The case q=2 (m=4,k=1) (Section 12.1)

The Chebyshev polynomials

>    T:=proc(m) subs(cos(x)=x,expand(cos(m*x))): end:

>    T(4);

8*x^4-8*x^2+1

>    m:=4: k:=1: 2+2*cos(2*k*Pi/m);

2

>    I2:=normal(subs(C=CC,II=I1,q=2,yinv=1/y,M(1,y)=M(y),normal(sqrt(C)^m*subs(x=xsubs,T(m))))):

>    indets(I2);

{nu, t, w, y, M(y)}

>    factor(series(normal(I2),y=1,1));

series(t^4*(nu^2+2*nu-1)*(nu^2-2*nu-1)*(y-1)^(-4)+O((y-1)^(-3)),y=-(-1),-3)

>    eqinv:=subs(M(1,y)=M(y),q=2,I2-add(c[i]*I1^i,i=0..4)):

>   

>    indets(eqinv);

{M(y), y, w, t, nu, c[4], c[0], c[1], c[2], c[3]}

>   

>   

Determination of the series c_i. An  equation with a single catalytic variable and two additional unknown functions , M(1,1) et M'(1,y=1)

>   

>    eqinvM;

4*y^4*t^2*w^2*nu*(y-1)^2*(nu+1)*M(y)^3+y^2*t*w*(y-1)*(4*nu^2*y^2*t-y^2+nu^2*y^2+4*nu*y^2*t-6*y*nu-2*y*nu^2+nu^2+1+6*nu)*M(y)^2+(-2*y^3*t^2*w*nu*(nu+1)*(y-1)*M(1)+1-y+nu+4*y^4*t*w*nu-8*y^3*t*w*nu+4*y^2*...
4*y^4*t^2*w^2*nu*(y-1)^2*(nu+1)*M(y)^3+y^2*t*w*(y-1)*(4*nu^2*y^2*t-y^2+nu^2*y^2+4*nu*y^2*t-6*y*nu-2*y*nu^2+nu^2+1+6*nu)*M(y)^2+(-2*y^3*t^2*w*nu*(nu+1)*(y-1)*M(1)+1-y+nu+4*y^4*t*w*nu-8*y^3*t*w*nu+4*y^2*...
4*y^4*t^2*w^2*nu*(y-1)^2*(nu+1)*M(y)^3+y^2*t*w*(y-1)*(4*nu^2*y^2*t-y^2+nu^2*y^2+4*nu*y^2*t-6*y*nu-2*y*nu^2+nu^2+1+6*nu)*M(y)^2+(-2*y^3*t^2*w*nu*(nu+1)*(y-1)*M(1)+1-y+nu+4*y^4*t*w*nu-8*y^3*t*w*nu+4*y^2*...
4*y^4*t^2*w^2*nu*(y-1)^2*(nu+1)*M(y)^3+y^2*t*w*(y-1)*(4*nu^2*y^2*t-y^2+nu^2*y^2+4*nu*y^2*t-6*y*nu-2*y*nu^2+nu^2+1+6*nu)*M(y)^2+(-2*y^3*t^2*w*nu*(nu+1)*(y-1)*M(1)+1-y+nu+4*y^4*t*w*nu-8*y^3*t*w*nu+4*y^2*...

>   

We will now solve this equation in one catalytic variable, following the approach used by Tutte for colored planar triangulations

>    RHS:=subs(c[4]=c4sol,c[3]=c3sol,c[2]=c2sol,c[1]=c1sol,c[0]=c0sol,add(c[i]*X^i,i=0..4));

