The case q=3 and nu=0 : asymptotics (Theorem 25)

Expression of M(1)

>    M0c;

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

>    algS;

(1+2*S)^3*t+S*(-1+2*S^3)

>    implicitplot(algS,t=0..0.1,S=0..1,numpoints=10000);

[Maple Plot]

>    factor(discrim(algS,S));

-4*(100*t^2-132*t+9)*(108*t^2-28*t+3)

>    solve(%);

7/54+2/27*I*2^(1/2), 7/54-2/27*I*2^(1/2), 33/50+6/25*6^(1/2), 33/50-6/25*6^(1/2)

>    evalf(%);

.1296296296+.1047565601*I, .1296296296-.1047565601*I, 1.247877538, .721224617e-1

The radius is 0.07...

>    tc:=33/50-6/25*6^(1/2); evalf (tc);

tc := 33/50-6/25*6^(1/2)

.721224617e-1

The value of mu

>    rationalize(1/tc);

22/3+8/3*6^(1/2)

>    factor(subs(t=tc,algS));

1/100*(50*S^2+32*S+2*S*6^(1/2)+21+6*6^(1/2))*(-2*S-2+6^(1/2))^2

>    Sc:=solve(2*S+2-6^(1/2));evalf(Sc);

Sc := -1+1/2*6^(1/2)

.224744872

>    algeqtoseries(subs(t=tc*(1-x),algS),x,S,2,true);

[series(RootOf(50*_Z^2+(32+2*6^(1/2))*_Z+6*6^(1/2)+21)+(11/304-13/304*6^(1/2)+23/456*RootOf(50*_Z^2+(32+2*6^(1/2))*_Z+6*6^(1/2)+21)-21/304*RootOf(50*_Z^2+(32+2*6^(1/2))*_Z+6*6^(1/2)+21)*6^(1/2))*x+O(x^...
[series(RootOf(50*_Z^2+(32+2*6^(1/2))*_Z+6*6^(1/2)+21)+(11/304-13/304*6^(1/2)+23/456*RootOf(50*_Z^2+(32+2*6^(1/2))*_Z+6*6^(1/2)+21)-21/304*RootOf(50*_Z^2+(32+2*6^(1/2))*_Z+6*6^(1/2)+21)*6^(1/2))*x+O(x^...
[series(RootOf(50*_Z^2+(32+2*6^(1/2))*_Z+6*6^(1/2)+21)+(11/304-13/304*6^(1/2)+23/456*RootOf(50*_Z^2+(32+2*6^(1/2))*_Z+6*6^(1/2)+21)-21/304*RootOf(50*_Z^2+(32+2*6^(1/2))*_Z+6*6^(1/2)+21)*6^(1/2))*x+O(x^...

>   

Equation for M0=M(1)

>    algM0:=collect(factor(1/320*resultant(numer(M-M0c),algS,S)),M,factor);

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

>    factor(discrim(algM0,M));

-16*t^12*(2700*t^2-404*t+23)^2*(100*t^2-132*t+9)^3*(108*t^2-28*t+3)^3*(2*t+1)^4

>    fsolve(2700*z^2-404*z+23,z,complex);

.7481481481e-1-.5404870029e-1*I, .7481481481e-1+.5404870029e-1*I

>    fsolve((100*z^2-132*z+9),z,complex);

.7212246173e-1, 1.247877538

>    fsolve(108*z^2-28*z+3,z,complex);

.1296296296-.1047565602*I, .1296296296+.1047565602*I

>    with(plots):

>    implicitplot(algM0,t=0..0.08,M=0..1.3,numpoints=10000);

[Maple Plot]

Expansion of M0 near its singularity

>    op(2,algeqtoseries(subs(t=tc*(1-x), algM0),x, M,3));

38/45+8/45*6^(1/2)+(-14/45-14/45*6^(1/2))*x+RootOf(-176-72*6^(1/2)+81*_Z^2)*x^(3/2)+O(x^(7/4))

>   

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