0YkNiMiIiEiIiVGKCwmKiQpRioiIiNGKEYoRihGKEYoRihGLS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRig=

x1[0]:=I:

n:=n+1: factor(series(subs(Sx=Sex(n),B1=B1ex(n),B2=B2ex(n),R0=R0ex(n),R1=R1ex(n), x=X1ser(n),deqS),t,n+1));

KydJInRHNiIqJiomIiInIiIiXiNGKEYoRigmSSN4MUdGJDYjRihGKEYoLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJEYsIiIj

solve(convert(%,polynom),x1[n]):x1[n]:=%;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjeDFHRig2IyIiIiIiITcjRjE=

for i from 1 to 10 do n:=n+1: factor(series(subs(Sx=Sex(n),B1=B1ex(n),B2=B2ex(n),R0=R0ex(n),R1=R1ex(n), x=X1ser(n),deqS),t,n+1)):solve(convert(%,polynom),x1[n]): x1[n]:=%; od:

X1ser(n);

LDheIyIiIkYkKiQpSSJ0RzYiIiIjRiRGJComRilGJClGJyIiJEYkRiQqJiwmIiM+RiQqJiNGJEYpRiRGI0YkISIiRiQpRiciIiVGJEYkKiYsJiIjIilGJComRilGJEYjRiRGMkYkKUYnIiImRiRGJComLCYiJHEmRiQqJiIjQEYkRiNGJEYyRiQpRiciIidGJEYkKiYsJiIlKkckRiQqJiIkPiJGJEYjRiRGMkYkKUYnIiIoRiRGJComLCYiJiYzQUYkKiYjIiUsdCIiKUYkRiNGJEYyRiQpRidGT0YkRiQqJiwmIid1RjlGJComIiVwZkYkRiNGJEYyRiQpRiciIipGJEYkKiYsJiInVzUoKkYkKiYiJid5VUYkRiNGJEYyRiQpRiciIzVGJEYkKiYsJiIoOVdnJ0YkKiYjIic4OWZGKUYkRiNGJEYyRiQpRiciIzZGJEYk

The other solution with non-zero constant term must be obtained by replacing I by -I. We conjecture that X1*X2 = 1

series(X1ser(n)*subs(I=-I,X1ser(n)),t,n-1);

KydJInRHNiIiIiIiIiEtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjNQ==

The other roots are expected to be series in s=sqrt(t)

unassign('x3'):

X3ser := proc(N) add(x3[n]*s^n,n=0..N): end:

Warning, (in X3ser) `n` is implicitly declared local

n:=0: factor(series(subs(Sx=Sex(n),B1=B1ex(n),B2=B2ex(n),R0=R0ex(n),R1=R1ex(n), x=X3ser(n),t=s^2,deqS),s,n+1));

KydJInNHNiIsJCooIiIkIiIiKSZJI3gzR0YkNiMiIiEiIiVGKCwmKiQpRioiIiNGKEYoRihGKEYoRihGLS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRjI=

x3[0]:=0:

n:=4: factor(series(subs(Sx=Sex(n),B1=B1ex(n),B2=B2ex(n),R0=R0ex(n),R1=R1ex(n), x=X3ser(n),t=s^2,deqS),s,n+1));

KydJInNHNiIsJCoqIiIkIiIiLCYmSSN4M0dGJDYjRihGKEYoISIiRigsJkYqRihGKEYoRigsJiokKUYqIiIjRihGKEYoRihGKEYoIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJEYsIiIm

x3[1]:=1:

for i from 1 to 10 do n:=n+1: factor(series(subs(Sx=Sex(n),B1=B1ex(n),B2=B2ex(n),R0=R0ex(n),R1=R1ex(n), x=X3ser(n),t=s^2,deqS),s,n+1));isolate(convert(%,polynom),x3[n-3]): assign(%); od:

X3ser(6);

LC5JInNHNiIiIiIqJClGIyIiI0YlRiUqJiMiIiRGKEYlKUYjRitGJUYlKiZGK0YlKUYjIiIlRiVGJSomIyIjXiIiKUYlKUYjIiImRiVGJSomIiM5RiUpRiMiIidGJUYl

Another series (called X4) is obtained by replacing s by -s

Next we compute X5

unassign('x5'):

X5ser := proc(N) add(x5[n]*s^n,n=0..N): end:

Warning, (in X5ser) `n` is implicitly declared local

n:=0: factor(series(subs(Sx=Sex(n),B1=B1ex(n),B2=B2ex(n),R0=R0ex(n),R1=R1ex(n), x=X5ser(n),t=s^2,deqS),s,n+1));

KydJInNHNiIsJCooIiIkIiIiKSZJI3g1R0YkNiMiIiEiIiVGKCwmKiQpRioiIiNGKEYoRihGKEYoRihGLS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRjI=

x5[0]:=0:

n:=4: factor(series(subs(Sx=Sex(n),B1=B1ex(n),B2=B2ex(n),R0=R0ex(n),R1=R1ex(n), x=X5ser(n),t=s^2,deqS),s,n+1));

KydJInNHNiIsJCoqIiIkIiIiLCYmSSN4NUdGJDYjRihGKEYoISIiRigsJkYqRihGKEYoRigsJiokKUYqIiIjRihGKEYoRihGKEYoIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJEYsIiIm

x5[1]:=I:

for i from 1 to 10 do n:=n+1: factor(series(subs(Sx=Sex(n),B1=B1ex(n),B2=B2ex(n),R0=R0ex(n),R1=R1ex(n), x=X5ser(n),t=s^2,deqS),s,n+1));isolate(convert(%,polynom),x5[n-3]): assign(%); od:

X5ser(10);

LDQqJl4jIiIiRiVJInNHNiJGJUYlKiYsJEYkRiVGJSlGJiIiJEYlISIiKiYjRiUiIiNGJSlGJiIiJUYlRiwqJiomI0YrRi9GJUYkRiVGJSlGJiIiJkYlRiUqJClGJiIiJ0YlRiwqJiwkKiYjIiNcIiIpRiVGJEYlRiVGJSlGJiIiKEYlRiwqJkZBRiUpRiZGP0YlRiwqJiomIyIjRkYxRiVGJEYlRiUpRiYiIipGJUYlKiYiI05GJSlGJiIjNUYlRiw=

The 6th root is obtained by replacing s by -s, or equivalently i by -i. Now we have the first terms of our 6 roots X_i

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

factor(diff(eqcat,x)): nops(%);

IiRSIg==

deqx:=collect(diff(eqcat,x),[Sx,R0,R1,B1,B2],factor); indets(%);

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

PClJI0IxRzYiSSNCMkdGJEkjUjBHRiRJI1IxR0YkSSNTeEdGJEkidEdGJEkieEdGJA==

degree(eqcat,B1);degree(deqS,B1);degree(deqx,B1); degree(eqcat,B2);degree(deqS,B2);degree(deqx,B2);

IiIi

coeff(eqcat,B1);coeff(deqS,B1);coeff(deqx,B1);

LDYqKCIiJCIiIkkjU3hHNiJGJSlJInhHRiciIidGJSEiIiooRipGJUYmRiUpRikiIiZGJUYrKiRGKEYlRisqKCIiKkYlRiZGJSlGKSIiJUYlRisqJEYtRiVGKyooRipGJUYmRiUpRilGJEYlRisqJkYkRiVGMkYlRisqKEYkRiVGJkYlKUYpIiIjRiVGKyomRjpGJUY2RiVGKyomRjpGJUY5RiVGKw==

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LCYqLCIiJyIiIkkieEc2IkYlLCgqJClGJiIiI0YlRiVGJkYlRiVGJUYlLCgqJiIiJEYlRipGJUYlKiZGK0YlRiZGJUYlRiVGJUYlSSNTeEdGJ0YlISIiKiZGJkYlLCwqJkYkRiUpRiYiIiVGJUYlKiYiIiZGJSlGJkYuRiVGJSomIiM3RiVGKkYlRiUqJkYkRiVGJkYlRiVGNkYlRiVGMQ==

We take a linear combination of the 3 equations to eliminate B1 and B2

e1:=numer(factor(subs(isolate(deqS,B1),eqcat))): nops(%);

IiRaIg==

e2:=numer(factor(subs(isolate(deqS,B1),deqx))): nops(%);

IiQ+Iw==

degree(e1,B2); degree(e2,B2);

IiIi

e3:=numer(factor(subs(isolate(e1,B2),e2))): nops(%);

IiIl

seq(indets(op(i,e3)),i=1..4);

NiY8IjwjSSJ4RzYiPCRJInRHRiZGJTwnSSNSMEdGJkkjUjFHRiZJI1N4R0YmRihGJQ==

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

We want to decide which factor vanishes, for each value of x=X_i. Only 2 factors are likely to be cancelled

Careful here, the definition of e3a and e3b may change depending on the version of Maple!

e3a:=op(3,e3): e3b:=op(4,e3):

collect(e3a,t,factor);

LCYqKCwmKiQpSSJ4RzYiIiIjIiIiRipGKkYqRiopLCZGJ0YqRipGKkYpRipJInRHRihGKkYqRiUhIiI=

n:=6: normal(series(subs(x=X1ser(n),e3a),t,4)); normal(series(subs(x=X3ser(n),t=s^2,e3a),s,4)); normal(series(subs(x=X5ser(n),t=s^2,e3a),s,4));

KytJInRHNiIiIiIiIiEsJComIiIjRiVeI0YlRiUhIiJGKSwmIiIlRisqJkYtRiVGKkYlRisiIiQtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJUYt

KyVJInNHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiU=

KylJInNHNiIiIiNGJSomRiUiIiJeI0YnRiciIiQtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJyIiJQ==

n:=6: normal(series(subs(R0=R0ex(n),R1=R1ex(n),Sx=Sex(n),x=X1ser(n),e3b),t,4)); normal(series(subs(R0=R0ex(n),R1=R1ex(n),Sx=Sex(n),x=X3ser(n),t=s^2,e3b),s,n)); normal(series(subs(R0=R0ex(n),R1=R1ex(n),Sx=Sex(n),x=X5ser(n),t=s^2,e3b),s,n));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiU=

KylJInNHNiIiIiVGJSIjNyIiJi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIiJw==

KyVJInNHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIic=

Hence X3 and X4 satisfy e3a, and the other four roots satisfy e3b

In particular, we have found the minimal equation satisfied by X3 and its twin X4 (obtained by replacing sqrt(t) by -sqrt(t)). Genus zero

algX3:=e3a;collect(%,t,factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGdYM0dGKCwuKiZJInRHRigiIiIpSSJ4R0YoIiIlRjFGMSooIiIjRjFGMEYxKUYzIiIkRjFGMSooRjZGMUYwRjEpRjNGNkYxRjEqKEY2RjFGMEYxRjNGMUYxKiRGOkYxISIiRjBGMTcjRi4=

LCYqKCwmKiQpSSJ4RzYiIiIjIiIiRipGKkYqRiopLCZGJ0YqRipGKkYpRipJInRHRihGKkYqRiUhIiI=

factor(isolate(algX3,t));

L0kidEc2IiooKUkieEdGJCIiIyIiIiwmKiRGJkYpRilGKUYpISIiKSwmRidGKUYpRilGKEYs

An equation for X1 and X5

eqX15:=collect(e3b,[Sx,R0,R1],factor);

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

The discriminant

factor(coeff(eqcat,Sx,3));

LCQqKiIiJCIiIiksKCokKUkieEc2IiIiI0YlRiVGKkYlRiVGJUYsRiUsKiomSSJ0R0YrRiVGKUYlRiUqKEYsRiVGL0YlRipGJUYlRi9GJUYqRiVGJSwqKihGJEYlRi9GJUYpRiVGJUYwRiUqJkYkRiVGL0YlRiVGKiEiIkYlRjQ=

Disc0:=factor(discrim(eqcat,Sx)): nops(Disc0);

IiIm

map(indets,[seq(op(i,Disc0),i=1..nops(Disc0))]);

Nyc8IjwjSSJ4RzYiPCRJInRHRiZGJUYnPChJI0IxR0YmSSNCMkdGJkkjUjBHRiZJI1IxR0YmRihGJQ==

factor(Disc0/op(5,Disc0)/Delta);

LCQqKCIjRiIiIiksKCokKUkieEc2IiIiI0YlRiVGKkYlRiVGJUYsRiVGKUYlRiU=

Let us check that the two simple factors do not cancel at X_i

op(3,Disc0); op(4,Disc0);

LCoqJkkidEc2IiIiIilJInhHRiUiIiNGJkYmKihGKUYmRiRGJkYoRiZGJkYkRiZGKEYm

LCoqKCIiJCIiIkkidEc2IkYlKUkieEdGJyIiI0YlRiUqKEYqRiVGJkYlRilGJUYlKiZGJEYlRiZGJUYlRikhIiI=

n:=6: normal(series(subs(x=X1ser(n),op(3,Disc0)),t,4)); normal(series(subs(x=X3ser(n),t=s^2,op(3,Disc0)),s,4)); normal(series(subs(x=X5ser(n),t=s^2,op(3,Disc0)),s,4));

Ky1JInRHNiJeIyIiIiIiISomIiIjRiZGJUYmRiZGJkYpLCYiIiVGJkYoRiYiIiQtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJkYr

KytJInNHNiIiIiJGJSIiI0YmIyIiKEYmIiIkLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIiU=

KytJInNHNiJeIyIiIkYmRiYiIiNGJSIiJC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YmIiIl

n:=6: normal(series(subs(x=X1ser(n),op(4,Disc0)),t,4)); normal(series(subs(x=X3ser(n),t=s^2,op(4,Disc0)),s,4)); normal(series(subs(x=X5ser(n),t=s^2,op(4,Disc0)),s,4));

Ky1JInRHNiIsJF4jIiIiISIiIiIhKiYiIiNGJ0YmRidGJ0YoRisqJiIiJ0YnRiZGJyIiJC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YnIiIl

KytJInNHNiIhIiIiIiIiIiNGJyNGJkYnIiIkLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiYiIiU=

KytJInNHNiIsJF4jIiIiISIiRiciIiQiIiMqJkYpRidGJkYnRiktSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJyIiJQ==

Hence all X_i's are double roots of

Disc1:=op(5,Disc0):

degree(Disc1,x);

IiND

This one is symmetric in x and 1/x

factor(subs(x=1/x,Disc1)/Disc1);

KiQpSSJ4RzYiIiNDISIi

We want to write Disc1/x^12 as a polynomial in 1+x+1/x=y

res:=Disc1/x^(12):discy:=0:

for i from 0 to 12 do discy:=discy+y^(12-i)*coeff(res,x,12-i): res:= expand(res-(1+x+1/x)^(12-i)*coeff(res,x,12-i)) od:

factor(subs(y=1+x+1/x,discy)/Disc1);

KiQpSSJ4RzYiIiM3ISIi

Disc2:=discy:

degree(Disc2,y);

IiM3

Let us examine the series Y_i =1+X_i+1/X_i. Could it be that Y_1=Y_2 ?

n:=6: normal(series(subs(x=X1ser(n),1+x+1/x),t,4)); subs(I=-I,%); normal(series(subs(x=X3ser(n),1+x+1/x),s,4)); subs(s=-s,%); normal(series(subs(x=X5ser(n),1+x+1/x),s,4)); subs(I=-I,%);

KytJInRHNiIiIiIiIiEiIiNGJyIiJSIiJC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlRig=

KylJInNHNiIiIiIhIiIjRiUiIiNGJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlRig=

KyksJEkic0c2IiEiIiIiIkYmI0YnIiIjRictSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJ0Yp

KylJInNHNiIsJF4jIiIiISIiRihGJyIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YnIiIj

KylJInNHNiJeIyIiIiEiIkYmIiIhLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiYiIiM=

This is indeed real, and we suspect that X1*X2=1

n:=10: normal(series(subs(x=X1ser(n),1+x+1/x),t,n));

KzdJInRHNiIiIiIiIiEiIiNGJyIiJSIiJCIjUUYoIiRpIiIiJiIlUzYiIiciJXlsIiIoIiZxVCUiIikiJ1tiRyIiKi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiM1

collect(algX3/x^2,t,expand);

LCYiIiIhIiIqJiwsKiQpSSJ4RzYiIiIjRiNGIyomRitGI0YpRiNGI0YrRiMqJkYrRiNGKUYkRiMqJEYoRiRGI0YjSSJ0R0YqRiNGIw==

expand(%-t*(x+1/x+1)^2);

LCZJInRHNiIhIiIiIiJGJQ==

We know the series Y3 and Y4

n:=10: factor(series((X3ser(n)+1/X3ser(n)+1)^2,s,n))

KylJInNHNiIiIiIhIiNGJSIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIo

Expansion of the other series

n:=6: factor(series(X1ser(n)+1/X1ser(n)+1,t,n))

Ky9JInRHNiIiIiIiIiEiIiNGJyIiJSIiJCIjUUYoIiRpIiIiJi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIn

n:=8: factor(series(X5ser(n)+1/X5ser(n)+1,s,n))