RHS := -32*w*nu*(nu+1)*(nu-1)^2*t^3*D(M)(1)-64*nu*w^2*(nu+1)*(nu-1)^2*t^3*M(1)^2+(-32*w*nu*(nu+1)*(nu-1)^2*t^3+32*w*(3*nu+1)*(nu-1)^2*t^2)*M(1)-4+4*(nu-1)^2*(-nu^2-12*w*nu+2*nu-1+4*w^2-4*w)*t^2+8*(nu-1...
RHS := -32*w*nu*(nu+1)*(nu-1)^2*t^3*D(M)(1)-64*nu*w^2*(nu+1)*(nu-1)^2*t^3*M(1)^2+(-32*w*nu*(nu+1)*(nu-1)^2*t^3+32*w*(3*nu+1)*(nu-1)^2*t^2)*M(1)-4+4*(nu-1)^2*(-nu^2-12*w*nu+2*nu-1+4*w^2-4*w)*t^2+8*(nu-1...
RHS := -32*w*nu*(nu+1)*(nu-1)^2*t^3*D(M)(1)-64*nu*w^2*(nu+1)*(nu-1)^2*t^3*M(1)^2+(-32*w*nu*(nu+1)*(nu-1)^2*t^3+32*w*(3*nu+1)*(nu-1)^2*t^2)*M(1)-4+4*(nu-1)^2*(-nu^2-12*w*nu+2*nu-1+4*w^2-4*w)*t^2+8*(nu-1...
RHS := -32*w*nu*(nu+1)*(nu-1)^2*t^3*D(M)(1)-64*nu*w^2*(nu+1)*(nu-1)^2*t^3*M(1)^2+(-32*w*nu*(nu+1)*(nu-1)^2*t^3+32*w*(3*nu+1)*(nu-1)^2*t^2)*M(1)-4+4*(nu-1)^2*(-nu^2-12*w*nu+2*nu-1+4*w^2-4*w)*t^2+8*(nu-1...

>    factor(T(4)-1);factor(T(4)+1);

8*x^2*(x-1)*(x+1)

2*(2*x^2-1)^2

The following two polynomials in X have a double root named P_+ and P_- in the paper)

>    pol1:=factor(subs(q=2,II=X, RHS-CC^2));
pol2:=factor(subs(q=2,II=X,RHS+CC^2));

pol1 := -8-32*nu^4*w*t^3*M(1)-16*t+16*X+64*w^2*t^3*M(1)^2*nu^2+32*t*nu-32*t^3*w*nu*M(1)+32*t^3*w*nu^3*M(1)-160*t^2*w*nu^2*M(1)+32*t^3*w*M(1)*nu^2+16*X^3*nu-8*X^4*nu^2+16*X*nu-8*X^2*nu^2+16*X^3*nu^2+32*...
pol1 := -8-32*nu^4*w*t^3*M(1)-16*t+16*X+64*w^2*t^3*M(1)^2*nu^2+32*t*nu-32*t^3*w*nu*M(1)+32*t^3*w*nu^3*M(1)-160*t^2*w*nu^2*M(1)+32*t^3*w*M(1)*nu^2+16*X^3*nu-8*X^4*nu^2+16*X*nu-8*X^2*nu^2+16*X^3*nu^2+32*...
pol1 := -8-32*nu^4*w*t^3*M(1)-16*t+16*X+64*w^2*t^3*M(1)^2*nu^2+32*t*nu-32*t^3*w*nu*M(1)+32*t^3*w*nu^3*M(1)-160*t^2*w*nu^2*M(1)+32*t^3*w*M(1)*nu^2+16*X^3*nu-8*X^4*nu^2+16*X*nu-8*X^2*nu^2+16*X^3*nu^2+32*...
pol1 := -8-32*nu^4*w*t^3*M(1)-16*t+16*X+64*w^2*t^3*M(1)^2*nu^2+32*t*nu-32*t^3*w*nu*M(1)+32*t^3*w*nu^3*M(1)-160*t^2*w*nu^2*M(1)+32*t^3*w*M(1)*nu^2+16*X^3*nu-8*X^4*nu^2+16*X*nu-8*X^2*nu^2+16*X^3*nu^2+32*...
pol1 := -8-32*nu^4*w*t^3*M(1)-16*t+16*X+64*w^2*t^3*M(1)^2*nu^2+32*t*nu-32*t^3*w*nu*M(1)+32*t^3*w*nu^3*M(1)-160*t^2*w*nu^2*M(1)+32*t^3*w*M(1)*nu^2+16*X^3*nu-8*X^4*nu^2+16*X*nu-8*X^2*nu^2+16*X^3*nu^2+32*...

