The case q=3 and nu=0 : conclusion (Section 13.1, the end)
Let us go back to the invariant equation, with its 3 unknown series
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> | eqMcat := 36*t^3*y^6*(2*y+1)*(y-1)^3*M(y)^4+2*t^2*y^4*(y-1)^2*(42*t*y^3-26*y^3+12*t*y^2-39*y^2+39*y+26)*M(y)^3-(y-1)*t*y^2*(36*y^5*M0*t^2-36*M0*t^2*y^4-2*y^5*t-32*t^2*y^5-8*y^5+120*y^4*t-38*y^4-4*t^2*y^4+25*y^3-78*t*y^3-40*t*y^2+71*y^2-25*y-25)*M(y)^2-2*(18*t^3*y^7*M0^2+18*y^7*M0*t^3-24*M0*t^3*y^6+6*y^5*M0*t^3-36*t^3*y^6*M0^2+42*y^6*M0*t^2-27*y^7*M0*t^2+18*t^3*y^5*M0^2-12*y^6*M1*t^3+6*y^7*M1*t^3-3*y^5*M0*t^2-12*M0*t^2*y^4+6*y^5*M1*t^3-2-y-8*y^4+47*y^4*t-7*t*y^3-2*t^2*y^4-16*y^5*t-14*t^2*y^5+y^5+13*y^7*t-5*y^7*t^2-2*t^3*y^7-11*t*y^2+8*y^2+21*y^6*t^2-26*t*y^6+2*y^6)*M(y)-2*t^2*y^3*(y-1)*(18*t*M0*y^2-18*M0*t*y+6*t*y^2-11*y^2-4*t*y+9*y+2)*M1-36*t^3*y^4*(y-1)^2*M0^3-2*t^2*(y-1)*y^3*(22*t*y^2-33*y^2-16*t*y+27*y+6)*M0^2-2*t*y^2*(6*t^2*y^4+13*t*y^2-7*t*y^3+5*y-6*t^2*y^3-23*y^3-4*y^4*t-2*t*y+2+16*y^4+2*t^2*y^2)*M0-2*t^3*y^4*(y-1)^2*M2+(y-1)^3*(11*t*y^3+4*y^3+t*y^2+14*y^2+14*y+4); |
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The conjectured values of the unknown series
> | algS := collect(8*t*S^3+12*t*S^2+6*t*S+t+2*S^4-S,t,factor); |
> | with(gfun): |
> | Sser:=op(2,algeqtoseries(algS,t,S,15,true)); |
> | tS:=factor(solve(algS,t)); |
Conjectured value of M(1) (now denoted M0...)
> | M0c := -(2*S+1)*(4*S^4+4*S^3+2*S^2-1)/(2*S^3-1)^2; |
Equation for M0
> | algM0:=collect(-factor(resultant(numer(M0-M0c),algS,S))/10/32,M0,factor); |
> | DeltaS:=(1+2*S)*(1+2*S+4*S^2); |
Conjectured value of the first derivative
> | M1c:=(1+2*S)*(32*S^12+32*S^11+12*S^10-32*S^9-16*S^8+18*S^7+14*S^6-28*S^5-58*S^4-49*S^3-25*S^2-7*S-1 +(1-2*S^3)*(4*S^2+2*S+1)*(1+S)^3*sqrt(Delta))/(2*S^3-1)^4/S^2; |
Conjectured value of the second derivative
> | M2c:= (2*(2*S+1)*(4*S^2+2*S+1)*(8*S^7+8*S^6+12*S^5-20*S^4-48*S^3-42*S^2-19*S-5)*(1+S)^3/(2*S^3-1)^5/S^3*Delta^(1/2)-2*(2*S+1)*(192*S^19+192*S^18+64*S^17-96*S^16+1728*S^15+6016*S^14+10080*S^13+8336*S^12-12*S^11-8724*S^10-10468*S^9-4970*S^8+1944*S^7+5310*S^6+4768*S^5+2721*S^4+1082*S^3+300*S^2+54*S+5)/S^3/(2*S^3-1)^6); |
An equation of degree 4 in M=M(y), in terms of S, delta and y
> | eqMS:=subs(M(y)=M,collect(numer(factor(subs(t=tS,M0=M0c,M1=M1c,M2=M2c,eqMcat))),[M,Delta],factor)); |
> | nops(factor(eqMS)); indets(eqMS); |
There is a unique solution that is finite when S=0 (and delta=1)
> | collect(subs(S=0,Delta=1,eqMS),M,factor); |
> | factor(%); |
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Once the correct parametrization is guessed (this requires to determine it for various values of S, using the 'parametrization' function)
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