KzFJInNHNiIsJF4jIiIiISIiRihGJyIiISNGKCIiI0YrLCQqJiNGJ0YrRidGJkYnRigiIiQjISImRisiIiUsJComIyIjPiIiKUYnRiZGJ0YoIiImLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiciIic=

Now D2(z) has 5 double roots (at least). It is also almost symmetric in y

seq(factor(coeff(Disc2,y,12-i)/coeff(Disc2,y,i)),i=0..5);

NigqJilJInRHNiIiIiciIiIpLCZGJUYoRihGKEYnISIiKiYpRiUiIiZGKClGKkYuRisqJilGJSIiJUYoKUYqRjJGKyomKUYlIiIkRigpRipGNkYrKiYpRiUiIiNGKClGKkY6RisqJkYlRihGKkYr

Let us write D2 as a polynomial in ty+(t+1)/y

res:=Disc2/y^(6):discz:=0:

for i from 0 to 6 do discz:=discz+(z/t)^(6-i)*coeff(res,y,6-i): res:= expand(res-(y+(t+1)/t/y)^(6-i)*coeff(res,y,6-i)) od:

factor(subs(z=t*y+(t+1)/y,discz)/Disc2);

KiQpSSJ5RzYiIiInISIi

Disc3:=discz: degree(%,z);

IiIn

What are the series Z_i, for i=2... 5 ?

n:=4: series(subs(y=X1ser(n)+1/X1ser(n)+1, t*y+(t+1)/y),t,n+1);

Ky9JInRHNiIiIiIiIiEiIiNGJSEiI0YnISIlIiIkISNNIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIiY=

We have determined Z3=- 2sqrt(t(1+t))

n:=12: series(subs(y=X3ser(n)+1/X3ser(n)+1,t=s^2, (t*y+(t+1)/y)^2),s,n+1);

KylJInNHNiIiIiUiIiNGJUYlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiM5

This is a series in t, not just in s, which suggests that Z5=Z6, ie Y5*Y6=1+1/t

n:=12: series(subs(y=X5ser(n)+1/X5ser(n)+1,t=s^2, t*y+(t+1)/y),s,n+1);

KzFJInNHNiIiIiNGJSEiIiIiJSEiJiIiJyEjPyIiKSEkPiIiIzUsJiIjcUYmKiYsJComRiUiIiJeI0YzRjNGM0YzLCYqJkY0RjMmSSN4NUdGJDYjIiM3RjNGMyomIyIkcCJGJUYzRjRGM0YmRjNGJkY6LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRjMiIzg=

Let us check : seems OK

series((X5ser(n)+1/X5ser(n)+1)*subs(I=-I,X5ser(n)+1/X5ser(n)+1),s,n);

KylJInNHNiIiIiIhIiNGJSIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIq

We know two roots of D3, and that there are in addition two double roots

D3form:=add(d[i]*z^i,i=0..6);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdEM2Zvcm1HRigsMComKUkiekdGKCIiJyIiIiZJImRHRig2I0YyRjNGMyomKUYxIiImRjMmRjU2I0Y5RjNGMyomKUYxIiIlRjMmRjU2I0Y+RjNGMyomKUYxIiIkRjMmRjU2I0ZDRjNGMyomKUYxIiIjRjMmRjU2I0ZIRjNGMyomRjFGMyZGNTYjRjNGM0YzJkY1NiMiIiFGMzcjRi4=

collect(D3form-(z^2-4*t*(1+t))*(alpha*z^2+beta*z+g)^2,z,factor);

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

sol:=factor(solve({seq(coeff(%,z,i),i=4..6)},{alpha,beta,g}));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzb2xHRig8JS9JJmFscGhhR0YoLUknUm9vdE9mR0YlNiMsJiokKUkjX1pHRiUiIiMiIiJGOSZJImRHRig2IyIiJyEiIi9JJWJldGFHRigsJCoqI0Y5RjhGOSZGOzYjIiImRjlGMUY5RjpGPkY5L0kiZ0dGKCwkKiojRjkiIilGOSwqKigiIztGOSlJInRHRihGOEY5KUY6RjhGOUY5KihGT0Y5RlFGOUZSRjlGOSooIiIlRjkmRjs2I0ZVRjlGOkY5RjkqJClGREY4RjlGPkY5RlJGPkYxRjlGOTcjRi4=

D3formfinal:=subs(RootOf(_Z^2-d[6])=sqrt(d[6]),factor(expand(subs(sol,(z^2-4*t*(1+t))*(alpha*z^2+beta*z+g)^2))));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSxEM2Zvcm1maW5hbEdGKCwkKiojIiIiIiNrRjEpJkkiZEdGKDYjIiInIiIkISIiLCgqJiIiJUYxKUkidEdGKCIiI0YxRjEqJClJInpHRihGP0YxRjkqJkY8RjFGPkYxRjFGMSksLiooIiM7RjFGPUYxKUY0Rj9GMUYxKigiIilGMUZBRjFGSEYxRjEqKEZHRjFGPkYxRkhGMUYxKipGPEYxRkJGMSZGNTYjIiImRjFGNEYxRjEqKEY8RjEmRjU2I0Y8RjFGNEYxRjEqJClGTUY/RjFGOUY/RjFGOTcjRi4=

factor(%/(z^2-4*t*(1+t)));

LCQqKCMiIiIiI2tGJSksLiooIiM7RiUpSSJ0RzYiIiIjRiUpJkkiZEdGLTYjIiInRi5GJUYlKigiIilGJSlJInpHRi1GLkYlRi9GJUYlKihGKkYlRixGJUYvRiVGJSoqIiIlRiVGN0YlJkYxNiMiIiZGJUYwRiVGJSooRjpGJSZGMTYjRjpGJUYwRiVGJSokKUY7Ri5GJSEiIkYuRiUpRjAiIiRGQ0Yl

coeffs(collect(D3form-D3formfinal,z,factor),z);

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

syst:=factor(subs(seq(d[i]=coeff(Disc3,z,i),i=0..6), {%})):

nops(syst);

IiIl

No need to remove any factors this time

map(nops,syst)

PCYiIz4iI1giIyUqIiRsIg==

systfinal:=[op(syst)]:

map(nops,systfinal);map(indets,systfinal); map(degree,systfinal,R0);map(degree,systfinal,R1);map(degree,systfinal,B1);map(degree,systfinal,B2);

NyYiIz4iI1giIyUqIiRsIg==

NyY8J0kjQjFHNiJJI0IyR0YlSSNSMEdGJUkjUjFHRiVJInRHRiVGI0YjRiM=

NyYiIiMiIiQiIiVGJQ==

NyYiIiQiIiUiIiYiIic=

NyYiIiJGIyIiIyIiJA==

The smallest equation

systfinal[1];

LEgqKCIkMyIiIiIpSSNSMEc2IiIiI0YlSSJ0R0YoRiUhIiIqKiIkOyNGJUYnRiVJI1IxR0YoRiVGKkYlRisqKCIjQ0YlRidGJSlGKkYpRiVGJSomIiRDJEYlKUYuIiIkRiVGJSooRjNGJSlGLkYpRiVGKkYlRisqKCIjc0YlRi5GJUYxRiVGJSomIiM3RiUpRipGNUYlRiUqKEY7RiVJI0IxR0YoRiVGKkYlRiUqKEY7RiVJI0IyR0YoRiVGKkYlRiUqJiIjYUYlRiZGJUYlKihGJEYlRidGJUYuRiVGJSooIiInRiVGJ0YlRipGJUYrKiYiJGkiRiVGN0YlRisqKCIjISpGJUYuRiVGKkYlRiUqJiIjPUYlRjFGJUYrKiZGRUYlRj5GJUYrKiZGRUYlRkBGJUYrKiYiI0lGJUYnRiVGKyomRktGJUYuRiVGJQ==

Checking the system

n:=10: for i from 1 to 4 do normal(series(subs(R0=R0ex(n),R1=R1ex(n),B1=B1ex(n),B2=B2ex(n),op(i,systfinal)),t,n+1)):od;

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

save(systfinal,systfinalR0R1B1B2_with_diag):

<Text-field style="Heading 2" layout="Heading 2">Now we can obtain individual equations for each series !</Text-field>

map(degree,[op(systfinal)],B1);map(degree,[op(systfinal)],B2);

NyYiIiJGIyIiIyIiJA==

map(degree,[op(systfinal)],R0);map(degree,[op(systfinal)],R1);

NyYiIiMiIiQiIiVGJQ==

NyYiIiQiIiUiIiYiIic=

Let us eliminate B1 and B2K

factor(coeff(systfinal[1],B1,1)); factor(coeff(systfinal[1],B2,1))

LCYqJiIjNyIiIkkidEc2IkYlRiUiIichIiI=

syst2:=map(numer,factor(subs(isolate(systfinal[1],B2),[systfinal[2], systfinal[3], systfinal[4]]))):

map(nops,%);

NyUiI1wiJE8iIiQrJA==

map(degree,syst2,B1);

NyUiIiIiIiMiIiQ=

res1:=factor(resultant(syst2[1],syst2[2],B1)): nops(res1);

IiIl

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

NyY8IjwjSSJ0RzYiPCRJI1IxR0YmRiU8JUkjUjBHRiZGKEYl

NyYiIiIiIiMiIiQiJGIi

The factor(s) that vanishes

n:=8: series(subs(R0=R0ex(n),R1=R1ex(n),op(3,res1)),t,n+1); series(subs(R0=R0ex(n),R1=R1ex(n),op(4,res1)),t,n+1);

KzVJInRHNiIhIiMiIiFGJSIiIiIiJCIiIyIjRiIiJSIjIikiIiYiJCJvIiInIiVSTyIiKCImQmAjIiIpLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiciIio=

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIio=

A first equation between R0 and R1

res2:=op(4,res1):

res3:=factor(resultant(syst2[1],syst2[3],B1)): nops(res3);

IiIk

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

NyU8IjwjSSJ0RzYiPCVJI1IwR0YmSSNSMUdGJkYl

NyUiIiIiIiMiJEQl

The factor(s) that vanish

n:=12: series(subs(R0=R0ex(n),R1=R1ex(n),op(3,res3)),t,n+1);

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzg=

Another equation between R0 and R1

res4:=op(3,res3):

degree(res2,R0);degree(res2,R1);degree(res4,R0);degree(res4,R1);

IiIj

IiIq

IiIk

IiM3

res5:=factor(resultant(res2,res4,R0)): nops(res5):

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

Nyk8IjwjSSJ0RzYiPCRJI1IxR0YmRiVGJ0YnRidGJw==

NykiIiIiIiMiIiQiIiciJEIkRiRGJA==

n:=12: series(subs(R0=R0ex(n),R1=R1ex(n),op(3,res5)),t,n+1); series(subs(R0=R0ex(n),R1=R1ex(n),op(4,res5)),t,n+1); series(subs(R0=R0ex(n),R1=R1ex(n),op(5,res5)),t,n+1); series(subs(R0=R0ex(n),R1=R1ex(n),op(6,res5)),t,n+1); series(subs(R0=R0ex(n),R1=R1ex(n),op(7,res5)),t,n+1);

Kz1JInRHNiIhIiIiIiEhIiMiIiIiIiNGKSIjPSIiJSIjYSIiJiIkYSUiIiciJUVDIiIoIiYjKW8iIiIpIicpSDIiIiIqIid5aXQiIzUiKHM/KlwiIzYiKUtteU0iIzctSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIjOA==

Kz9JInRHNiIiIigiIiEiIikiIiIhIzkiIiMhIzciIiQhJF0iIiIlISQlZiIiJiElJT4lIiInISY1UiNGJSEndSs7RichKEk/LiIiIiohKEUqNHEiIzUhKU8nXHclIiM2ISpjInA0TCIjNy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiM4

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzg=

Kz9JInRHNiIiIiIiIiEhIiRGJSIiJyIiIyEjPiIiJCEjSSIiJSEkUCMiIiYhJSNmIkYoISV1KSkiIighJnc8JyIiKSEnJlsjUiIiKiEoJTQtRiIjNSEpXUtLPSIjNiEqcm8oeTciIzctSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIjOA==

Kz9JInRHNiIhJDcmIiIhISUzWSIiIiEmPzoiIiIjIiYpRzciIiQiJ2coXCIiIiUiJ1dsciIiJiIoO0N4JCIiJyIpUydvPiMiIigiKmtLN04iIiIpIipTdUJqKSIiKiIra0decmMiIzUiLF8kKWV2IVEiIzYiLUtvKGYxZyMiIzctSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIjOA==

algR1:=collect(op(5,res5),R1,factor):

res6:=factor(resultant(res2,algR1,R1)):nops(%)

IiIk

map(indets,[seq(op(i,res6),i=1..nops(res6))]); map(nops,[seq(op(i,res6),i=1..nops(res6))]);

NyU8IjwkSSNSMEc2IkkidEdGJkYk

NyUiIiIiJDkkIiU9Xg==

n:=12: series(subs(R0=R0ex(n),op(2,res6)),t,n+1); series(subs(R0=R0ex(n),op(3,res6)),t,n+1);

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzg=

Kz1JInRHNiIiNXdcJGYhRy0iUkY4IyIiIiE3JzR3KjNQQCNbXSdIKioiIiMiOl82PiY9aFZNWCRlQWgjIiIkITxXRFY0dil5MiczZkFYRyUiIiUiPiVRbz0nKVttLzVJKlxqLFQmIiImIUBjMTcmKmY0dU84aCRHMWswbCIiJyJCJXlsRiNvPnNKcClSKD1SQzdyKCIiKCFELyg9aW9SdUVlLUt3Z2Q0QVgiKSIiKSJGN1JRJmYlZXFVKnlVO3kzLCcpKWY7KCIiKiFIRWoxamI8XSZwTFZpP19RMURmJj0mIiM1IkoiZih6PiYpKSp5LWwmb0N4REAjKVtZSj1KIiM2IUwjb0sqPnZGMil5Vz5RIXBibSM9JFFrPWUiIiM3LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiYiIzg=

Careful here with the choice

algR0:=collect(op(2,res6),R0,factor): degree(%,R0)

IiND

syst2[1]; indets(%)

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

PCZJI0IxRzYiSSNSMEdGJEkjUjFHRiRJInRHRiQ=

Let us form an equation between B1 and R1

map(degree,[op(syst2)],R0);map(degree,[op(syst2)],R1);

NyUiIiMiIiUiIic=

NyUiIiYiIigiIio=

res7:=numer(factor(rem(syst2[2],syst2[1], R0))): nops(%); indets(%%)

IiRZIw==

PCZJI0IxRzYiSSNSMEdGJEkjUjFHRiRJInRHRiQ=

res8:=numer(factor(rem(syst2[3],syst2[1], R0))): nops(%); indets(%%)

IiRKKQ==

PCZJI0IxRzYiSSNSMEdGJEkjUjFHRiRJInRHRiQ=

res9:=factor(resultant(res7,res8,R0)): nops(%)

IiIl

map(indets,[seq(op(i,res9),i=1..nops(res9))]); map(nops,[seq(op(i,res9),i=1..nops(res9))]);

NyY8IjwjSSJ0RzYiPCRJI1IxR0YmRiU8JUkjQjFHRiZGKEYl

NyYiIiIiIiNGJCIkaCk=

n:=12: series(subs(R1=R1ex(n),B1=B1ex(n),op(3,res9)),t,n+1); series(subs(R1=R1ex(n),B1=B1ex(n),op(4,res9)),t,n+1);

Kz9JInRHNiIhI0siIiEhJGciIiIiISMhKSIiIyIkUyciIiQiJT9GIiIlIiYpKVEiIiImIiZnJnoiIiciJ1MtWyIiKCIoISpSLiQiIikiKSFSOCg+IiIqIiokRyhISiIiIzUiKj82QyEqKSIjNiIrYmJqSWgiIzctSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIjOA==

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzg=

res10:=factor(resultant(op(4,res9),algR1,R1)): nops(res10)

IiIm

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

Nyc8IjwjSSJ0RzYiRiQ8JEkjQjFHRiZGJUYn

NyciIiIiIiNGJCIkOiciJj1FIg==

n:=8: series(subs(B1=B1ex(n),op(4,res10)),t,n+1); series(subs(B1=B1ex(n),op(5,res10)),t,n+1);

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIio=

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

Careful here

algB1:=collect(op(4,res10),B1,factor): degree(%,B1)

IiND

Let us form an equation between B2 and R1

syst3:=map(numer,factor(subs(isolate(systfinal[1],B1),[systfinal[2], systfinal[3], systfinal[4]]))):

map(degree,[op(syst3)],R0);map(degree,[op(syst3)],R1);

NyUiIiMiIiUiIic=

NyUiIiYiIigiIio=

res7:=numer(factor(rem(syst3[2],syst3[1], R0))): nops(%); indets(%%)

IiRYIw==

PCZJI0IyRzYiSSNSMEdGJEkjUjFHRiRJInRHRiQ=

res8:=numer(factor(rem(syst3[3],syst3[1], R0))): nops(%); indets(%%)

IiRKKQ==

PCZJI0IyRzYiSSNSMEdGJEkjUjFHRiRJInRHRiQ=

res9:=factor(resultant(res7,res8,R0)): nops(%)