pol2 := -2*(nu-1)^2*(16*t+16*t^2*w+32*w^2*t^3*M(1)^2*nu^2+16*t^3*w*nu*M(1)+16*t^3*w*M(1)*nu^2+4*X^2*t+4*X^3*nu-X^4*nu^2-48*t^2*w*nu*M(1)-16*X*t-16*t^2*w^2+16*t^3*w*nu*D(M)(1)+16*t^2*w*nu-8*X^2*w*t+8*X^...
pol2 := -2*(nu-1)^2*(16*t+16*t^2*w+32*w^2*t^3*M(1)^2*nu^2+16*t^3*w*nu*M(1)+16*t^3*w*M(1)*nu^2+4*X^2*t+4*X^3*nu-X^4*nu^2-48*t^2*w*nu*M(1)-16*X*t-16*t^2*w^2+16*t^3*w*nu*D(M)(1)+16*t^2*w*nu-8*X^2*w*t+8*X^...
pol2 := -2*(nu-1)^2*(16*t+16*t^2*w+32*w^2*t^3*M(1)^2*nu^2+16*t^3*w*nu*M(1)+16*t^3*w*M(1)*nu^2+4*X^2*t+4*X^3*nu-X^4*nu^2-48*t^2*w*nu*M(1)-16*X*t-16*t^2*w^2+16*t^3*w*nu*D(M)(1)+16*t^2*w*nu-8*X^2*w*t+8*X^...

The first discriminant

>    eq1:=factor(discrim(pol1,X));

eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...
eq1 := 67108864*(nu-1)^6*nu^5*w^2*t^3*(-96*t^2*w^2*nu^5+2-408*nu^4*w*t^3*M(1)-6*nu-240*nu^5*w^2*t^3+160*nu^6*w*t^3+16*t^3*nu^3-48*t*w*nu^3-2*nu^3-15*t*nu-6*nu^2*M(1)-6*nu*w*t^2*M(1)^3-64*nu^5*t^5*M(1)*...

>    nops(eq1);

6

>    seq(op(i,eq1),i=1..5);

67108864, (nu-1)^6, nu^5, w^2, t^3

>    nops(op(6,eq1));

421

Here is the factor of eq1 that vanishes

>    eq1:=op(6,eq1):

The second discriminant

>    eq2:=factor(discrim(pol2,X)):

>    nops(eq2);

6

>    op(6,eq2);

(nu-1)^12

>    nops(op(3,eq2));

401

>    eq2:=op(3,eq2):

Let us check that eq1 and eq2 are indeed 0, using the expansion of M

>    N:=7:normal(series(subs(M(1)=subs(x=1,y=1,q=2,z=1,Mser(N)), D(M)(1)=subs(y=1,diff(subs(x=1,q=2,z=1,Mser(N)),y)), eq1),t,N));

series(O(t^7),t,7)

>    N:=7:normal(series(subs(M(1)=subs(x=1,y=1,q=2,z=1,Mser(N)), D(M)(1)=subs(y=1,diff(subs(x=1,q=2,z=1,Mser(N)),y)), eq2),t,N));

series(O(t^7),t,7)

We eliminate M'(1). The series M(1) is now denoted M0 (it is the zero-th derivatiove of M....)

>    res:=factor(resultant(subs(D(M)(1)=M1, M(1)=M0,eq1),subs(D(M)(1)=M1,M(1)=M0,eq2),M1)):

>    nops(res);

7

>    seq( nops(op(i,res)),i=1..7);

1, 2, 2, 2, 2, 1913, 2

>    seq(op(i,res),i=1..5);

-16, t^14, nu^6, w^2, (nu+1)^6

There is one big factor (the 6th one). The 7th one is the square of a polynomial that might vanish

>    op(6,res): op(7,res);

