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

>   

>    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);

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^3-39*y^2+12*t*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*t*y^5-32*t^2*y^5-8*y^5+120*y^4*t-38*y^4-4*t^2*y...
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^3-39*y^2+12*t*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*t*y^5-32*t^2*y^5-8*y^5+120*y^4*t-38*y^4-4*t^2*y...
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^3-39*y^2+12*t*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*t*y^5-32*t^2*y^5-8*y^5+120*y^4*t-38*y^4-4*t^2*y...
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^3-39*y^2+12*t*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*t*y^5-32*t^2*y^5-8*y^5+120*y^4*t-38*y^4-4*t^2*y...
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^3-39*y^2+12*t*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*t*y^5-32*t^2*y^5-8*y^5+120*y^4*t-38*y^4-4*t^2*y...
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^3-39*y^2+12*t*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*t*y^5-32*t^2*y^5-8*y^5+120*y^4*t-38*y^4-4*t^2*y...
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^3-39*y^2+12*t*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*t*y^5-32*t^2*y^5-8*y^5+120*y^4*t-38*y^4-4*t^2*y...

>   

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);

algS := (1+2*S)^3*t+S*(-1+2*S^3)

>    with(gfun):

>    Sser:=op(2,algeqtoseries(algS,t,S,15,true));

Sser := series(1*t+6*t^2+48*t^3+442*t^4+4428*t^5+46920*t^6+517424*t^7+5878560*t^8+68343600*t^9+809244016*t^10+9725962656*t^11+118345000128*t^12+1455070212096*t^13+18049788630144*t^14+225624841265664*t^...
Sser := series(1*t+6*t^2+48*t^3+442*t^4+4428*t^5+46920*t^6+517424*t^7+5878560*t^8+68343600*t^9+809244016*t^10+9725962656*t^11+118345000128*t^12+1455070212096*t^13+18049788630144*t^14+225624841265664*t^...

>    tS:=factor(solve(algS,t));

tS := -(-1+2*S^3)*S/(1+2*S)^3

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;

M0c := -(1+2*S)*(4*S^4+4*S^3+2*S^2-1)/(-1+2*S^3)^2

Equation for M0

>    algM0:=collect(-factor(resultant(numer(M0-M0c),algS,S))/10/32,M0,factor);

algM0 := -12500*M0^4*t^6+24*t^4*(1000*t-71)*M0^3-2*t^2*(39-1020*t+7216*t^2+3600*t^3)*M0^2+(-1+42*t-536*t^2+1712*t^3+9040*t^4-864*t^5)*M0-40*t+540*t^2-2720*t^3+432*t^4+1
algM0 := -12500*M0^4*t^6+24*t^4*(1000*t-71)*M0^3-2*t^2*(39-1020*t+7216*t^2+3600*t^3)*M0^2+(-1+42*t-536*t^2+1712*t^3+9040*t^4-864*t^5)*M0-40*t+540*t^2-2720*t^3+432*t^4+1

>    DeltaS:=(1+2*S)*(1+2*S+4*S^2);

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;

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)*(1+2*S+4*S^2)*(S+1)^3*Delta^(1/2))/(-1+2*S^3)^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);

M2c := 2*(1+2*S)*(1+2*S+4*S^2)*(8*S^7+8*S^6+12*S^5-20*S^4-48*S^3-42*S^2-19*S-5)*(S+1)^3/(-1+2*S^3)^5/S^3*Delta^(1/2)-2*(1+2*S)*(192*S^19+192*S^18+64*S^17-96*S^16+1728*S^15+6016*S^14+10080*S^13+8336*S^1...
M2c := 2*(1+2*S)*(1+2*S+4*S^2)*(8*S^7+8*S^6+12*S^5-20*S^4-48*S^3-42*S^2-19*S-5)*(S+1)^3/(-1+2*S^3)^5/S^3*Delta^(1/2)-2*(1+2*S)*(192*S^19+192*S^18+64*S^17-96*S^16+1728*S^15+6016*S^14+10080*S^13+8336*S^1...
M2c := 2*(1+2*S)*(1+2*S+4*S^2)*(8*S^7+8*S^6+12*S^5-20*S^4-48*S^3-42*S^2-19*S-5)*(S+1)^3/(-1+2*S^3)^5/S^3*Delta^(1/2)-2*(1+2*S)*(192*S^19+192*S^18+64*S^17-96*S^16+1728*S^15+6016*S^14+10080*S^13+8336*S^1...

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));

eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...
eqMS := -36*S^3*y^6*(2*y+1)*(y-1)^3*(-1+2*S^3)^4*M^4-2*S^2*y^4*(y-1)^2*(-1+2*S^3)^3*(114*S*y^3+84*S^4*y^3+312*S^2*y^3+208*S^3*y^3+26*y^3+468*S^2*y^2+312*S^3*y^2+39*y^2+222*S*y^2+24*S^4*y^2-468*S^2*y-31...

>    nops(factor(eqMS)); indets(eqMS);

473

{M, S, y, Delta, Delta^(1/2)}

There is a unique solution that is finite when S=0 (and delta=1)

>    collect(subs(S=0,Delta=1,eqMS),M,factor);

2*(2*y+1)*(y+2)*(y+1)*(y-1)^3*M-2*(2*y+1)*(y+2)*(y+1)*(y-1)^3

>    factor(%);

2*(2*y+1)*(y+2)*(y+1)*(y-1)^3*(-1+M)

>   

Once the correct parametrization is guessed (this requires to determine it for various values of S, using the 'parametrization' function)

>   

>   

>   

>