IiIl

map(indets,[seq(op(i,res9),i=1..nops(res9))]); map(nops,[seq(op(i,res9),i=1..nops(res9))]);

NyY8IjwjSSJ0RzYiPCRJI1IxR0YmRiU8JUkjQjJHRiZGKEYl

NyYiIiIiIiNGJCIkaCk=

n:=12: series(subs(R1=R1ex(n),B2=B2ex(n),op(3,res9)),t,n+1); series(subs(R1=R1ex(n),B2=B2ex(n),op(4,res9)),t,n+1);

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzg=

res10:=factor(resultant(op(4,res9),algR1,R1)): nops(res10)

IiIm

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

Nyc8IjwjSSJ0RzYiRiQ8JEkjQjJHRiZGJUYn

NyciIiIiIiNGJCImQ0UiIiRAJw==

n:=8: series(subs(B2=B2ex(n),op(4,res10)),t,n+1); series(subs(B2=B2ex(n),op(5,res10)),t,n+1);

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

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIio=

Careful again...

algB2:=collect(op(5,res10),B2,factor): degree(%,B2)

IiND

The system of 4 equations obtained for R0 R1 B1 and B2

read(systfinalR0R1B1B2_with_diag):

indets(systfinal[1]);

PCdJI0IxRzYiSSNCMkdGJEkjUjBHRiRJI1IxR0YkSSJ0R0Yk

systfinal:=[op(systfinal)]:

map(degree,systfinal,R0);map(degree,systfinal,R1);map(degree,systfinal,B1);map(degree,systfinal,B2);map(degree,systfinal,t);

NyYiIiMiIiQiIiVGJQ==

NyYiIiQiIiUiIiYiIic=

NyYiIiJGIyIiIyIiJA==

NyYiIiRGIyIiJiIiJw==

map(Z->nops(expand(Z)),systfinal);

NyYiIz4iI1giIyUqIiRsIg==

But this time we've been able to solve the system and obtain individual equations

map(nops,[algR0,algR1,algB1,algB2])

NyYiI0RGI0YjRiM=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">The algebraic structure of univariate algebraic series, as stated in Section 4.1</Text-field>

with(algcurves):

A quartic series u

algu:= (1-3*u)^3*(1+u)*t^2+(1+18*u^2-27*u^4)*t-u; genus(%,t,u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVhbGd1R0YoLCgqKCksJiIiIkYyKiYiIiRGMkkidUdGKEYyISIiRjRGMiwmRjJGMkY1RjJGMilJInRHRigiIiNGMkYyKiYsKComIiNGRjIpRjUiIiVGMkY2KiYiIz1GMilGNUY6RjJGMkYyRjJGMkY5RjJGMkY1RjY3I0Yu

IiIj

collect(algu/(t*(t+1)),u,factor);

LCoiIiJGIyomIiNGRiMpSSJ1RzYiIiIlRiMhIiIqJiIjPUYjKUYnIiIjRiNGIyoqLCYqJiIiKUYjKUkidEdGKEYuRiNGI0YjRiNGI0Y0RiosJkY0RiNGI0YjRipGJ0YjRio=

A cubic series over Q(u)

algvu:=u*(1+3*v-v^3)- v*(v^2+v+1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGd2dUdGKCwmKiZJInVHRigiIiIsKCokKUkidkdGKCIiJEYxISIiKiZGNkYxRjVGMUYxRjFGMUYxRjEqJkY1RjEsKCokKUY1IiIjRjFGMUY1RjFGMUYxRjFGNzcjRi4=

uvsol:=solve(%,u);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZ1dnNvbEdGKCwkKihJInZHRigiIiIsKCokKUYwIiIjRjFGMUYwRjFGMUYxRjEsKCokKUYwIiIkRjFGMSomRjlGMUYwRjEhIiJGMUY7RjtGOzcjRi4=

algv:=collect(factor(resultant(algvu,algu,u)),v,factor);

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

And finally a quadratic series over Q(v)

algwv := 4*v^4+4*v^3-4*v-1+w^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZhbGd3dkdGKCwsKiYiIiUiIiIpSSJ2R0YoRjBGMUYxKiZGMEYxKUYzIiIkRjFGMSokKUkid0dGKCIiI0YxRjEqJkYwRjFGM0YxISIiRjFGPDcjRi4=

genus(algwv,v,w);

IiIi

Series expansions

user:=proc(n) option remember: if n=0 then 0 else series(subs(u=user(n-1),algu+u),t,n+1);fi: end:

user(4);

Ky1JInRHNiIiIiJGJUYlIiIjIiM1IiIkIiNZIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIiY=

algvu;

LCYqJkkidUc2IiIiIiwoKiQpSSJ2R0YlIiIkRiYhIiIqJkYrRiZGKkYmRiZGJkYmRiZGJiomRipGJiwoKiQpRioiIiNGJkYmRipGJkYmRiZGJkYs

vser:=proc(n) option remember: if n=0 then 0 else series(subs(u=user(n), v=vser(n-1), u*(-v^3+3*v+1)/(v^2+v+1)),t,n+1): fi: end:

vser(5);

Ky9JInRHNiIiIiJGJSIiJCIiIyIjOkYmIiMiKSIiJSIkKlsiIiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIiJw==

wser:=proc(n) series(sqrt(subs(w=0,v=vser(n),-algwv)),t,1+n): end:

wser(5);

KzFJInRHNiIiIiIiIiEiIiNGJSIiJUYnIiM/IiIkIiMlKkYoIiRVJiIiJi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIn

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

algu:= (1-3*u)^3*(1+u)*t^2+(1+18*u^2-27*u^4)*t-u; algvu:=u*(1+3*v-v^3)- v*(v^2+v+1);

algv:=collect(factor(resultant(algvu,algu,u)),v,factor);

algwv := 4*v^4+4*v^3-4*v-1+w^2;

uv:=solve(algvu,u)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN1dkdGKCwkKihJInZHRigiIiIsKCokKUYwIiIjRjFGMUYwRjFGMUYxRjEsKCokKUYwIiIkRjFGMSomRjlGMUYwRjEhIiJGMUY7RjtGOzcjRi4=

Series expansions

user:=proc(n) option remember: if n=0 then 0 else series(subs(u=user(n-1),algu+u),t,n+1);fi: end:

user(4);

Ky1JInRHNiIiIiJGJUYlIiIjIiM1IiIkIiNZIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIiY=

vser:=proc(n) option remember: if n=0 then 0 else series(subs(u=user(n), v=vser(n-1), u*(-v^3+3*v+1)/(v^2+v+1)),t,n+1): fi: end:

vser(5);

Ky9JInRHNiIiIiJGJSIiJCIiIyIjOkYmIiMiKSIiJSIkKlsiIiYtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIiJw==

wser:=proc(n) series(sqrt(subs(w=0,v=vser(n),-algwv)),t,1+n): end:

wser(5);

KzFJInRHNiIiIiIiIiEiIiNGJSIiJUYnIiM/IiIkIiMlKkYoIiRVJiIiJi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIn

alias(troot=RootOf(algv,t))

SSZ0cm9vdEc2Ig==

map(series,[solve(algv,t)],v,4)

NyQrLUkidkc2IiEiIiIiISEiKiIiIiEjWCIiIyEkciIiIiQtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGKSIiJSsrRiRGKUYpISIkRitGLUYtRi5GNA==

tv:=op(2,[solve(algv,t)]):

factor(evala(Normal(subs(t=troot,t*(1+t)/(1-2*t)^2)))); factor(evala(Normal(subs(t=troot,(1-8*t)/(1-2*t)^2))))

KixJInZHNiIiIiIsKCokKUYjIiIjRiVGJUYjRiVGJUYlRiUpLCgqJClGIyIiJEYlRiUqJkYuRiVGI0YlISIiRiVGMEYuRiUsKiomIiIlRiUpRiNGM0YlRiUqJkYzRiVGLUYlRiUqJkYzRiVGI0YlRjBGJUYwRjApLCwqJEY0RiVGJSomIiIpRiVGLUYlRiUqJiIiJ0YlRihGJUYlKiZGKUYlRiNGJUYlRiVGJUYpRjA=

KiosKCokKUkidkc2IiIiIyIiIkYpKiYiIiVGKUYmRilGKUYpRilGKSksKComRitGKSlGJiIiJEYpRikqJkYwRilGJUYpRilGKSEiIkYwRiksKiomRitGKSlGJkYrRilGKUYuRilGKkYyRilGMkYyKSwsKiRGNUYpRikqJiIiKUYpRi9GKUYpKiYiIidGKUYlRilGKSomRihGKUYmRilGKUYpRilGKEYy

<Text-field style="Heading 1" layout="Heading 1">From the individual equations for R0, etc. to a complete algebraic structure</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Expression of R0</Text-field>

nops(algR0);

IiNE

We first compute an equation in R0 and u

res:=factor(resultant(algR0,algu,t)): nops(%);

IiIk

seq(indets(op(i,res)),i=1..3)

NiU8IjwkSSNSMEc2IkkidUdGJkYk

n:=20: series(subs(R0=R0ex(n),u=user(n),op(2,res)),t,n); series(subs(R0=R0ex(n),u=user(n),op(3,res)),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIz8=

K01JInRHNiIiKSIpZS9aIiIhISs1TjEnPSciIiIiLWooW1snKXkkIiIjIS9BTiQ9IylRVyIiIiQiMGhAdjEpZmxRIiIlITF5PSQ+X3pEeigiIiYiMypcYztJK3QlUjciIichNHdlZ0ddK0kwaCIiIigiNXZIIlxdTT1db3YiIiIpITYtTSxeMjNUN1xrIiIiKiI3YXQnR3BXem9wZ00iIiM1ITdPIkdSMm0hWy01cigqIiM2IjhaSzZuJUdUdXU5bmoiIzchOU9lOTsiKXopPWA7I2ZQIiM4Ijp3VSpvdlBbT1dWYUM/IiM5ITpnJD5Nb0Y7SFgzdCcpKioiIzoiOztoQk1dKWZuPEdYb14lIiM7ITwxKDRPYy4rYDp1SCdwJz0iIzwiPDQ+ViFcIjMnej95eiVwKHAiIz0hPSVbKmZzIlEpNFsjUlJIXUgjIiM+LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIz8=

algR0u:=op(2,res): nops(%);

IiQjRw==

degree(algR0u,R0);

IiM3

Now in R0 and v

res:=numer(factor(subs(u=uv,algR0u))): nops(%)

IiIj

degree(op(1,res),R0);degree(op(2,res),R0);

IiIl

IiIp

nops(op(2,res));

IiQmUg==

n:=15: series(subs(R0=R0ex(n),v=vser(n),op(1,res)),t,n); series(subs(R0=R0ex(n),v=vser(n),op(2,res)),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

K0NJInRHNiIiIzsiIiEiJHMnIiIiIiZXYCIiIiMiJ3NpRCIiJCIoW1lgJCIiJSIpJzRMSCUiIiYiKl9uOnglIiInIitvST1zXCIiKCIsOGBieCRcIiIpIi0vaFZBRVoiIioiLmBKP2FhUiUiIzUiLz1zV19UJipSIiM2IjB4RSwpKilRbE4iIzciMW8pXGc+OFM4JCIjOCIybmZRcSFbbj9GIiM5LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIzo=

algR0v:=collect(op(1,res),R0,factor);

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

algvp:=collect(-numer(factor(subs(t=tp/lcoeff(algv,t),algv))),tp,factor);

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

res:=factor(numer(subs(RootOf(algvp,tp)=tp,tp=t*lcoeff(algv,t),factor(algR0v,RootOf(algvp,tp))))):nops(res);

IiIl

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

NyY8I0kidkc2IkYjPCVJI1IwR0YlSSJ0R0YlRiRGJg==

n:=15: series(subs(R0=R0ex(n),v=vser(n),op(3,res)),t,n); series(subs(R0=R0ex(n),v=vser(n),op(4,res)),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

KztJInRHNiIiIiciIiUiJC8jIiImIiU7UkYlIiZfcSYiIigiJ1V0cSIiKSIoJyo0IXoiIioiKU94NyMpIiM1IiolZSVbNSkiIzYiKylIOlNwKCIjNyIsOy9LKSozKCIjOCItMVNpVSNRJyIjOS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIjOg==

nops(op(3,res));

IiNl

algR0vt:=collect(op(3,res),[R0],factor);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShhbGdSMHZ0R0YoLCgqKiwoKiQpSSJ2R0YoIiIjIiIiRjUqJiIiJUY1RjNGNUY1RjVGNUY1LCwqJClGM0Y3RjVGNSomIiIpRjUpRjMiIiRGNUY1KiYiIidGNUYyRjVGNSomRjRGNUYzRjVGNUY1RjVGNSksKComRjdGNUY9RjVGNSomRj5GNUYyRjVGNUY1ISIiRj5GNSlJI1IwR0YoRjRGNUY1Ki5GM0Y1LCZGQUY1RjVGNUY1RjBGNUZCRjUsJiomRjRGNUkidEdGKEY1RjVGNUZGRjVGSEY1RjUqKkY6RjUsLComIiImRjVGOkY1RjVGO0Y1RjFGNUY2RkZGNUZGRjVGOEY1LCYqJkY8RjVGTUY1RjVGNUZGRjVGRjcjRi4=

This suggests to introduce R0/(1-2t)=V0

collect(subs(R0=(1-2*t)*V0, algR0vt/(1-2*t)^2),V0,factor)

LCgqKiwoKiQpSSJ2RzYiIiIjIiIiRioqJiIiJUYqRidGKkYqRipGKkYqLCwqJClGJ0YsRipGKiomIiIpRiopRiciIiRGKkYqKiYiIidGKkYmRipGKiomRilGKkYnRipGKkYqRipGKiksKComRixGKkYyRipGKiomRjNGKkYmRipGKkYqISIiRjNGKilJI1YwR0YoRilGKkYqKixGN0YqRiRGKkYnRiosJkY2RipGKkYqRipGPUYqRjsqLEYvRiosLComIiImRipGL0YqRipGMEYqRiVGKkYrRjtGKkY7RipGLUYqLCYqJkYxRipJInRHRihGKkYqRipGO0YqKSwmKiZGKUYqRkZGKkYqRipGO0YpRjtGOw==

algV0v:=collect(op(3,numer(factor(evala(Normal(subs(t=troot,%)))))),V0,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdhbGdWMHZHRigsKCooLCoqJiIiJSIiIilJInZHRihGMkYzRjMqJkYyRjMpRjUiIiRGM0YzKiZGMkYzRjVGMyEiIkYzRjpGMyksLCokRjRGM0YzKiYiIilGM0Y3RjNGMyomIiInRjMpRjUiIiNGM0YzKiZGQ0YzRjVGM0YzRjNGM0ZDRjMpSSNWMEdGKEZDRjNGMyosLCZGREYzRjNGM0YzRjxGM0YwRjNGNUYzRkZGM0Y6KiZGNEYzLCwqJiIiJkYzRjRGM0YzRj5GMyokRkJGM0YzRjlGOkYzRjpGM0YzNyNGLg==

V0solens:=(simplify(factor([solve(algV0v,V0)]),symbolic))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlWMHNvbGVuc0dGKDckLCQqKiMiIiIiIilGMiwmKiYsKComIiIjRjIpSSJ2R0YoIiIlRjJGMiomRjtGMilGOkY4RjIhIiJGOkY+RjIpLCoqJkY7RjJGOUYyRj4qJkY7RjIpRjoiIiRGMkY+KiZGO0YyRjpGMkYyRjJGMiNGMkY4RjJGMioqRjNGMiwqKiRGOUYyRjIqJEZDRjJGMkY6Rj4jRjJGO0Y+RjIsJkY6RjJGRkYyRjJGOkYyRjJGMkZIRj4sLEZJRjIqJkYzRjJGQ0YyRjIqJiIiJ0YyRj1GMkYyKiZGOEYyRjpGMkYyRjJGMkY+RjIsJCoqRjFGMiwmKiYsKEY3Rj5GPEYyRjpGMkYyRj9GMkYyRkdGMkYyRkhGPkZNRj5GMjcjRi4=

map(series,%,v,6)

NyQrLUkidkc2IiIiIkYmISIoIiIkIiIpIiIlIiNAIiImLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRiYiIicrK0YkRiZGKCEiJUYqRjNGLEYtRjM=

R0ex(4);vser(4);

KiYpSSJ0RzYiIiIjIiIiLCoqJiIkMSJGJylGJCIiJUYnRicqJiIjP0YnKUYkIiIkRidGJyomRjBGJ0YjRidGJ0YkRidGJw==

Ky1JInRHNiIiIiJGJSIiJCIiIyIjOkYmIiMiKSIiJS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YlIiIm

V0solv:=collect(((-2*v^4+4*v^2+v)*w+8*v*(v+1/2)*(v^4+v^3-v-1/4))/((8*(v^4+8*v^3+6*v^2+2*v+1))*(v^4+v^3-v-1/4)),w,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdWMHNvbHZHRigsJiouIyIiIiIiI0YxSSJ2R0YoRjEsKComRjJGMSlGMyIiJEYxRjEqJiIiJUYxRjNGMSEiIkYxRjpGMSwsKiQpRjNGOUYxRjEqJiIiKUYxRjZGMUYxKiYiIidGMSlGM0YyRjFGMSomRjJGMUYzRjFGMUYxRjFGOiwqKiZGOUYxRj1GMUYxKiZGOUYxRjZGMUYxRjhGOkYxRjpGOkkid0dGKEYxRjoqKkYwRjFGM0YxLCZGQ0YxRjFGMUYxRjtGOkYxNyNGLg==