(-8*t^4*M0^2*nu^9*w^2-1+2*t+2*nu+112*nu^6*w*t^3+16*t^3*nu^3+2*nu^3+64*t^3*w^2*nu^3+160*t^3*w^2*nu^4-32*t^2*w^2*nu^2+48*t^2*w^2*nu^4-64*nu^4*w*t^3-16*t*nu^3+16*t^3*nu^7-16*t^2*nu^6-80*nu^3*w*t^3-16*t^3*...
(-8*t^4*M0^2*nu^9*w^2-1+2*t+2*nu+112*nu^6*w*t^3+16*t^3*nu^3+2*nu^3+64*t^3*w^2*nu^3+160*t^3*w^2*nu^4-32*t^2*w^2*nu^2+48*t^2*w^2*nu^4-64*nu^4*w*t^3-16*t*nu^3+16*t^3*nu^7-16*t^2*nu^6-80*nu^3*w*t^3-16*t^3*...
(-8*t^4*M0^2*nu^9*w^2-1+2*t+2*nu+112*nu^6*w*t^3+16*t^3*nu^3+2*nu^3+64*t^3*w^2*nu^3+160*t^3*w^2*nu^4-32*t^2*w^2*nu^2+48*t^2*w^2*nu^4-64*nu^4*w*t^3-16*t*nu^3+16*t^3*nu^7-16*t^2*nu^6-80*nu^3*w*t^3-16*t^3*...
(-8*t^4*M0^2*nu^9*w^2-1+2*t+2*nu+112*nu^6*w*t^3+16*t^3*nu^3+2*nu^3+64*t^3*w^2*nu^3+160*t^3*w^2*nu^4-32*t^2*w^2*nu^2+48*t^2*w^2*nu^4-64*nu^4*w*t^3-16*t*nu^3+16*t^3*nu^7-16*t^2*nu^6-80*nu^3*w*t^3-16*t^3*...
(-8*t^4*M0^2*nu^9*w^2-1+2*t+2*nu+112*nu^6*w*t^3+16*t^3*nu^3+2*nu^3+64*t^3*w^2*nu^3+160*t^3*w^2*nu^4-32*t^2*w^2*nu^2+48*t^2*w^2*nu^4-64*nu^4*w*t^3-16*t*nu^3+16*t^3*nu^7-16*t^2*nu^6-80*nu^3*w*t^3-16*t^3*...
(-8*t^4*M0^2*nu^9*w^2-1+2*t+2*nu+112*nu^6*w*t^3+16*t^3*nu^3+2*nu^3+64*t^3*w^2*nu^3+160*t^3*w^2*nu^4-32*t^2*w^2*nu^2+48*t^2*w^2*nu^4-64*nu^4*w*t^3-16*t*nu^3+16*t^3*nu^7-16*t^2*nu^6-80*nu^3*w*t^3-16*t^3*...
(-8*t^4*M0^2*nu^9*w^2-1+2*t+2*nu+112*nu^6*w*t^3+16*t^3*nu^3+2*nu^3+64*t^3*w^2*nu^3+160*t^3*w^2*nu^4-32*t^2*w^2*nu^2+48*t^2*w^2*nu^4-64*nu^4*w*t^3-16*t*nu^3+16*t^3*nu^7-16*t^2*nu^6-80*nu^3*w*t^3-16*t^3*...

Which of these two factors vanishes? Here is the big guy:

>    alg1:=collect(op(6,res),M0,factor): nops(alg1);

351

and the smaller one:

>    alg2:=collect(op(1,op(7,res)),M0,factor);