V0solv:=v*(2*v^3-4*v-1)/((2*(v^4+8*v^3+6*v^2+2*v+1))*w)+(2*v+1)*v/(2*(v^4+8*v^3+6*v^2+2*v+1)); factor(%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdWMHNvbHZHRigsJiosIyIiIiIiI0YxSSJ2R0YoRjEsKComRjJGMSlGMyIiJEYxRjEqJiIiJUYxRjNGMSEiIkYxRjpGMSwsKiQpRjNGOUYxRjEqJiIiKUYxRjZGMUYxKiYiIidGMSlGM0YyRjFGMSomRjJGMUYzRjFGMUYxRjFGOkkid0dGKEY6RjEqKCwmRkNGMUYxRjFGMUYzRjEsLComRjJGMUY9RjFGMSomIiM7RjFGNkYxRjEqJiIjN0YxRkJGMUYxRjhGMUYyRjFGOkYxNyNGLg==

LCQqLCMiIiIiIiNGJUkidkc2IkYlLCwqJkYmRiUpRiciIiRGJUYlKihGJkYlRidGJUkid0dGKEYlRiUqJiIiJUYlRidGJSEiIkYuRiVGJUYxRiUsLCokKUYnRjBGJUYlKiYiIilGJUYrRiVGJSomIiInRiUpRidGJkYlRiUqJkYmRiVGJ0YlRiVGJUYlRjFGLkYxRiU=

R0w:=V0solv*(1-2*t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRSMHdHRigqJiwmKiwjIiIiIiIjRjJJInZHRihGMiwoKiZGM0YyKUY0IiIkRjJGMiomIiIlRjJGNEYyISIiRjJGO0YyLCwqJClGNEY6RjJGMiomIiIpRjJGN0YyRjIqJiIiJ0YyKUY0RjNGMkYyKiZGM0YyRjRGMkYyRjJGMkY7SSJ3R0YoRjtGMiooLCZGREYyRjJGMkYyRjRGMiwsKiZGM0YyRj5GMkYyKiYiIztGMkY3RjJGMiomIiM3RjJGQ0YyRjJGOUYyRjNGMkY7RjJGMiwmRjJGMiomRjNGMkkidEdGKEYyRjtGMjcjRi4=

n:=8:series(subs(w=wser(n),v=vser(n),R0w),t,n)

Ky9JInRHNiIiIiIiIiRGJiIiJSIjPyIiJiIkMSIiIiciJHgnIiIoLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiUiIik=

series(%-R0ex(n),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIik=

<Text-field style="Heading 2" layout="Heading 2">Expression of R1</Text-field>

nops(algR1);

IiNE

res:=factor(resultant(algR1,algu,t)): nops(%);

IiIk

seq(indets(op(i,res)),i=1..3)

NiU8IjwkSSNSMUc2IkkidUdGJkYk

n:=5: series(subs(R1=R1ex(n),u=user(n),op(2,res)),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiY=

series(subs(R1=R1ex(n),u=user(n),op(3,res)),t,n)

Ky9JInRHNiIiJWZvIiIhIScjZihHIiIiIihjIilHJyIiIyEpKSk0RSMqIiIkIishXHMpcDciIiUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIiJg==

algR1u:=op(2,res): nops(%);

IiQ8JA==

degree(algR1u,R1);indets(algR1u);

IiM3

PCRJI1IxRzYiSSJ1R0Yk

res:=numer(factor(subs(u=uv,algR1u))): nops(%)

IiIj

degree(op(1,res),R1);degree(op(2,res),R1);

IiIl

IiIp

nops(op(2,res));

IiRKJQ==

n:=15: series(subs(R1=R1ex(n),v=vser(n),op(1,res)),t,n); series(subs(R1=R1ex(n),v=vser(n),op(2,res)),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

K0NJInRHNiIiIzsiIiEiJF8oIiIiIiZ3Jj4iIiMiJ19mUCIiJCIoT1InZiIiJSIpKT0mKkcpIiImIis1cCVmLyIiIiciLGNPJT1FNyIiKCItXiMpeURkOCIiKSIuUyM+TzVOOSIiKiIvJ1EjKTM8QFkiIiM1IjA1b2EtcVtXIiIjNiIxaV1rblsvI1IiIiM3IjJxKjNdVEsmR0oiIiM4IjNYXCMpRyNbISlmQCIiIzktSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIjOg==

An equation of degree 4 over Q(v)

algR1v:=collect(op(1,res),R1,factor);

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

We try to use t and w, and obtain an equation of degree 1

res:=numer(subs(RootOf(algwv,w)=w,RootOf(algvp,tp)=tp,tp=t*lcoeff(algv,t),factor(algR1v,{RootOf(algvp,tp),RootOf(algwv,w)}))): nops(res)

IiIl

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

NyY8JkkjUjFHNiJJInRHRiVJInZHRiVJIndHRiVGI0YjRiM=

n:=15: series(subs(R1=R1ex(n),v=vser(n),w=wser(n),op(1,res)),t,n); series(subs(R1=R1ex(n),v=vser(n),w=wser(n),op(2,res)),t,n); series(subs(R1=R1ex(n),v=vser(n),w=wser(n),op(3,res)),t,n); series(subs(R1=R1ex(n),v=vser(n),w=wser(n),op(4,res)),t,n);

K0NJInRHNiIhIiMiIiEhIzkiIiIiIzkiIiMiJSU0IiIiJCImV1siIiIlIictUzoiIiYiKDshRzkiIiciKW82WjciIigiKlFVMjAiIiIpIip1IjReJykiIioiKz85PjhxIiM1IiwlZTJkQ2MiIzYiLSFIWCVmd1ciIzciLitqQ2FMYSQiIzgiLzddVC9WJHojRiktSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIjOg==

K0NJInRHNiIhIiMiIiEhIz0iIiIhIyEqIiIjISQpUiIiJCElPzsiIiUhJWlfIiImISU3UCIiJyInXy87IiIoIigxUEAjIiIpIilZajVBIiIqIipLNkonPiIjNSIrPyVRV2siIiM2IixZPjNCTCIiIzciLSF5O2h0MCIiIzgiLWM0PiMpeiMpIiM5LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIzo=

K0NJInRHNiIiIiUiIiEiIykpIiIiIiUrNiIiIyImdzQiIiIkIiZnKCkqRiUiJzs2JSkiIiYiKGtnJHAiIiciKSc+YWcmIiIoIipXdDRaJSIiKSIrJSlcKFxgJCIiKiIsL1g9JHlGIiM1Ii0nKnBMc3VAIiM2Ii5bWkIudnAiIiM3Ii9DO0AnKltBOCIjOCIwT1pjJHAtSDUiIzktSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIjOg==

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

algR1vwt:=collect(op(4,res),[R1,w],factor)

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

R1w:=collect(solve(algR1vwt,R1),[w],factor);

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

subs(troot=t,collect(evala(Normal(subs(t=troot,R1w/(1-2*t)))),w,factor))

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

co:=collect(op(2,R1w),t,factor)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjb0dGKCwmKiojIiIiIiIkRjEsLCokKUkidkdGKCIiJUYxRjEqJiIiKUYxKUY2RjJGMUYxKiYiIzdGMSlGNiIiI0YxRjEqJiIiJkYxRjZGMUYxRjFGMUYxLCxGNEYxRjhGMSomIiInRjFGPUYxRjEqJkY+RjFGNkYxRjFGMUYxISIiSSJ0R0YoRjFGMSooI0YxRkNGMSwqRjRGMUY4RjFGNkZFRjFGMUYxRkFGRUYxNyNGLg==

subs(troot=t,factor(evala(Normal(subs(t=troot,co/(1-8*t))))))

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

n:=8:series(subs(w=wser(n),v=vser(n),R1w),t,n)

Ky9JInRHNiIjIiIiIiIkIiIjRiciIiUiIioiIiYjIiRGI0YnIiInIyIlODdGJyIiKC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YmIiIp

series(%-R1ex(n),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIik=

JSFH

<Text-field style="Heading 2" layout="Heading 2">Expression of B1</Text-field>

nops(algB1);

IiNE

res:=factor(resultant(algB1,algu,t)): nops(%);

IiIk

seq(indets(op(i,res)),i=1..3)

NiU8IjwkSSNCMUc2IkkidUdGJkYk

n:=5: series(subs(B1=B1ex(n),u=user(n),op(2,res)),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIio=

series(subs(B1=B1ex(n),u=user(n),op(3,res)),t,n)

Ky9JInRHNiIiN145Y1hNKTNLWWsmKSkiIiEiNyYzWEVAcXNoKFtLdCIiIiE5KCplWUxnbklvaiQ9cCoiIiMhO3dILlYkZVlnOGJDQSo9IiIkITwoMzJubSoqNGBxLi0jSE4jIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIiY=

algB1u:=op(2,res): nops(%);

IiRlJg==

degree(algB1u,B1);indets(algB1u);

IiM3

PCRJI0IxRzYiSSJ1R0Yk

res:=numer(factor(subs(u=uv,algB1u))): nops(%)

IiIj

degree(op(1,res),B1);degree(op(2,res),B1);

IiIl

IiIp

nops(op(2,res));

IiR5KA==

n:=15: series(subs(B1=B1ex(n),v=vser(n),op(1,res)),t,n); series(subs(B1=B1ex(n),v=vser(n),op(2,res)),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

Kz9JInRHNiIiJEgoIiIjIiY+NCkiIiQiKFJoZSUiIiUiKltPangiIiImIitGMV8xYCIiJyIteEIvWjI4IiIoIi5DWEEwS3gjIiIpIi8uSmYzNjtfIiIqIjArSTJWPzUqKSkiIzUiMkszbXVFNWlSIiIjNiIzaUIjUlJkIzNZPyIjNyI0MyFvJnpUZSMpbyNHIiM4IjVhTUcvZGdKdDdQIiM5LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiM6

An equation of degree 4 over Q(v)

algB1v:=collect(op(1,res),B1,factor);

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

We try to use t and w, and obtain an equation of degree 1

res:=numer(subs(RootOf(algwv,w)=w,RootOf(algvp,tp)=tp,tp=t*lcoeff(algv,t),factor(algB1v,{RootOf(algvp,tp),RootOf(algwv,w)}))): nops(res)

IiIl

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

NyY8JkkjQjFHNiJJInRHRiVJInZHRiVJIndHRiVGI0YjRiM=

n:=15: series(subs(B1=B1ex(n),v=vser(n),w=wser(n),op(1,res)),t,n); series(subs(B1=B1ex(n),v=vser(n),w=wser(n),op(2,res)),t,n); series(subs(B1=B1ex(n),v=vser(n),w=wser(n),op(3,res)),t,n); series(subs(B1=B1ex(n),v=vser(n),w=wser(n),op(4,res)),t,n);

Kz9JInRHNiIhI1siIiMhJU06IiIkISY1diMiIiUhJz9hUCIiJiEoKyMpUSUiIichKW8kUmwlIiIoISpNYEhpJSIiKSEraVlAJVElIiIqISwjPSEpZj1TIiM1IS01SFF6KmUkIiM2IS5FUj1BTTkkIiM3IS9JQCYqb2Q0RiIjOCEwPXU2a0lqSSMiIzktSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

Kz9JInRHNiIhI1siIiMhJSllIiIiJCEmUSNIIiIlISdVdVMiIiYhKGlZJVsiIichKTdvNl8iIighKlVFJlJfIiIpISslPWssLSYiIiohLGFGM0NrJSIjNSEtV2IwNnpUIiM2IS45ViNbRyVvJCIjNyEvVVY9XiNbPiQiIzghMD1IPkpiUXQjIiM5LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjIiIiIiM6

K0FJInRHNiIhIiMiIiIhI1EiIiMhJGskIiIkISVBQyIiJSEmM0QiIiImISZJeCUiIichJitzJiIiKCIoWS9NIiIiKSIpSUUiKT4iIioiKnFIbyo+IiM1IitzZmRdPCIjNiIsJ0hjJilHOSIjNyItZSU+SXk2IiIjOCItI1whZV80JikiIzktSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJiIjOg==

algB1vwt:=collect(op(2,res),[B1,w],factor)

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

B1w:=collect(solve(algB1vwt,B1),[w,t],factor);

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

factor(subs(troot=t,evala(Normal(subs(t=troot,coeff(B1w,w)/(1-2*t)))))); factor(subs(troot=t,evala(Normal(subs(t=troot,coeff(B1w,w,0)/(1-2*t))))))

LCQqLCMiIiIiIiVGJUkidkc2IkYlLEAqKCIjNUYlSSJ0R0YoRiUpRiciIihGJUYlKigiIztGJUYsRiUpRiciIidGJSEiIiomIiIpRiVGLUYlRjMqKEY1RiVGLEYlKUYnIiImRiVGMyomRjBGJUYxRiVGMyooIiNURiVGLEYlKUYnRiZGJUYlKiZGNUYlRjdGJUYzKihGMkYlRixGJSlGJyIiJEYlRjMqJiIjQkYlRjxGJUYlKigiI0FGJUYsRiUpRiciIiNGJUYzKiYiI1tGJUY/RiVGJSomIiNLRiVGRUYlRiVGLEYlKiYiIipGJUYnRiVGJUYlRiVGJSwqKiZGJkYlRjxGJUYlKiZGJkYlRj9GJUYlKiZGJkYlRidGJUYzRiVGM0YzKSwsKiRGPEYlRiUqJkY1RiVGP0YlRiUqJkYyRiVGRUYlRiUqJkZGRiVGJ0YlRiVGJUYlRkZGM0Yz

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

B1w:=(1-2*t)*(w*%%+%)

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n:=8:series(subs(w=wser(n),v=vser(n),B1w),t,n)

KzFJInRHNiIjISIkIiIjRicjISIiRiciIiQjISM4RiciIiUjISNaRiciIiYhJGYiIiInIyElYjxGJyIiKC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIiKQ==

series(%-B1ex(n),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIik=

JSFH

<Text-field style="Heading 2" layout="Heading 2">Expression of B2</Text-field>

nops(algB2);

IiNE

res:=factor(resultant(algB2,algu,t)): nops(%);

IiIk

seq(indets(op(i,res)),i=1..3)

NiU8IjwkSSNCMkc2IkkidUdGJkYk

n:=5: series(subs(B2=B2ex(n),u=user(n),op(2,res)),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIic=

series(subs(B2=B2ex(n),u=user(n),op(3,res)),t,n)

Ky9JInRHNiIiNFxwQTc2J1s3KmUjIiIhIjZSVzw3Zz55LEMrJCIiIiI5OlxPZl4wKGZCWmBUJiIiIyI7ayY+V2BSI3l6XXE2OEEiIiQiPCMpbyEpZUhid3EqPUZnL1IiIiUtSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKCIiJg==

algB2u:=op(2,res): nops(%);

IiQ/Jw==

degree(algB2u,B2);

IiM3

res:=numer(factor(subs(u=uv,algB2u))): nops(%)

IiIj

degree(op(1,res),B2);degree(op(2,res),B2);

IiIl

IiIp

nops(op(2,res));

IiRdKQ==

n:=15: series(subs(B2=B2ex(n),v=vser(n),op(1,res)),t,n); series(subs(B2=B2ex(n),v=vser(n),op(2,res)),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

Kz9JInRHNiIiIyIpIiIjIiUoUikiIiQiJzBvVyIiJSIpLyRHaiIiIiYiKngnekJZIiInIixqLmNZMyIiIigiLWkpSCE+Kj4jIiIpIi56LTxfJm9SIiIqIi9qR0RJMTZsIiM1IjBwekF6UShwKSoiIzYiMnZCc3osRCcqUiIiIzciM3lfcDRdV0Z2PSIjOCI0MmtRTSZmMiNIUiMiIzktSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

An equation of degree 4 over Q(v)

algB2v:=collect(op(1,res),B2,factor);

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

We try to use t and w, and obtain an equation of degree 1

res:=numer(subs(RootOf(algwv,w)=w,RootOf(algvp,tp)=tp,tp=t*lcoeff(algv,t),factor(algB2v,{RootOf(algvp,tp),RootOf(algwv,w)}))): nops(res)

IiIl

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

NyY8JkkjQjJHNiJJInRHRiVJInZHRiVJIndHRiVGI0YjRiM=

n:=15: series(subs(B2=B2ex(n),v=vser(n),w=wser(n),op(1,res)),t,n); series(subs(B2=B2ex(n),v=vser(n),w=wser(n),op(2,res)),t,n); series(subs(B2=B2ex(n),v=vser(n),w=wser(n),op(3,res)),t,n); series(subs(B2=B2ex(n),v=vser(n),w=wser(n),op(4,res)),t,n);

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KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