alg2 := -1+2*t+2*nu+112*nu^6*w*t^3+16*t^3*nu^3+4*w*(nu+1)*nu*t^2*(t*nu^7+5*t*nu^6+16*nu^5*t*w-5*nu^5*t-t*nu^4+8*t*w*nu^3-t*nu^3-5*t*nu^2-8*t*w*nu+5*t*nu+t-4*nu^4+4*nu^3+4*nu^2-4*nu)*M0+2*nu^3+64*t^3*w^...
alg2 := -1+2*t+2*nu+112*nu^6*w*t^3+16*t^3*nu^3+4*w*(nu+1)*nu*t^2*(t*nu^7+5*t*nu^6+16*nu^5*t*w-5*nu^5*t-t*nu^4+8*t*w*nu^3-t*nu^3-5*t*nu^2-8*t*w*nu+5*t*nu+t-4*nu^4+4*nu^3+4*nu^2-4*nu)*M0+2*nu^3+64*t^3*w^...
alg2 := -1+2*t+2*nu+112*nu^6*w*t^3+16*t^3*nu^3+4*w*(nu+1)*nu*t^2*(t*nu^7+5*t*nu^6+16*nu^5*t*w-5*nu^5*t-t*nu^4+8*t*w*nu^3-t*nu^3-5*t*nu^2-8*t*w*nu+5*t*nu+t-4*nu^4+4*nu^3+4*nu^2-4*nu)*M0+2*nu^3+64*t^3*w^...
alg2 := -1+2*t+2*nu+112*nu^6*w*t^3+16*t^3*nu^3+4*w*(nu+1)*nu*t^2*(t*nu^7+5*t*nu^6+16*nu^5*t*w-5*nu^5*t-t*nu^4+8*t*w*nu^3-t*nu^3-5*t*nu^2-8*t*w*nu+5*t*nu+t-4*nu^4+4*nu^3+4*nu^2-4*nu)*M0+2*nu^3+64*t^3*w^...
alg2 := -1+2*t+2*nu+112*nu^6*w*t^3+16*t^3*nu^3+4*w*(nu+1)*nu*t^2*(t*nu^7+5*t*nu^6+16*nu^5*t*w-5*nu^5*t-t*nu^4+8*t*w*nu^3-t*nu^3-5*t*nu^2-8*t*w*nu+5*t*nu+t-4*nu^4+4*nu^3+4*nu^2-4*nu)*M0+2*nu^3+64*t^3*w^...
alg2 := -1+2*t+2*nu+112*nu^6*w*t^3+16*t^3*nu^3+4*w*(nu+1)*nu*t^2*(t*nu^7+5*t*nu^6+16*nu^5*t*w-5*nu^5*t-t*nu^4+8*t*w*nu^3-t*nu^3-5*t*nu^2-8*t*w*nu+5*t*nu+t-4*nu^4+4*nu^3+4*nu^2-4*nu)*M0+2*nu^3+64*t^3*w^...

>    degree(alg1,M0);degree(alg2,M0);

8

2

To decide, we expand alg1 and alg2 in t. Alas... the correct equation for M0 turns out to be the big guy, alg1

>    n:=6:normal(series(subs(M0=subs(x=1,y=1,q=2,z=1,Mser(n)),alg1),t,n+1));

series(O(t^7),t,7)

>    n:=4:factor(series(subs(M0=subs(x=1,y=1,q=2,z=1,Mser(n)),alg2),t,n+1));

series((-(nu^2+1)*(nu-1)^2)+2*(nu-1)*(nu^5+6*nu^4*w+nu^4+4*nu^3-4*nu^2-nu+2*w-1)*t+(-1-32*w^2*nu^2+2*nu^3+32*nu^2*w-32*nu^4*w-16*nu^2-2*nu^7+16*nu^3*w+48*w^2*nu^4+34*nu^4-16*nu^6-16*nu^7*w-nu^8-48*nu^6...
series((-(nu^2+1)*(nu-1)^2)+2*(nu-1)*(nu^5+6*nu^4*w+nu^4+4*nu^3-4*nu^2-nu+2*w-1)*t+(-1-32*w^2*nu^2+2*nu^3+32*nu^2*w-32*nu^4*w-16*nu^2-2*nu^7+16*nu^3*w+48*w^2*nu^4+34*nu^4-16*nu^6-16*nu^7*w-nu^8-48*nu^6...
series((-(nu^2+1)*(nu-1)^2)+2*(nu-1)*(nu^5+6*nu^4*w+nu^4+4*nu^3-4*nu^2-nu+2*w-1)*t+(-1-32*w^2*nu^2+2*nu^3+32*nu^2*w-32*nu^4*w-16*nu^2-2*nu^7+16*nu^3*w+48*w^2*nu^4+34*nu^4-16*nu^6-16*nu^7*w-nu^8-48*nu^6...
series((-(nu^2+1)*(nu-1)^2)+2*(nu-1)*(nu^5+6*nu^4*w+nu^4+4*nu^3-4*nu^2-nu+2*w-1)*t+(-1-32*w^2*nu^2+2*nu^3+32*nu^2*w-32*nu^4*w-16*nu^2-2*nu^7+16*nu^3*w+48*w^2*nu^4+34*nu^4-16*nu^6-16*nu^7*w-nu^8-48*nu^6...
series((-(nu^2+1)*(nu-1)^2)+2*(nu-1)*(nu^5+6*nu^4*w+nu^4+4*nu^3-4*nu^2-nu+2*w-1)*t+(-1-32*w^2*nu^2+2*nu^3+32*nu^2*w-32*nu^4*w-16*nu^2-2*nu^7+16*nu^3*w+48*w^2*nu^4+34*nu^4-16*nu^6-16*nu^7*w-nu^8-48*nu^6...