K0FJInRHNiIhIzciIiIhJGskIiIjISU7ayIiJCEmWXYpIiIlISh1Kkg1IiImISltJz01IiIiJyEqZ1ZYNSIiIighK0FuemM1IiIpIStfJnlzdyoiIiohLE1mOEB6KSIjNSEtXyI+JDRgeCIjNiEubSVSMDBFbiIjNyEvNWpxWnFlZCIjOCEwJ1tBMCpSeChbIiM5LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiYiIzo=

K0FJInRHNiIhIiMiIiIhI0UiIiMhJGMiIiIkISRBJiIiJSIkOyMiIiYiJkE8IyIiJyInKUdZIyIiKCIoVSlHQCIiKSIpeWtaOyIiKiIqUSMqUj8iIiM1IipTKGYzJikiIzYiK0NZdCopZSIjNyIsZS9USC0lIiM4Ii0jXEw6UHMjIiM5LUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiYiIzo=

Careful here

algB2vwt:=collect(op(2,res),[B2,w],factor)

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

B2w:=collect(solve(algB2vwt,B2),[w,t],factor);

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factor(subs(troot=t,evala(Normal(subs(t=troot,coeff(B2w,w)/(1-2*t)))))); factor(subs(troot=t,evala(Normal(subs(t=troot,coeff(B2w,w,0)/(1-2*t))))))

LCQqLCMiIiIiIiVGJSwmKiYiIiNGJUkidkc2IkYlRiVGJUYlRiUsQCooRiZGJUkidEdGK0YlKUYqIiIoRiVGJSooIiNFRiVGLkYlKUYqIiInRiUhIiIqJkYmRiVGL0YlRjUqKCIjekYlRi5GJSlGKiIiJkYlRjUqJkY0RiVGM0YlRjUqKCIkVSJGJUYuRiUpRipGJkYlRjUqJEY5RiVGNSooIiRBIkYlRi5GJSlGKiIiJEYlRjUqJiIjN0YlRj5GJUYlKigiI11GJUYuRiUpRipGKUYlRjUqJiIjPUYlRkJGJUYlKigiIzpGJUYuRiVGKkYlRjUqJkYwRiVGSEYlRiUqJkYpRiVGLkYlRjVGKkYlRiUsKiomRiZGJUY+RiVGJSomRiZGJUZCRiVGJSomRiZGJUYqRiVGNUYlRjVGNSksLCokRj5GJUYlKiYiIilGJUZCRiVGJSomRjRGJUZIRiVGJUYoRiVGJUYlRilGNUYl

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

B2w:=(1-2*t)*(w*%%+%)

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

n:=8:series(subs(w=wser(n),v=vser(n),B2w),t,n)

KzFJInRHNiIjISIiIiIjRicjISIkRiciIiQjISIoRiciIiUjISNaRiciIiYhJDciIiInIyElSjlGJyIiKC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIiIiKQ==

series(%-B2ex(n),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIik=

JSFH

R0w:=(v*(2*v^3-4*v-1)/((2*(v^4+8*v^3+6*v^2+2*v+1))*w)+(2*v+1)*v/(2*v^4+16*v^3+12*v^2+4*v+2))*(1-2*t)

R1w:=-(7*v^4+8*v^3+6*v^2+5*v+1)*(2*t-1)*w/((6*(v^4+8*v^3+6*v^2+2*v+1))*(4*v^4+4*v^3-4*v-1))+(2*t*v^4+16*t*v^3+v^4+24*t*v^2+8*v^3+10*t*v+2*t-v+1)/(6*(v^4+8*v^3+6*v^2+2*v+1))

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

B1w:=(1-2*t)*(-v*(10*t*v^7-16*t*v^6-8*v^7-8*t*v^5-16*v^6+41*t*v^4-8*v^5-6*t*v^3+23*v^4-22*t*v^2+48*v^3+32*v^2+t+9*v+1)*w/((4*(4*v^4+4*v^3-4*v-1))*(v^4+8*v^3+6*v^2+2*v+1)^2)+v*(2*v+1)*(20*t*v^8-12*t*v^7-16*v^8-8*t*v^6-48*v^7+32*t*v^5-80*v^6+69*t*v^4-58*v^5-20*t*v^3-3*v^4-70*t*v^2+52*v^3-12*t*v+56*v^2+t+15*v+1)/((4*(4*v^4+4*v^3-4*v-1))*(v^4+8*v^3+6*v^2+2*v+1)^2))

B2w:=(1-2*t)*((2*v+1)*(4*t*v^7-26*t*v^6-4*v^7-79*t*v^5-6*v^6-142*t*v^4-v^5-122*t*v^3+12*v^4-50*t*v^2+18*v^3-15*t*v+7*v^2-2*t+v)*w/((4*(4*v^4+4*v^3-4*v-1))*(v^4+8*v^3+6*v^2+2*v+1)^2)+(8*t*v^12+136*t*v^11+712*t*v^10+1556*t*v^9+24*v^10+1746*t*v^8+36*v^9+792*t*v^7-36*v^8-474*t*v^6-108*v^7-1029*t*v^5-36*v^6-876*t*v^4+99*v^5-458*t*v^3+144*v^4-142*t*v^2+90*v^3-25*t*v+27*v^2-2*t+3*v)/((4*(4*v^4+4*v^3-4*v-1))*(v^4+8*v^3+6*v^2+2*v+1)^2))

n:=15: series(subs(v=vser(n),w=wser(n),R0w-R0ex(n)),t,n); series(subs(v=vser(n),w=wser(n),R1w-R1ex(n)),t,n); series(subs(v=vser(n),w=wser(n),B1w-B1ex(n)),t,n); series(subs(v=vser(n),w=wser(n),B2w-B2ex(n)),t,n)

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzo=

<Text-field style="Heading 2" layout="Heading 2">Back to S(x)</Text-field>

We start from the equation in St and the 4 univariate series that we have now determined

eqStcatz

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

Now we replace the 4 univariate series by their expressions

res:=numer(factor(subs(R0=R0w, R1=R1w,B1=B1w,B2=B2w,eqStcatz))): nops(res)

IiQ0Jw==

degree(res,w);

IiIl

We use the defining equation of w:. This time w remains in the equation... so we expect St to have degree 72

res1:=numer(factor(rem(res,algwv,w))): nops(res1)

IiIk

seq(indets(op(i,res1)),i=1..nops(res1));

NiU8IjwjSSJ2RzYiPCdJI1N0R0YmSSJ0R0YmRiVJIndHRiZJInpHRiY=

nops(op(3,res1));

IiR6Iw==

res2:=op(3,res1):

degree(res2,t); degree(res2,v);

IiIj

IiM3

We lower the degree in t

res3:=factor(rem(res2,algv,t)): nops(res3)

IiIm

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

Nyc8IjwjSSJ2RzYiPCdJI1N0R0YmSSJ0R0YmRiVJIndHRiZJInpHRiZGJEYk

res4:=op(3,res3): nops(%);

IiQjUQ==

Here is an equation for St with coefficients in Q(t,v,w,z), of degree 1 in t

algStvw:=collect(res4,[St,z],factor);indets(%)

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

PCdJI1N0RzYiSSJ0R0YkSSJ2R0YkSSJ3R0YkSSJ6R0Yk

(coeff(algStvw,w))

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

We now look at the coefficients of z^i. They depend on t in a very simple fashion

degree(algStvw,z)

IiIj

coeff(algStvw,z,2); indets(%)

LCYqLiIkaSIiIiIsKCokKUkidkc2IiIiI0YlRiUqJiIiJUYlRilGJUYlRiVGJUYlKSwoKiZGLUYlKUYpIiIkRiVGJSomRjJGJUYoRiVGJUYlISIiRjJGJSwsKiQpRilGLUYlRiUqJiIiKUYlRjFGJUYlKiYiIidGJUYoRiVGJSomRitGJUYpRiVGJUYlRiVGJSwqKiZGLUYlRjdGJUYlRjBGJUYsRjRGJUY0RiUpSSNTdEdGKkYyRiVGJSosRjtGJUYmRiVGLkYlRjVGJUY9RiVGJQ==

PCRJI1N0RzYiSSJ2R0Yk

coeff(algStvw,z,1): subs(t=1/2,%); indets(factor(coeff(algStvw,z,1)/(1-2*t)))

IiIh

PCVJI1N0RzYiSSJ2R0YkSSJ3R0Yk

coeff(algStvw,z,0): subs(t=1/8,%); indets(factor(coeff(algStvw,z,0)/(1-8*t)))

IiIh

PCVJI1N0RzYiSSJ2R0YkSSJ3R0Yk

Recall the following identity, obtained from the minimal polynomial of v

factor(evala(Normal(subs(t=troot,(1-8*t)/(1-2*t)^2))))

We introduce zt=z/(1-2t). Then we have an equation with no t (but v and w)

algStvwzt:=collect(op(3,numer(factor(evala(Normal(subs(t=troot,subs(z=zt*(1-2*t), algStvw/(1-2*t)^2))))))), [St,zt],factor)

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

indets(algStvwzt); degree(algStvwzt,St)

PCZJI1N0RzYiSSJ2R0YkSSJ3R0YkSSN6dEdGJA==

IiIk

degree(algStvwzt,w)

IiIi

coeff(algStvwzt,w); coeff(algStvwzt,w,0)

LCYqJiwmKioiIioiIiIsLCokKUkidkc2IiIiJUYnRicqJiIiKUYnKUYrIiIkRidGJyomIiInRicpRisiIiNGJ0YnKiZGNUYnRitGJ0YnRidGJ0YnLCwqJiIjQEYnRipGJyEiIiomIiNDRidGMEYnRjoqJiIjPUYnRjRGJ0Y6KiYiIzpGJ0YrRidGOkYxRjpGJ0kjenRHRixGJ0Y6KihGJkYnLCgqJEYwRidGJyomRjFGJ0YrRidGOkYnRjpGJywsKiYiI0ZGJylGKyIiJkYnRicqJiIjc0YnRipGJ0YnKiYiJDMiRidGMEYnRicqJiIjYUYnRjRGJ0YnKiZGJkYnRitGJ0YnRidGJ0YnSSNTdEdGLEYnRicqKEYoRicsLComRiZGJ0YqRidGJyomRkxGJ0YwRidGJyomIiRpIkYnRjRGJ0YnKiZGTEYnRitGJ0YnRiZGJ0YnRkFGJ0Yn

LCwqJixAKioiJGkiIiIiLCoqJiIiJUYnKUkidkc2IkYqRidGJyomRipGJylGLCIiJEYnRicqJkYqRidGLEYnISIiRidGMkYnKSwsKiRGK0YnRicqJiIiKUYnRi9GJ0YnKiYiIidGJylGLCIiI0YnRicqJkY7RidGLEYnRidGJ0YnRjtGJylJI3p0R0YtRjtGJ0YnKioiJDMiRidGKEYnRjNGJ0Y+RidGJyomIiUhMyJGJylGLCIjN0YnRjIqJiIlT1hGJylGLCIjNkYnRjIqJiImZ0giRicpRiwiIzVGJ0YyKiYiJkt1I0YnKUYsIiIqRidGMiomIiZxaCVGJylGLEY3RidGMiomIiZvRiVGJylGLCIiKEYnRjIqJkZGRicpRixGOUYnRjIqJiImN1olRicpRiwiIiZGJ0YnKiYiJiE9akYnRitGJ0YnKiYiJm90JEYnRi9GJ0YnKiYiJm8uIkYnRjpGJ0YnKiYiJSdIIkYnRixGJ0YnIiNhRidGJylJI1N0R0YtRjBGJ0YnKiYsJioqRlBGJ0Y0RicsNComRipGJ0ZTRidGJyomIiNPRidGVkYnRicqJiIkRyJGJ0ZZRidGJyomIiRPIkYnRmZuRidGJyomIiM6RidGK0YnRicqJiIkKyJGJ0YvRidGMiomIiNvRidGOkYnRjIqJkZfcEYnRixGJ0YyRidGMkYnRj5GJ0YyKihGUEYnLCgqJEYvRidGJyomRjBGJ0YsRidGMkYnRjJGJyw2KiZGKkYnRk9GJ0YnKiZGaW9GJ0ZTRidGJyomIiMlKUYnRlZGJ0YnKiYiJC0iRidGWUYnRicqJiIkTiJGJ0ZmbkYnRicqJiIkUSJGJ0YrRidGJyomRkBGJ0YvRidGJyomIiNzRidGOkYnRicqJiIjQEYnRixGJ0YnRjtGJ0YnRidGJ0Zib0YnRicqKkY5RidGKEYnRjNGJ0Y9RidGJyooRjRGJyw0RmdvRidGaG9GJyomIiNjRidGWUYnRicqJiIjR0YnRmZuRidGJyomRmhxRidGK0YnRjIqJkZfckYnRi9GJ0YyKiYiIzlGJ0Y6RidGMiomRjlGJ0YsRidGMkYnRjJGJ0Y+RidGJyooRipGJywwKiZGV0YnRllGJ0YnKiZGX3BGJ0ZmbkYnRicqJiIjRkYnRitGJ0YnRmFyRicqJiIjSUYnRjpGJ0YnKiZGUEYnRixGJ0YnRidGJ0YnKUZmcEY7RidGMg==

IiIj

Decomposition of St as Se+wSf, where Se and Sf have degree 3 over Q(t,v,x)

subs(St=Se+w*Sf, algStvwzt): res:=factor(rem(%,algwv,w)): res1:=factor(coeff(%,w,1)): nops(%)

IiRVIg==

res2:=factor(coeff(res,w,0)):nops(%);

IiQhPQ==

res3:=factor(resultant(res1,res2,Sf)): nops(%)

IiIm

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

Nyc8IjwjSSJ2RzYiPCRGJUkjenRHRiY8JUkjU2VHRiZGJUYoRik=

NyciIiIiIiMiI1IiJDwiIiVeOg==

NyciIiFGI0YjIiIkIiIn

res4:=factor(resultant(res1,res2,Se)): nops(%)

IiIl

map(indets,[seq(op(i,res4),i=1..nops(res4))]); map(nops,[seq(op(i,res4),i=1..nops(res4))]); map(degree,[seq(op(i,res4),i=1..nops(res4))],Sf);

NyY8IjwkSSJ2RzYiSSN6dEdGJjwlSSNTZkdGJkYlRidGKA==

NyYiIiIiI1IiJCsiIiU8OQ==

NyYiIiFGIyIiJCIiJw==

We take the two simplest equations, of degree 3

algSevzt:=collect(op(4,res3),Se,factor); algSfvzt:=collect(op(3,res4),Sf,factor)

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

We will need expansions of Se and Sf

n:=10:eqe:=numer(factor(convert(normal(series(subs(zt=zx/(1-2*t),v=vser(n),algSevzt),t,n+1)),polynom))):

Seserens:=map(Z->map(factor,Z),map(allvalues,algeqtoseries(eqe,t,Se,5)))

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

eqf:=numer(factor(convert(normal(series(subs(zt=zx/(1-2*t),v=vser(n),algSfvzt),t,n+1)),polynom))):

Sfserens:=map(Z->map(simplify,Z,symbolic),map(allvalues,algeqtoseries(eqf,t,Sf,5)))

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

Correct values?

Stser:=proc(n) normal(series(subs(S(x)=Sex(n),StS),t,n+1)): end:

Stser(4)

Ky1JInRHNiIsJCooIyIiIiIiJEYoLCYqJiIiI0YoKUkieEdGJEYsRihGKEYoRihGKCwmKiRGLUYoRihGKCEiIkYxRigiIiEqKEYuRigsKEYwRihGLkYoRihGKEYoLCZGLkYoRihGMUYxRiwqKkYuRihGNEYoLChGK0YoKiZGLEYoRi5GKEYoRilGKEYoRi9GMUYpLCQqLEYsRihGLkYoRjRGKCwqKiZGKUYoKUYuRilGKEYoKiYiIidGKEYtRihGKComIiM3RihGLkYoRigiIzVGKEYoRi9GMUYoIiIlLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRigiIiY=

for i from 1 to 3 do: normal(series(Seserens[i]+wser(n)*Sfserens[1]-Stser(n),t,3)); od;

KytJInRHNiIsJCoqIyIiIiIiI0YoSSJ4R0YkRigsJiomRilGKEYqRihGKEYoISIiRigsJiokKUYqRilGKEYoRihGLUYtRi0iIiEsJCoqRidGKCwsKiQpRioiIiVGKEYoKiQpRioiIiRGKEYoRi9GLUYsRi1GKUYtRihGKkYtRi5GLUYtRigsJCoqRidGKCwuKiYiIiZGKClGKkY/RihGKComRj9GKEY2RihGKComRj9GKEY5RihGKComRjdGKEYwRihGLSomRjdGKEYqRihGLUY3Ri1GKEYwRi0sJkYqRihGKEYtRi1GLUYpLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGOg==

KylJInRHNiIsJComLCYqJClJInhHRiQiIiMiIiJGLEYsRixGLCwmRihGLEYsISIiRi5GLiIiISwkKigsLiokKUYqIiImRixGLCokKUYqIiIlRixGLCokKUYqIiIkRixGLEYoRixGKkYsRixGLEYsRilGLiwmRipGLEYsRi5GLkYuRistSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGLEY7

KytJInRHNiIqJkkieEdGJCIiIiwmKiQpRiYiIiNGJ0YnRichIiJGLCIiISwkKiojRidGK0YnLCwqJClGJiIiJUYnRicqJClGJiIiJEYnRicqJkYrRidGKkYnRidGJkYnRidGJ0YnRiZGLEYoRixGJ0YnLCQqKkYwRicsLiokKUYmIiImRidGJ0YyRidGNUYnRilGJ0YmRidGJ0YnRidGKkYsLCZGJkYnRidGLEYsRidGKy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YnRjc=

for i from 1 to 3 do: normal(series(Seserens[i]+wser(n)*Sfserens[2]-Stser(n),t,3)); od;

KytJInRHNiIsJCoqIyIiIiIiI0YoSSJ4R0YkRigsJiomRilGKEYqRihGKEYoRihGKCwmKiQpRipGKUYoRihGKCEiIkYwRjAiIiEsJCoqRidGKCwsKiZGKUYoKUYqIiIlRihGKComRilGKClGKiIiJEYoRihGLkYoRipGMEYoRjBGKEYqRjBGLUYwRjBGKCwkKiojRjpGKUYoLC4qJkYpRigpRioiIiZGKEYoRjVGKEY4RihGLkYwRipGMEYoRjBGKEYvRjAsJkYqRihGKEYwRjBGMEYpLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGOg==

KytJInRHNiIsJComLCgqJClJInhHRiQiIiMiIiJGLEYqRixGLEYsRiwsJkYoRixGLCEiIkYuRi4iIiEsJCoqI0YsRitGLCwsKiQpRioiIiVGLEYsKiQpRioiIiRGLEYsKiZGK0YsRilGLEYsRipGLEYsRixGLEYqRi5GLUYuRi5GLCwkKiojRjlGK0YsLC4qJClGKiIiJkYsRixGNEYsRjdGLEYoRixGKkYsRixGLEYsRilGLiwmRipGLEYsRi5GLkYuRistSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGLEY5

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiQ=

for i from 1 to 3 do: normal(series(Seserens[i]+wser(n)*Sfserens[3]-Stser(n),t,3)); od;

KylJInRHNiIsJComKUkieEdGJCIiIyIiIiwmKiRGJ0YqRipGKiEiIkYtRi0iIiEsJCooLC4qJkYpRiopRigiIiZGKkYqKiZGKUYqKUYoIiIlRipGKiomRilGKilGKCIiJEYqRipGLEYtRihGLUYqRi1GKkYnRi0sJkYoRipGKkYtRi1GLUYpLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRipGOg==

KytJInRHNiIsJCooIyIiIiIiI0YoLCgqJkYpRigpSSJ4R0YkRilGKEYoRi1GKEYpRihGKCwmKiRGLEYoRihGKCEiIkYwRjAiIiEsJCosRidGKCwsKiQpRi0iIiVGKEYoKiQpRi0iIiRGKEYoRi9GMComRilGKEYtRihGMEYpRjBGKEYtRjAsJkYtRihGKEYwRjAsJkYtRihGKEYoRjBGKEYoLCQqKkYnRigsLiokKUYtIiImRihGKEY1RihGOEYoKiZGN0YoRixGKEYoKiZGN0YoRi1GKEYoRjdGKEYoRixGMEY8RjBGMEYpLUkiT0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRihGOg==

KytJInRHNiIsJCooIyIiIiIiI0YoSSJ4R0YkRigsJiokKUYqRilGKEYoRighIiJGLkYoIiIhLCQqLkYnRigsKEYsRihGKkYoRihGKEYoLCYqJkYpRihGLUYoRihGKEYuRihGKkYuLCZGKkYoRihGLkYuLCZGKkYoRihGKEYuRihGKCwkKipGJ0YoLC4qJkYpRigpRioiIiZGKEYoKiZGKUYoKUYqIiIlRihGKComRilGKClGKiIiJEYoRihGLEYuRipGLkYoRi5GKEYtRi5GNUYuRihGKS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRkI=

Seser:=Seserens[3]; Sfser:=Sfserens[2];

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

n:=6: normal(series(Seser+wser(n)*Sfser-Stser(n),t,5));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiY=

<Text-field style="Heading 2" layout="Heading 2">The degree of S(x) is 72, even at x=2 </Text-field>

zx;StS

LCYqJkkidEc2IiIiIiwoSSJ4R0YlRiYqJEYoISIiRiZGJkYmRiZGJiomLCZGJEYmRiZGJkYmRidGKkYm

KiYsJkkieEc2IiIiIiokRiQhIiJGKEYoLCgqJiwoRiRGJkYnRiZGJkYmRiYtSSJTR0YlNiNGJEYmRiYqJiMiIiMiIiRGJkYkRiZGJiomI0YmRjJGJkYkRihGJkYm

St is not well defined at x=1

n:=2: factor(series(subs(S(x)=Sex(n),R0=R0ex(n),StS),t,n))

KydJInRHNiIsJCoqIyIiIiIiJEYoLCYqJiIiI0YoKUkieEdGJEYsRihGKEYoRihGKCwmRi5GKEYoISIiRjAsJkYuRihGKEYoRjBGKCIiIS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoRiw=

The degree of S(x), at x=2. We first eliminate w

res:=factor(resultant(numer(factor(subs(St=StS,S(x)=S,zt=zx/(1-2*t),x=2,algStvwzt))),algwv,w)): nops(res)

IiR1KQ==

indets(res);degree(res,S);

PCVJIlNHNiJJInRHRiRJInZHRiQ=

IiIn

Now we eliminate v

res1:=factor(resultant(res,algv,v)): nops(res1)

IiIk

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

NyU8IjwjSSJ0RzYiPCRJIlNHRiZGJQ==

res2:=op(3,res1): nops(%)

IiV2Tg==

degree(res2,S)

IiNz

n:=18:series(subs(S=subs(x=2,Sex(n)),res2),t,n);

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIz0=

algSof2:=res2:

<Text-field style="Heading 2" layout="Heading 2">Parametrization of algStvwzt</Text-field>

We try to recycle the parametrization already obtained for the other variant oif this model

ztP0:=(4*v^12+8*P0*v^10+8*v^11+4*P0^2*v^8+12*P0*v^9+25*v^10+4*P0^2*v^7+32*P0*v^8+49*v^9+12*P0^2*v^6+52*P0*v^7+53*v^8+4*P0^3*v^4+16*P0^2*v^5+33*P0*v^6+7*v^7+4*P0^3*v^3+7*P0^2*v^4-37*P0*v^5-59*v^6-12*P0^2*v^3-86*P0*v^4-61*v^5-4*P0^3*v-11*P0^2*v^2-54*P0*v^3-23*v^4-P0^3-2*P0^2*v-13*P0*v^2-3*v^3-P0*v)/((v^4+8*v^3+6*v^2+2*v+1)*(4*v^8+8*P0*v^6+8*v^7+4*P0^2*v^4+12*P0*v^5+5*v^6+4*P0^2*v^3+4*P0*v^4-3*v^5-8*P0*v^3-8*v^4-4*P0^2*v-6*P0*v^2-5*v^3-P0^2-P0*v-v^2));

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

P0paramser := t/x+(x^2+4)*t^2/x+(x^4+4*x^3-x^2+20*x)*t^3/x^2+(2*x^6+7*x^5+28*x^4-7*x^3+114*x^2)*t^4/x^3+(4*x^8+16*x^7+52*x^6+160*x^5-56*x^4+697*x^3)*t^5/x^4+(9*x^10+41*x^9+143*x^8+356*x^7+1058*x^6-397*x^5+4472*x^4)*t^6/x^5+(21*x^12+108*x^11+402*x^10+1066*x^9+2471*x^8+6823*x^7-2894*x^6+29644*x^5)*t^7/x^6+(51*x^14+294*x^13+1182*x^12+3371*x^11+8105*x^10+17108*x^9+46157*x^8-20773*x^7+201416*x^6)*t^8/x^7+(127*x^16+812*x^15+3513*x^14+10828*x^13+27447*x^12+59395*x^11+119536*x^10+314143*x^9-150623*x^8+1394371*x^7)*t^9/x^8+(323*x^18+2272*x^17+10552*x^16+35064*x^15+94814*x^14+215320*x^13+437370*x^12+840764*x^11+2183471*x^10-1093420*x^9+9798590*x^8)*t^10/x^9+(835*x^20+6412*x^19+31843*x^18+113556*x^17+327438*x^16+789555*x^15+1666134*x^14+3197244*x^13+5959785*x^12+15324589*x^11-7986035*x^10+69694888*x^9)*t^11/x^10+(2188*x^22+18223*x^21+96428*x^20+367415*x^19+1127541*x^18+2889558*x^17+6429524*x^16+12738556*x^15+23437113*x^14+42540465*x^13+108862718*x^12-58554217*x^11+500728130*x^10)*t^12/x^11+O(t^13):

res:=numer(factor(subs(zt=ztP0, algSevzt))): nops(%);

IiIj

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

NyQ8JUkjUDBHNiJJI1NlR0YlSSJ2R0YlRiM=

NyQiI0YiJDQj

NyQiIiIiIiM=

Let us choose the correct factor

n:=3: normal(series(subs(Se=Seser,P0=P0paramser,v=vser(n),op(1,res)),t,n)); normal(series(subs(Se=Seser,P0=P0paramser,v=vser(n),op(2,res)),t,n))

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiQ=

KydJInRHNiIsJCosIiM9IiIiSSJ4R0YkISIiLCoqJiIiI0YoKUYpIiIkRihGKComRi9GKClGKUYtRihGKComRi9GKEYpRihGKEYoRihGKCwmRilGKEYoRipGKiwmRilGKEYoRihGKkYoRi0tSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGKEYv

SevP0:=factor(solve(op(1,res),Se))

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

res:=numer(factor(subs(zt=ztP0, algSfvzt))): nops(%);

IiIj

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

NyQ8JUkjUDBHNiJJI1NmR0YlSSJ2R0YlRiM=

NyQiIzwiJDwj

NyQiIiIiIiM=

n:=3: normal(series(subs(Sf=Sfser,P0=P0paramser,v=vser(n),op(1,res)),t,n)); normal(series(subs(Sf=Sfser,P0=P0paramser,v=vser(n),op(2,res)),t,n))

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIiQ=

KydJInRHNiIsJComIiIjIiIiLCYqJClJInhHRiRGJ0YoRihGKCEiIkYtRi1GJy1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YoIiIk

SfvP0:=factor(solve(op(1,res),Sf))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZTZnZQMEdGKCwkKiojIiIiIiIjRjFJInZHRihGMSwwKiZGMkYxKUYzIiImRjFGMSooRjJGMUkjUDBHRihGMSlGMyIiJEYxRjEqJClGMyIiJUYxRjEqJkYyRjFGOkYxISIiKihGPkYxRjlGMUYzRjFGQCokKUYzRjJGMUZARjlGQEYxLDYqJkY+RjEpRjMiIilGMUZAKiZGR0YxKUYzIiIoRjFGQCooRj5GMSlGOUYyRjFGPUYxRjEqJiIjNkYxKUYzIiInRjFGQCooRj5GMUZMRjFGOkYxRjEqJiIjOUYxRj1GMUYxKihGPkYxRkxGMUYzRjFGQComRkdGMUY6RjFGMSokRkxGMUZARkJGMUZARjE3I0Yu

Let us check again the two series together

n:=10: factor(series(subs(P0=P0paramser,v=vser(n),w=wser(n),SevP0+w*SfvP0-subs(S(x)=Sex(n),StS)),t,n))

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIik=

StP0:=SevP0+w*SfvP0

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

factor(SevP0+1/3)

LCQqLCMiIiIiIiNGJUkidkc2IkYlLCYqJkYmRiVGJ0YlRiVGJUYlRiUsNiomIiIlRiUpRiciIidGJUYlKihGLUYlSSNQMEdGKEYlKUYnRi1GJUYlKiZGL0YlKUYnIiImRiVGJSooRi1GJUYxRiUpRiciIiRGJUYlKiZGNUYlRjJGJUYlKiZGLUYlRjdGJSEiIiooRi1GJUYxRiVGJ0YlRjsqJiIiKkYlKUYnRiZGJUY7RjFGO0YqRjtGJSw2KiZGLUYlKUYnIiIpRiVGOyomRkNGJSlGJyIiKEYlRjsqKEYtRiUpRjFGJkYlRjJGJUYlKiYiIzZGJUYuRiVGOyooRi1GJUZIRiVGN0YlRiUqJiIjOUYlRjJGJUYlKihGLUYlRkhGJUYnRiVGOyomRkNGJUY3RiVGJSokRkhGJUY7KiRGP0YlRiVGO0Y7

StP0:=w*v*(2*v^5 + 2*P0*v^3 + v^4 - 2*v^3 - 4*P0*v - v^2 - P0)/(2*(-4*v^8 - 8*v^7 + 4*P0^2*v^4 - 11*v^6 + 4*P0^2*v^3 + 14*v^4 - 4*P0^2*v + 8*v^3 - P0^2 + v^2)) - (16*v^8 + 24*P0*v^6 + 32*v^7 + 8*P0^2*v^4 + 36*P0*v^5 + 26*v^6 + 8*P0^2*v^3 + 12*P0*v^4 - 9*v^5 - 24*P0*v^3 - 38*v^4 - 8*P0^2*v - 18*P0*v^2 - 23*v^3 - 2*P0^2 - 3*P0*v - 4*v^2)/(6*(-4*v^8 - 8*v^7 + 4*P0^2*v^4 - 11*v^6 + 4*P0^2*v^3 + 14*v^4 - 4*P0^2*v + 8*v^3 - P0^2 + v^2))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVTdFAwR0YoLCYqKkkid0dGKCIiIkkidkdGKEYxLDAqJiIiI0YxKUYyIiImRjFGMSooRjVGMUkjUDBHRihGMSlGMiIiJEYxRjEqJClGMiIiJUYxRjEqJkY1RjFGOkYxISIiKihGPkYxRjlGMUYyRjFGQCokKUYyRjVGMUZARjlGQEYxLDYqJiIiKUYxKUYyRkZGMUZAKiYiIztGMSlGMiIiKEYxRkAqKEZGRjEpRjlGNUYxRj1GMUYxKiYiI0FGMSlGMiIiJ0YxRkAqKEZGRjFGTUYxRjpGMUYxKiYiI0dGMUY9RjFGMSooRkZGMUZNRjFGMkYxRkAqJkZJRjFGOkYxRjEqJkY1RjFGTUYxRkAqJkY1RjFGQ0YxRjFGQEYxKiYsRComRklGMUZHRjFGMSooIiNDRjFGOUYxRlBGMUYxKiYiI0tGMUZKRjFGMUZMRjEqKCIjT0YxRjlGMUY2RjFGMSomIiNFRjFGUEYxRjFGUkYxKigiIzdGMUY5RjFGPUYxRjEqJiIiKkYxRjZGMUZAKihGZ25GMUY5RjFGOkYxRkAqJiIjUUYxRj1GMUZARlVGQCooIiM9RjFGOUYxRkNGMUZAKiYiI0JGMUY6RjFGQEZXRkAqKEY7RjFGOUYxRjJGMUZAKiZGPkYxRkNGMUZARjEsNiomRmduRjFGR0YxRkAqJiIjW0YxRkpGMUZAKihGZ25GMUZNRjFGPUYxRjEqJiIjbUYxRlBGMUZAKihGZ25GMUZNRjFGOkYxRjEqJiIjJSlGMUY9RjFGMSooRmduRjFGTUYxRjJGMUZAKiZGXnBGMUY6RjFGMSomRlFGMUZNRjFGQComRlFGMUZDRjFGMUZARkA3I0Yu

<Text-field style="Heading 2" layout="Heading 2">Back to R(x)</Text-field>

The series Sh and Rh

eqRS := (S(x)-2*S(1/x)-1/(x^2+x+1))*sqrt(De)-(3*(x^2+x+1))*R(x)/x-3*R1+3*R0/x-(t*x^2+t-x)/(x*(x^2+x+1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcVJTR0YoLCwqJiwoLUkiU0dGKDYjSSJ4R0YoIiIiKiYiIiNGNS1GMjYjKiRGNCEiIkY1RjsqJCwoKiQpRjRGN0Y1RjVGNEY1RjVGNUY7RjtGNSlJI0RlR0YoI0Y1RjdGNUY1KioiIiRGNUY9RjUtSSJSR0YoRjNGNUY0RjtGOyomRkRGNUkjUjFHRihGNUY7KihGREY1SSNSMEdGKEY1RjRGO0Y1KigsKComSSJ0R0YoRjVGP0Y1RjVGTkY1RjRGO0Y1RjRGO0Y9RjtGOzcjRi4=

In terms of St

collect(subs(isolate(St(x)-StS,S(x)),isolate(St(1/x)-subs(x=1/x,StS),S(1/x)), eqRS),[De,R,R1,R0],factor);