>    algM0:=collect(alg1,M0,factor): degree(algM0,M0);

8

>   

Let us not count vertices

>    res:=factor(subs(w=1,algM0)):nops(res);

4

>    op(1,res);op(2,res);op(4,res);

-1

nu

(-1+t*M0+t*nu*M0)^2

here is an equation of degree 6 for M0

>    algMw1:=collect(op(3,factor(subs(w=1,algM0))),M0,factor);

algMw1 := 1-11*t-2*nu-13985*t^3*nu^3-15*t*nu-29*t*nu^3-14193*t^3*nu^7+824*t^2*nu^6-16*nu^5*t+(-1+12*t+2*nu+12367*t^3*nu^3+14*t*nu+30*t*nu^3+12809*t^3*nu^7-824*t^2*nu^6+16*nu^5*t+34879*t^3*nu^5+14490*t^...
algMw1 := 1-11*t-2*nu-13985*t^3*nu^3-15*t*nu-29*t*nu^3-14193*t^3*nu^7+824*t^2*nu^6-16*nu^5*t+(-1+12*t+2*nu+12367*t^3*nu^3+14*t*nu+30*t*nu^3+12809*t^3*nu^7-824*t^2*nu^6+16*nu^5*t+34879*t^3*nu^5+14490*t^...
algMw1 := 1-11*t-2*nu-13985*t^3*nu^3-15*t*nu-29*t*nu^3-14193*t^3*nu^7+824*t^2*nu^6-16*nu^5*t+(-1+12*t+2*nu+12367*t^3*nu^3+14*t*nu+30*t*nu^3+12809*t^3*nu^7-824*t^2*nu^6+16*nu^5*t+34879*t^3*nu^5+14490*t^...
algMw1 := 1-11*t-2*nu-13985*t^3*nu^3-15*t*nu-29*t*nu^3-14193*t^3*nu^7+824*t^2*nu^6-16*nu^5*t+(-1+12*t+2*nu+12367*t^3*nu^3+14*t*nu+30*t*nu^3+12809*t^3*nu^7-824*t^2*nu^6+16*nu^5*t+34879*t^3*nu^5+14490*t^...
algMw1 := 1-11*t-2*nu-13985*t^3*nu^3-15*t*nu-29*t*nu^3-14193*t^3*nu^7+824*t^2*nu^6-16*nu^5*t+(-1+12*t+2*nu+12367*t^3*nu^3+14*t*nu+30*t*nu^3+12809*t^3*nu^7-824*t^2*nu^6+16*nu^5*t+34879*t^3*nu^5+14490*t^...
algMw1 := 1-11*t-2*nu-13985*t^3*nu^3-15*t*nu-29*t*nu^3-14193*t^3*nu^7+824*t^2*nu^6-16*nu^5*t+(-1+12*t+2*nu+12367*t^3*nu^3+14*t*nu+30*t*nu^3+12809*t^3*nu^7-824*t^2*nu^6+16*nu^5*t+34879*t^3*nu^5+14490*t^...
algMw1 := 1-11*t-2*nu-13985*t^3*nu^3-15*t*nu-29*t*nu^3-14193*t^3*nu^7+824*t^2*nu^6-16*nu^5*t+(-1+12*t+2*nu+12367*t^3*nu^3+14*t*nu+30*t*nu^3+12809*t^3*nu^7-824*t^2*nu^6+16*nu^5*t+34879*t^3*nu^5+14490*t^...
algMw1 := 1-11*t-2*nu-13985*t^3*nu^3-15*t*nu-29*t*nu^3-14193*t^3*nu^7+824*t^2*nu^6-16*nu^5*t+(-1+12*t+2*nu+12367*t^3*nu^3+14*t*nu+30*t*nu^3+12809*t^3*nu^7-824*t^2*nu^6+16*nu^5*t+34879*t^3*nu^5+14490*t^...
algMw1 := 1-11*t-2*nu-13985*t^3*nu^3-15*t*nu-29*t*nu^3-14193*t^3*nu^7+824*t^2*nu^6-16*nu^5*t+(-1+12*t+2*nu+12367*t^3*nu^3+14*t*nu+30*t*nu^3+12809*t^3*nu^7-824*t^2*nu^6+16*nu^5*t+34879*t^3*nu^5+14490*t^...
algMw1 := 1-11*t-2*nu-13985*t^3*nu^3-15*t*nu-29*t*nu^3-14193*t^3*nu^7+824*t^2*nu^6-16*nu^5*t+(-1+12*t+2*nu+12367*t^3*nu^3+14*t*nu+30*t*nu^3+12809*t^3*nu^7-824*t^2*nu^6+16*nu^5*t+34879*t^3*nu^5+14490*t^...
algMw1 := 1-11*t-2*nu-13985*t^3*nu^3-15*t*nu-29*t*nu^3-14193*t^3*nu^7+824*t^2*nu^6-16*nu^5*t+(-1+12*t+2*nu+12367*t^3*nu^3+14*t*nu+30*t*nu^3+12809*t^3*nu^7-824*t^2*nu^6+16*nu^5*t+34879*t^3*nu^5+14490*t^...
algMw1 := 1-11*t-2*nu-13985*t^3*nu^3-15*t*nu-29*t*nu^3-14193*t^3*nu^7+824*t^2*nu^6-16*nu^5*t+(-1+12*t+2*nu+12367*t^3*nu^3+14*t*nu+30*t*nu^3+12809*t^3*nu^7-824*t^2*nu^6+16*nu^5*t+34879*t^3*nu^5+14490*t^...
algMw1 := 1-11*t-2*nu-13985*t^3*nu^3-15*t*nu-29*t*nu^3-14193*t^3*nu^7+824*t^2*nu^6-16*nu^5*t+(-1+12*t+2*nu+12367*t^3*nu^3+14*t*nu+30*t*nu^3+12809*t^3*nu^7-824*t^2*nu^6+16*nu^5*t+34879*t^3*nu^5+14490*t^...
algMw1 := 1-11*t-2*nu-13985*t^3*nu^3-15*t*nu-29*t*nu^3-14193*t^3*nu^7+824*t^2*nu^6-16*nu^5*t+(-1+12*t+2*nu+12367*t^3*nu^3+14*t*nu+30*t*nu^3+12809*t^3*nu^7-824*t^2*nu^6+16*nu^5*t+34879*t^3*nu^5+14490*t^...
algMw1 := 1-11*t-2*nu-13985*t^3*nu^3-15*t*nu-29*t*nu^3-14193*t^3*nu^7+824*t^2*nu^6-16*nu^5*t+(-1+12*t+2*nu+12367*t^3*nu^3+14*t*nu+30*t*nu^3+12809*t^3*nu^7-824*t^2*nu^6+16*nu^5*t+34879*t^3*nu^5+14490*t^...
algMw1 := 1-11*t-2*nu-13985*t^3*nu^3-15*t*nu-29*t*nu^3-14193*t^3*nu^7+824*t^2*nu^6-16*nu^5*t+(-1+12*t+2*nu+12367*t^3*nu^3+14*t*nu+30*t*nu^3+12809*t^3*nu^7-824*t^2*nu^6+16*nu^5*t+34879*t^3*nu^5+14490*t^...

>   

>    degree(algMw1,M0);

6

>    n:=6:normal(series(subs(M0=subs(x=1,y=1,q=2,w=1,z=1,Mser(n)),algMw1),t,n+1));

series(O(t^7),t,7)

>   

>   

Parametrization for w=1 (degree 6)

>   

>   

Asymptotics and phase transition (w=1)

>   

>   

>   

>