LCwqLCwmSSJ4RzYiIiIiRichIiJGJywmRiVGJ0YnRidGJywmLUkjU3RHRiY2I0YlRicqJiIiI0YnLUYsNiMqJEYlRihGJ0YnRicsKCokKUYlRi9GJ0YnRiVGJ0YnRidGKClJI0RlR0YmI0YnRi9GJ0YnKioiIiRGJ0YzRictSSJSR0YmRi1GJ0YlRihGKComRjpGJ0kjUjFHRiZGJ0YoKihGOkYnSSNSMEdGJkYnRiVGKEYnKigsKComSSJ0R0YmRidGNUYnRidGREYnRiVGKEYnRiVGKEYzRihGKA==

RhR:=-subs(De=0,eqRS);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRSaFJHRigsKioqIiIkIiIiLCgqJClJInhHRigiIiNGMUYxRjVGMUYxRjFGMS1JIlJHRig2I0Y1RjFGNSEiIkYxKiZGMEYxSSNSMUdGKEYxRjEqKEYwRjFJI1IwR0YoRjFGNUY6RjoqKCwoKiZJInRHRihGMUY0RjFGMUZCRjFGNUY6RjFGNUY6RjJGOkYxNyNGLg==

The monic equation satisfied by St with coefficients in Q(zt,v,w)

POL:=collect(algStvwzt/lcoeff(algStvwzt,St),St,factor): coeff(%,St,2); indets(POL)

IiIh

PCZJI1N0RzYiSSJ2R0YkSSJ3R0YkSSN6dEdGJA==

A property of the roots of a cubic depressed equation (same as before)

Here is an expression for sqrt(delta)*(s2-s3), where s=s1

exprsddiff := -6*p*s^2-4*p^2+9*q*s

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SStleHByc2RkaWZmR0YoLCgqKCIiJyIiIkkicEdGKEYxKUkic0dGKCIiI0YxISIiKiYiIiVGMSlGMkY1RjFGNiooIiIqRjFJInFHRihGMUY0RjFGMTcjRi4=

IiIh

Now we compute the discriminant of POL

disc:=factor((subs(RootOf(algwv,w)=w,factor(subs(w=RootOf(algwv,w),discrim(POL,St)),{RootOf(algwv,w)}))));

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We want to form the part of disc that is not a square -- and the part that IS a square.

numdisc:=numer(disc); nops(%)

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IiIk

op(2,numdisc)

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

Let us factor this over Q(t,v)

ztyt := yt+v*(v^2+v+1)*(v^3-3*v-1)^3/((4*v^4+4*v^3-4*v-1)*(v^4+8*v^3+6*v^2+2*v+1)^2*yt)

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factor(subs(zt=ztyt,op(2,numdisc)))

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

disc1:=factor(disc/op(2,numdisc)*%)

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In the denominator there is one term that is not a square

denomdisc:=denom(disc1);

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

sfdisc:=op(1,op(2,denomdisc))

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

Let us factor this as well

sfdiscyt:=factor(subs(zt=ztyt,%))

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Now consider the other discriminant, since we want to form the ratio Delta/disc

Delta

LCYqJCksJiIiIkYmKiZJInRHNiJGJiwmSSJ4R0YpRiYqJEYrISIiRiZGJkYtIiIjRiZGJiooIiIlRiYpRihGLkYmKSwoRitGJkYsRiZGJkYmRi5GJkYt

In terms of y and then yt

Deltay:=(1-t*(y-1))^2-4*t^2*y^2; Deltayt:=collect(subs(y=(1-2*t)*yt/t,Deltay),yt,factor); factor(%)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdEZWx0YXlHRigsJiokKSwmIiIiRjIqJkkidEdGKEYyLCZJInlHRihGMkYyISIiRjJGNyIiI0YyRjIqKCIiJUYyKUY0RjhGMilGNkY4RjJGNzcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShEZWx0YXl0R0YoLCgqKCIiJCIiIiksJiomIiIjRjFJInRHRihGMUYxRjEhIiJGNUYxKUkjeXRHRihGNUYxRjcqKkY1RjEsJkY2RjFGMUYxRjFGM0YxRjlGMUYxKiQpRjtGNUYxRjE3I0Yu

LCQqJiwqKigiIiciIiJJInRHNiJGJ0kjeXRHRilGJ0YnRihGJyomIiIkRidGKkYnISIiRidGJ0YnLCoqKCIiI0YnRihGJ0YqRidGJ0YoRi1GKkYtRidGLUYnRi0=

ytrootDelta:=[solve(Deltayt,yt)]

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSx5dHJvb3REZWx0YUdGKDckLCQqKCMiIiIiIiRGMiwmSSJ0R0YoRjJGMkYyRjIsJiomIiIjRjJGNUYyRjJGMiEiIkY5RjkqJkY0RjJGNkY5NyNGLg==

We expect to find these roots also in the bigger discriminant, and indeed

factor(rem(numer(factor(subs(yt=-(t+1)/(3*(2*t-1)),sfdiscyt))),algv,t))

IiIh

factor(rem(numer(factor(subs(yt=(t+1)/(2*t-1),sfdiscyt))),algv,t))

IiIh

We thus want to factor sfdiscyt further, using t and v. We see indeed 4 linear factors

(subs(troot=t,evala(Factor(numer(sfdiscyt),RootOf(algv,t)))))

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

factor(op(2,%))

KiYpLCoqJiIiJSIiIilJInZHNiJGJkYnRicqJkYmRicpRikiIiRGJ0YnKiZGJkYnRilGJyEiIkYnRi8iIiNGJyksLCokRihGJ0YnKiYiIilGJ0YsRidGJyomIiInRicpRilGMEYnRicqJkYwRidGKUYnRidGJ0YnRiZGJw==

rootsden:=solve(%%,yt)

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What are really these roots? We expect to see functions of t only

factor(resultant(numer(r1-rootsden[1]),algv,v))

LCQqKCI+bjUiW3cvWHRRPDQnUUlUIiIiIiksJiomIiIjRiVJInRHNiJGJUYlRiUhIiIiIzdGJSksKCooRilGJUkjcjFHRitGJUYqRiVGJUYxRiwqJiIiJEYlRipGJUYlRi1GJUYs

r1sol:=op(1,[solve(%,r1)])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyMXNvbEdGKCwkKigiIiQiIiJJInRHRihGMSwmKiYiIiNGMUYyRjFGMUYxISIiRjZGNjcjRi4=

factor(resultant(numer(r2-rootsden[2]),algv,v))

LCQqKCJDWnZEZWZyWVIhW3N6XF5tJTR2IiIiKSwmKiYiIiNGJUkidEc2IkYlRiVGJSEiIiIjN0YlKSwoKihGKUYlSSNyMkdGK0YlRipGJUYlRjFGLEYqRixGLUYlRiw=

r2sol:=op(1,[solve(%,r2)])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyMnNvbEdGKComSSJ0R0YoIiIiLCYqJiIiI0YwRi9GMEYwRjAhIiJGNDcjRi4=

factor(resultant(numer(r3-rootsden[3]),algv,v))

LCQqKCI+bjUiW3cvWHRRPDQnUUlUIiIiIiksJiomIiIjRiVJInRHNiJGJUYlRiUhIiIiIzdGJSksKiooIiInRiVJI3IzR0YrRiVGKkYlRiUqJiIiJEYlRjJGJUYsRipGJUYlRiVGLUYlRiw=

r3sol:=op(1,[solve(%,r3)])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyM3NvbEdGKCwkKigjIiIiIiIkRjEsJkkidEdGKEYxRjFGMUYxLCYqJiIiI0YxRjRGMUYxRjEhIiJGOEY4NyNGLg==

factor(resultant(numer(r4-rootsden[4]),algv,v))

LCQqKCI+bjUiW3cvWHRRPDQnUUlUIiIiIiksJiomIiIjRiVJInRHNiJGJUYlRiUhIiIiIzdGJSksKiooRilGJUkjcjRHRitGJUYqRiVGJUYxRixGKkYsRiVGLEYtRiVGLA==

r4sol:=op(1,[solve(%,r4)])

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyNHNvbEdGKComLCZJInRHRigiIiJGMUYxRjEsJiomIiIjRjFGMEYxRjFGMSEiIkY1NyNGLg==

factor(lcoeff(sfdiscyt,yt))

LCQqKCIiJCIiIiwqKiYiIiVGJSlJInZHNiJGKEYlRiUqJkYoRiUpRipGJEYlRiUqJkYoRiVGKkYlISIiRiVGL0YlKSwsKiRGKUYlRiUqJiIiKUYlRi1GJUYlKiYiIidGJSlGKiIiI0YlRiUqJkY4RiVGKkYlRiVGJUYlRjhGJUYl

We can thus rewrite the square free part of the discriminant as follows

factor(lcoeff(sfdiscyt,yt))

LCQqKCIiJCIiIiwqKiYiIiVGJSlJInZHNiJGKEYlRiUqJkYoRiUpRipGJEYlRiUqJkYoRiVGKkYlISIiRiVGL0YlKSwsKiRGKUYlRiUqJiIiKUYlRi1GJUYlKiYiIidGJSlGKiIiI0YlRiUqJkY4RiVGKkYlRiVGJUYlRjhGJUYl

sfdisc1yt:=%*(yt-r1sol)*(yt-r2sol)*(yt-r3sol)*(yt-r4sol)/yt^2

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Let us combine the two roots that do not come from Delta

pro:=factor(subs(yt=t*y/(1-2*t),y=1+x+1/x,(yt-r1sol)*(yt-r2sol)))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRwcm9HRigqLClJInRHRigiIiMiIiIpLCZJInhHRihGMkYyISIiRjFGMiksJkY1RjJGMkYyRjFGMilGNUYxRjYpLCYqJkYxRjJGMEYyRjJGMkY2RjFGNjcjRi4=

sfdisc2yt:=factor(sfdisc1yt/Deltayt/(yt-r1sol)/(yt-r2sol)*pro*DELTA)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpzZmRpc2MyeXRHRigsJCo0SSZERUxUQUdGKCIiIiksJkkieEdGKEYxRjFGMSIiI0YxKSwmRjRGMUYxISIiRjVGMSlJInRHRihGNUYxLCoqJiIiJUYxKUkidkdGKEY9RjFGMSomRj1GMSlGPyIiJEYxRjEqJkY9RjFGP0YxRjhGMUY4RjEpLCwqJEY+RjFGMSomIiIpRjFGQUYxRjEqJiIiJ0YxKUY/RjVGMUYxKiZGNUYxRj9GMUYxRjFGMUY1RjEpRjRGNUY4KUkjeXRHRihGNUY4KSwmKiZGNUYxRjpGMUYxRjFGOEY9RjhGODcjRi4=

Recall that sfdisc was cubed

disc1;sfdisc;

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

discfinal:=factor(disc1*sfdisc^3/sfdisc2yt^3)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSpkaXNjZmluYWxHRigsJCo6IyIiIiIiJUYxKSxeciooIiIpRjEpSSJ2R0YoIiM3RjEpSSN6dEdGKCIiI0YxRjEqKEYyRjFGN0YxRjtGMUYxKigiJE8iRjEpRjgiIzZGMUY6RjFGMSomRjlGMUY3RjEhIiIqKCIjb0YxRkBGMUY7RjFGMSooIiRPKEYxKUY4IiM1RjFGOkYxRjEqJiIjV0YxRkBGMUZDKigiJG8kRjFGSEYxRjtGMUYxKigiJSs5RjEpRjgiIipGMUY6RjFGMSomIiMjKkYxRkhGMUZDKigiJCsoRjFGUEYxRjtGMUYxKioiIiRGMSlGOEY2RjFJIndHRihGMUY7RjFGMSooIiVJN0YxRlhGMUY6RjFGMSomIiQ7IkYxRlBGMUZDKiZGWEYxRllGMUYxKigiJDonRjFGWEYxRjtGMUYxKioiI1tGMSlGOCIiKEYxRllGMUY7RjFGMSooIiRTI0YxRl1vRjFGOkYxRjEqJiIjJCpGMUZYRjFGQyooIiM7RjFGXW9GMUZZRjFGMSooIiQ/IkYxRl1vRjFGO0YxRjEqKiIkayNGMSlGOCIiJ0YxRllGMUY7RjFGMSooIiQvJkYxRmlvRjFGOkYxRkMqJkZcb0YxRl1vRjFGQyooIiMpKUYxRmlvRjFGWUYxRjEqKCIkXyNGMUZpb0YxRjtGMUZDKioiJDEnRjEpRjgiIiZGMUZZRjFGO0YxRjEqKCIkKydGMUZkcEYxRjpGMUZDKiYiIyUpRjFGaW9GMUZDKigiJC0jRjFGZHBGMUZZRjFGMSooIiQrJEYxRmRwRjFGO0YxRkMqKiIkcSZGMSlGOEYyRjFGWUYxRjtGMUYxKigiJD8lRjFGYHFGMUY6RjFGQyomIiR3I0YxRmRwRjFGQyooIiQhPkYxRmBxRjFGWUYxRjEqKCIkNSNGMUZgcUYxRjtGMUZDKipGXXFGMSlGOEZXRjFGWUYxRjtGMUYxKigiJCsjRjFGanFGMUY6RjFGQyomIiRRJUYxRmBxRjFGQyooIiQrIkYxRmpxRjFGWUYxRjEqKEZgckYxRmpxRjFGO0YxRkMqKkZmb0YxKUY4RjxGMUZZRjFGO0YxRjEqKCIja0YxRmNyRjFGOkYxRkMqJkZdcUYxRmpxRjFGQyooIiNTRjFGY3JGMUZZRjFGMSooIiNLRjFGY3JGMUY7RjFGQyoqIiNJRjFGOEYxRllGMUY7RjFGMSooRmRvRjFGOEYxRjpGMUZDKiZGYHJGMUZjckYxRkMqKEZJRjFGOEYxRllGMUYxKihGNkYxRjhGMUY7RjFGQyooRldGMUZZRjFGO0YxRjEqJkY8RjFGOkYxRkMqJkZkb0YxRjhGMUZDRllGMUY7RkNGMUZDRjxGMSksVCooRjJGMUY3RjEpSSN5dEdGKEY8RjFGMSooRkVGMUZARjFGZ3NGMUYxKiRGN0YxRkMqKEZNRjFGSEYxRmdzRjFGMSokRkBGMUZDKihGVUYxRlBGMUZnc0YxRjEqJkY2RjFGSEYxRjEqKEZqbkYxRlhGMUZnc0YxRjEqJkY5RjFGUEYxRjEqKEZmb0YxRl1vRjFGZ3NGMUYxKiYiIzpGMUZYRjFGQyooRmFwRjFGaW9GMUZnc0YxRkMqJiIjVUYxRl1vRjFGQyooRl1xRjFGZHBGMUZnc0YxRkMqJiIjQEYxRmlvRjFGQyooRmhxRjFGYHFGMUZnc0YxRkMqJiIjTEYxRmRwRjFGMSooRmByRjFGanFGMUZnc0YxRkMqJiIjZ0YxRmBxRjFGMSooRmpyRjFGY3JGMUZnc0YxRkMqJiIjUEYxRmpxRjFGMSooRjZGMUY4RjFGZ3NGMUZDKiZGSUYxRmNyRjFGMSokRmdzRjFGQ0Y4RjFGPEYxKSwqKiZGMkYxRmBxRjFGMSomRjJGMUZqcUYxRjEqJkYyRjFGOEYxRkNGMUZDRjJGQyksLCokRmBxRjFGMSomRjZGMUZqcUYxRjEqJkZqb0YxRmNyRjFGMSomRjxGMUY4RjFGMUYxRjFGNkZDKUZoc0YyRjEpSSZERUxUQUdGKEZXRkMpLCZJInhHRihGMUYxRjFGam9GQyksJkZmdkYxRjFGQ0Zqb0ZDKUkidEdGKEZqb0ZDKUZmdkZqb0YxKSwmKiZGPEYxRmp2RjFGMUYxRkNGOUYxRjE3I0Yu

Let us check this

n:=10: normal(series(subs(DELTA=Delta, zt=zx/(1-2*t),yt= (x+1/x+1)*t/(1-2*t), v=vser(n),discfinal-disc),t,n))

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIik=

The following ratio of discriminants is now a square in Q(t,x,w)

discratio:=factor(DELTA/discfinal): sqdiscratio:=factor(simplify(sqrt(%),symbolic))

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

Can we still simplify some pieces?

denom(sqdiscratio)

KiwsXnIqKCIiKSIiIilJInZHNiIiIzdGJilJI3p0R0YpIiIjRiZGJiooIiIlRiZGJ0YmRixGJkYmKigiJE8iRiYpRigiIzZGJkYrRiZGJiomRipGJkYnRiYhIiIqKCIjb0YmRjJGJkYsRiZGJiooIiRPKEYmKUYoIiM1RiZGK0YmRiYqJiIjV0YmRjJGJkY1KigiJG8kRiZGOkYmRixGJkYmKigiJSs5RiYpRigiIipGJkYrRiZGJiomIiMjKkYmRjpGJkY1KigiJCsoRiZGQkYmRixGJkYmKioiIiRGJilGKEYlRiZJIndHRilGJkYsRiZGJiooIiVJN0YmRkpGJkYrRiZGJiomIiQ7IkYmRkJGJkY1KiZGSkYmRktGJkYmKigiJDonRiZGSkYmRixGJkYmKioiI1tGJilGKCIiKEYmRktGJkYsRiZGJiooIiRTI0YmRlVGJkYrRiZGJiomIiMkKkYmRkpGJkY1KigiIztGJkZVRiZGS0YmRiYqKCIkPyJGJkZVRiZGLEYmRiYqKiIkayNGJilGKCIiJ0YmRktGJkYsRiZGJiooIiQvJkYmRltvRiZGK0YmRjUqJkZURiZGVUYmRjUqKCIjKSlGJkZbb0YmRktGJkYmKigiJF8jRiZGW29GJkYsRiZGNSoqIiQxJ0YmKUYoIiImRiZGS0YmRixGJkYmKigiJCsnRiZGZm9GJkYrRiZGNSomIiMlKUYmRltvRiZGNSooIiQtI0YmRmZvRiZGS0YmRiYqKCIkKyRGJkZmb0YmRixGJkY1KioiJHEmRiYpRihGL0YmRktGJkYsRiZGJiooIiQ/JUYmRmJwRiZGK0YmRjUqJiIkdyNGJkZmb0YmRjUqKCIkIT5GJkZicEYmRktGJkYmKigiJDUjRiZGYnBGJkYsRiZGNSoqRl9wRiYpRihGSUYmRktGJkYsRiZGJiooIiQrI0YmRlxxRiZGK0YmRjUqJiIkUSVGJkZicEYmRjUqKCIkKyJGJkZccUYmRktGJkYmKihGYnFGJkZccUYmRixGJkY1KipGaG5GJilGKEYtRiZGS0YmRixGJkYmKigiI2tGJkZlcUYmRitGJkY1KiZGX3BGJkZccUYmRjUqKCIjU0YmRmVxRiZGS0YmRiYqKCIjS0YmRmVxRiZGLEYmRjUqKiIjSUYmRihGJkZLRiZGLEYmRiYqKEZmbkYmRihGJkYrRiZGNSomRmJxRiZGZXFGJkY1KihGO0YmRihGJkZLRiZGJiooRiVGJkYoRiZGLEYmRjUqKEZJRiZGS0YmRixGJkYmKiZGLUYmRitGJkY1KiZGZm5GJkYoRiZGNUZLRiZGLEY1RiZGNUYmLFQqKEYvRiZGJ0YmKUkjeXRHRilGLUYmRiYqKEY3RiZGMkYmRmhyRiZGJiokRidGJkY1KihGP0YmRjpGJkZockYmRiYqJEYyRiZGNSooRkdGJkZCRiZGaHJGJkYmKiZGJUYmRjpGJkYmKihGUkYmRkpGJkZockYmRiYqJkYqRiZGQkYmRiYqKEZobkYmRlVGJkZockYmRiYqJiIjOkYmRkpGJkY1KihGY29GJkZbb0YmRmhyRiZGNSomIiNVRiZGVUYmRjUqKEZfcEYmRmZvRiZGaHJGJkY1KiYiI0BGJkZbb0YmRjUqKEZqcEYmRmJwRiZGaHJGJkY1KiYiI0xGJkZmb0YmRiYqKEZicUYmRlxxRiZGaHJGJkY1KiYiI2dGJkZicEYmRiYqKEZcckYmRmVxRiZGaHJGJkY1KiYiI1BGJkZccUYmRiYqKEYlRiZGKEYmRmhyRiZGNSomRjtGJkZlcUYmRiYqJEZockYmRjVGKEYmRiZGaHJGJilJInhHRilGSUYmKSwmKiZGLUYmSSJ0R0YpRiZGJkYmRjVGXG9GJg==

collect(op(2,denom(sqdiscratio)),yt,factor);

LCYqKCwqKiYiIiUiIiIpSSJ2RzYiRiZGJ0YnKiZGJkYnKUYpIiIkRidGJyomRiZGJ0YpRichIiJGJ0YvRicpLCwqJEYoRidGJyomIiIpRidGLEYnRicqJiIiJ0YnKUYpIiIjRidGJyomRjhGJ0YpRidGJ0YnRidGOEYnKUkjeXRHRipGOEYnRicqKCwoKiRGN0YnRidGKUYnRidGJ0YnKSwoKiRGLEYnRicqJkYtRidGKUYnRi9GJ0YvRi1GJ0YpRidGLw==

We recognize the factor Rat, which was in fact a fraction in t

Rat:= factor(evala(Normal(subs(t=troot,t*(1+t)/(1-2*t)^2))))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRSYXRHRigqLEkidkdGKCIiIiwoKiQpRi8iIiNGMEYwRi9GMEYwRjBGMCksKCokKUYvIiIkRjBGMComRjlGMEYvRjAhIiJGMEY7RjlGMCwqKiYiIiVGMClGL0Y+RjBGMComRj5GMEY4RjBGMComRj5GMEYvRjBGO0YwRjtGOyksLCokRj9GMEYwKiYiIilGMEY4RjBGMComIiInRjBGM0YwRjAqJkY0RjBGL0YwRjBGMEYwRjRGOzcjRi4=

sqdiscratio1:=factor(simplify(factor(subs(yt=t*y/(1-2*t),sqdiscratio*op(2,denom(sqdiscratio))/coeff(op(2,denom(sqdiscratio)),yt,2)/(yt^2-t*(1+t)/(1-2*t)^2)))));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS1zcWRpc2NyYXRpbzFHRigsJCo4IiIjIiIiKUkmREVMVEFHRihGMEYxLCoqJiIiJUYxKUkidkdGKEY2RjFGMSomRjZGMSlGOCIiJEYxRjEqJkY2RjFGOEYxISIiRjFGPUYxKSwsKiRGN0YxRjEqJiIiKUYxRjpGMUYxKiYiIidGMSlGOEYwRjFGMSomRjBGMUY4RjFGMUYxRjFGMEYxKSwmSSJ4R0YoRjFGMUYxRjtGMSksJkZJRjFGMUY9RjtGMSwoKiZJInRHRihGMSlJInlHRihGMEYxRjFGTkY9RjFGPUY9KSwmKiZGMEYxRk5GMUYxRjFGPUYwRj0pRklGO0Y9Rk9GPSxeciooRkJGMSlGOCIjN0YxKUkjenRHRihGMEYxRjEqKEY2RjFGV0YxRlpGMUYxKigiJE8iRjEpRjgiIzZGMUZZRjFGMSomRlhGMUZXRjFGPSooIiNvRjFGaG5GMUZaRjFGMSooIiRPKEYxKUY4IiM1RjFGWUYxRjEqJiIjV0YxRmhuRjFGPSooIiRvJEYxRl9vRjFGWkYxRjEqKCIlKzlGMSlGOCIiKkYxRllGMUYxKiYiIyMqRjFGX29GMUY9KigiJCsoRjFGZ29GMUZaRjFGMSoqRjtGMSlGOEZCRjFJIndHRihGMUZaRjFGMSooIiVJN0YxRl5wRjFGWUYxRjEqJiIkOyJGMUZnb0YxRj0qJkZecEYxRl9wRjFGMSooIiQ6J0YxRl5wRjFGWkYxRjEqKiIjW0YxKUY4IiIoRjFGX3BGMUZaRjFGMSooIiRTI0YxRmlwRjFGWUYxRjEqJiIjJCpGMUZecEYxRj0qKCIjO0YxRmlwRjFGX3BGMUYxKigiJD8iRjFGaXBGMUZaRjFGMSoqIiRrI0YxKUY4RkRGMUZfcEYxRlpGMUYxKigiJC8mRjFGZXFGMUZZRjFGPSomRmhwRjFGaXBGMUY9KigiIykpRjFGZXFGMUZfcEYxRjEqKCIkXyNGMUZlcUYxRlpGMUY9KioiJDEnRjEpRjgiIiZGMUZfcEYxRlpGMUYxKigiJCsnRjFGX3JGMUZZRjFGPSomIiMlKUYxRmVxRjFGPSooIiQtI0YxRl9yRjFGX3BGMUYxKigiJCskRjFGX3JGMUZaRjFGPSoqIiRxJkYxRjdGMUZfcEYxRlpGMUYxKigiJD8lRjFGN0YxRllGMUY9KiYiJHcjRjFGX3JGMUY9KigiJCE+RjFGN0YxRl9wRjFGMSooIiQ1I0YxRjdGMUZaRjFGPSoqRmhyRjFGOkYxRl9wRjFGWkYxRjEqKCIkKyNGMUY6RjFGWUYxRj0qJiIkUSVGMUY3RjFGPSooIiQrIkYxRjpGMUZfcEYxRjEqKEZpc0YxRjpGMUZaRjFGPSoqRmJxRjFGRUYxRl9wRjFGWkYxRjEqKCIja0YxRkVGMUZZRjFGPSomRmhyRjFGOkYxRj0qKCIjU0YxRkVGMUZfcEYxRjEqKCIjS0YxRkVGMUZaRjFGPSoqIiNJRjFGOEYxRl9wRjFGWkYxRjEqKEZgcUYxRjhGMUZZRjFGPSomRmlzRjFGRUYxRj0qKEZgb0YxRjhGMUZfcEYxRjEqKEZCRjFGOEYxRlpGMUY9KihGO0YxRl9wRjFGWkYxRjEqJkYwRjFGWUYxRj0qJkZgcUYxRjhGMUY9Rl9wRjFGWkY9RjFGPUY9RjE3I0Yu

n:=10: factor(series(subs(zt=ztyt,yt=t*y/(1-2*t),y=1+x+1/x,w=wser(n), v=vser(n),sqdiscratio-sqdiscratio1),t,n+1));

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzU=

We still need to adjust the sign

n:=2: factor(series(subs(s3=-s1-s2,s1=StS,s2=subs(x=1/x,StS),S(x)=Sex(n),S(1/x)=subs(x=1/x,Sex(n)),R0=R0ex(n),sqrt(Delta)/(s1-s2)/(s1-s3)/(s2-s3)),t,n))

KylJInRHNiIsJCoqKSwmSSJ4R0YkIiIiRipGKiIiJEYqKSwmRilGKkYqISIiRitGKilGKSIiI0YuLCYqJEYvRipGKkYqRipGLkYuIiIhKihGJ0YqRixGKilGKUYrRi5GKi1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YqRjA=

n:=2: factor(series(subs(DELTA=Delta,zt=ztyt,yt=t*y/(1-2*t),y=1+x+1/x,w=wser(n), v=vser(n),sqdiscratio),t,n+1));

KylJInRHNiIqKiksJkkieEdGJCIiIkYpRikiIiRGKSksJkYoRilGKSEiIkYqRikpRigiIiNGLSwmKiRGLkYpRilGKUYpRi0iIiEsJCooRiZGKUYrRikpRihGKkYtRi1GKS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YpRi8=

sqdiscratio2:=-sqdiscratio1:

Now we can reconstruct an expression for R-hat

exprsddiff := -6*p*s^2-4*p^2+9*q*s

Let us check the expression of Rhat

n:=4: factor(series(subs(R(x)=Rex(n), R1=R1ex(n), R0=R0ex(n),RhR),t,n))

Ky1JInRHNiIsJCokLCgqJClJInhHRiQiIiMiIiJGLEYqRixGLEYsISIiRi0iIiEqKCwmRihGLEYsRixGLEYqRi1GJ0YtRiwsKEYoRixGKkYsRitGLEYrLCoqJClGKiIiJEYsRixGKEYsKiYiIiVGLEYqRixGLEY1RixGNS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YsRjc=

res:=factor((x-1/x)/y*sqdiscratio2*subs(s=St,p=coeff(POL, St,1), q=coeff(POL, St,0),exprsddiff)): indets(%);

PCpJJkRFTFRBRzYiSSNTdEdGJEkidEdGJEkidkdGJEkid0dGJEkieEdGJEkieUdGJEkjenRHRiQ=

n:=5: factor(series(subs(St=StS, S(x)=Sex(n), R0=R0ex(n),DELTA=Delta, w=wser(n),zt=z/(1-2*t), z=zx, yt=y*t/(1-2*t),y=x+1/x+1,v=vser(n),res),t,4))

exprRh:=res:

denom(exprRh);

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

sfdisc; sfdisc2yt

LCQqNEkmREVMVEFHNiIiIiIpLCZJInhHRiVGJkYmRiYiIiNGJiksJkYpRiZGJiEiIkYqRiYpSSJ0R0YlRipGJiwqKiYiIiVGJilJInZHRiVGMkYmRiYqJkYyRiYpRjQiIiRGJkYmKiZGMkYmRjRGJkYtRiZGLUYmKSwsKiRGM0YmRiYqJiIiKUYmRjZGJkYmKiYiIidGJilGNEYqRiZGJiomRipGJkY0RiZGJkYmRiZGKkYmKUYpRipGLSlJI3l0R0YlRipGLSksJiomRipGJkYvRiZGJkYmRi1GMkYtRi0=

exprRh1:=factor(exprRh*sfdisc^2/sfdisc2yt^2):

In terms of P0

ztP0

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

factor((subs(RootOf(algwv,w)=w,factor(subs(w=RootOf(algwv,w),discrim(POL,St)),{RootOf(algwv,w)}))));

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

RhP0a:=subs(RootOf(algwv,w)=w,factor(subs(zt=ztP0,St=StP0,w=RootOf(algwv,w),exprRh1),{RootOf(algwv,w)}))

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

RhP0b:=factor(subs(yt=y*t/(1-2*t),factor(RhP0a)))

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

n:=4: factor(series(subs(R(x)=Rex(n), R1=R1ex(n), R0=R0ex(n),RhR),t,n))

n:=4: factor(series(subs( y=x+1/x+1,v=vser(n),w=wser(n),P0=P0paramser,RhP0b),t,8))

<Text-field style="Heading 2" layout="Heading 2">Degree (72) and equation for R(2)</Text-field>

ztP0; zx

LCYqJkkidEc2IiIiIiwoSSJ4R0YlRiYqJEYoISIiRiZGJkYmRiZGJiomLCZGJEYmRiZGJkYmRidGKkYm

RhP0b-Rhx; RhR;R0w

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

LCoqKiIiJCIiIiwoKiQpSSJ4RzYiIiIjRiVGJUYpRiVGJUYlRiUtSSJSR0YqNiNGKUYlRikhIiJGJSomRiRGJUkjUjFHRipGJUYlKihGJEYlSSNSMEdGKkYlRilGL0YvKigsKComSSJ0R0YqRiVGKEYlRiVGN0YlRilGL0YlRilGL0YmRi9GJQ==

KiYsJiosIyIiIiIiI0YmSSJ2RzYiRiYsKComRidGJilGKCIiJEYmRiYqJiIiJUYmRihGJiEiIkYmRjBGJiwsKiQpRihGL0YmRiYqJiIiKUYmRixGJkYmKiYiIidGJilGKEYnRiZGJiomRidGJkYoRiZGJkYmRiZGMEkid0dGKUYwRiYqKCwmRjlGJkYmRiZGJkYoRiYsLComRidGJkYzRiZGJiomIiM7RiZGLEYmRiYqJiIjN0YmRjhGJkYmRi5GJkYnRiZGMEYmRiYsJkYmRiYqJkYnRiZJInRHRilGJkYwRiY=

We use the expression of Rh that we have just obtained

res:=factor(resultant(subs(y=x+1+1/x,Rhx=RhR,R(x)=R,R0=R0w,R1=R1w,x=2,numer(RhP0b-Rhx)),numer(subs(x=2,zx/(1-2*t)-ztP0)),P0)):

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

Ny08IjwjSSJ2RzYiPCNJInRHRiZGJEYkRiRGJEYkRiQ8JkkiUkdGJkYoRiVJIndHRiY8I0Yr

IiM2

res1:=op(10,res): nops(%)

IiV0UA==

degree(res1,t);degree(res1,w);

IiIn

res2:=factor(resultant(res1,algwv,w)):nops(%)

IiIk

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

IidrTzo=

KiQpLCoqJiIiJSIiIilJInZHNiJGJkYnRicqJkYmRicpRikiIiRGJ0YnKiZGJkYnRilGJyEiIkYnRi8iIiZGJw==

indets(res2);degree(res,t)

PCVJIlJHNiJJInRHRiRJInZHRiQ=

IiM3

Before we go further let us reduce the degree in t

res3:=factor(rem(res2,algv,t)):nops(%)

IiIn

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

Nyg8IjwjSSJ2RzYiRiQ8JUkiUkdGJkkidEdGJkYlRiRGJA==

res4:=op(4,res3): nops(%);degree(res4,R)

IiV1PQ==

IiIn

algRof2vt:=res4:

res5:=factor(resultant(res4,algv,v)):nops(%)

IiIm

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

Nyc8IjwjSSJ0RzYiRiRGJDwkSSJSR0YmRiU=

res6:=op(5,res5): nops(%)

IiUocCM=

degree(res6,R)

IiNz

n:=10:series(subs(R=Rex(n),x=2,v=vser(n),res6),t,n+1);

KyVJInRHNiItSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiIiIzY=

algRof2:=res6:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=