The case q=2 (m=4,k=1) (Section 12.2)

>    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

The second invariant

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

>    indets(I2);

{Q(0,y), y, t, beta}

>    factor(series(normal(I2),y,1));

series((-4*t^8*(1+beta)^4)*y^(-8)+O(y^(-7)),y,-7)

The invariant equation

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

>   

Determination of the series c_i. An equation with one catalytic variable and 2 additional unknown series Q'(0) and Q'''(0)

>   

Solution using  Tutte's trick

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

RHS := -32/3*nu^2*(nu-1)^2*t^6*`@@`(D,3)(Q)(0)-32*(nu-1)^2*t^4*(nu^2-nu-2+2*nu^2*t*X)*D(Q)(0)+(nu-1)^4+4*(nu-1)^2*(2*nu^2+12*t^3*nu-3*nu+1)*t*X-112*nu^2*(nu-1)^2*t^6-8*(nu-4)*(nu-1)^3*t^3-4*nu^4*t^4*X^...
RHS := -32/3*nu^2*(nu-1)^2*t^6*`@@`(D,3)(Q)(0)-32*(nu-1)^2*t^4*(nu^2-nu-2+2*nu^2*t*X)*D(Q)(0)+(nu-1)^4+4*(nu-1)^2*(2*nu^2+12*t^3*nu-3*nu+1)*t*X-112*nu^2*(nu-1)^2*t^6-8*(nu-4)*(nu-1)^3*t^3-4*nu^4*t^4*X^...

>    CC;

t^2*q*(1+beta)^2*II^2+beta*(4*beta+q)*t*II+q*beta*(1+beta)*(q-4)*t^3+beta^2

Each of the following polynomials in X has a double root.
We denote by Q_i the series counting quasi-triangulations with outer degree i.

>    pol1:=collect(op(3,factor(subs(`@@`(D,3)(Q)(0)=6*Q3,D(Q)(0)=Q1,beta=nu-1,factor(subs(q=2,II=X,RHS-CC^2))))),[Q1,Q3,X],factor);
pol2:=collect(op(3,factor(subs(`@@`(D,3)(Q)(0)=6*Q3,D(Q)(0)=Q1,beta=nu-1,factor(subs(q=2,II=X,RHS+CC^2))))),[Q1,Q3,X],factor);

pol1 := (8*t^3*nu^2*(nu-1)^2*X+4*t^2*(nu+1)*(nu-2)*(nu-1)^2)*Q1+8*nu^2*(nu-1)^2*t^4*Q3+t^2*nu^4*X^4+2*nu^2*t*(nu-1)*X^3-(nu-1)*(4*t^3*nu^3-nu+1)*X^2-4*nu*t^2*(nu+1)*(nu-1)^2*X+4*t*(nu-1)^2*(4*t^3*nu^2-...
pol1 := (8*t^3*nu^2*(nu-1)^2*X+4*t^2*(nu+1)*(nu-2)*(nu-1)^2)*Q1+8*nu^2*(nu-1)^2*t^4*Q3+t^2*nu^4*X^4+2*nu^2*t*(nu-1)*X^3-(nu-1)*(4*t^3*nu^3-nu+1)*X^2-4*nu*t^2*(nu+1)*(nu-1)^2*X+4*t*(nu-1)^2*(4*t^3*nu^2-...

pol2 := (32*t^5*nu^2*X+16*t^4*(nu+1)*(nu-2))*Q1+32*t^6*Q3*nu^2-16*nu^2*t^3*X^3-4*nu*t^2*(5*nu-4)*X^2+4*t*(-8*t^3*nu+3*nu-2*nu^2+4*t^3*nu^2-1)*X-nu^2+2*nu+48*t^6*nu^2+8*t^3*nu^2-1-24*t^3*nu+16*t^3
pol2 := (32*t^5*nu^2*X+16*t^4*(nu+1)*(nu-2))*Q1+32*t^6*Q3*nu^2-16*nu^2*t^3*X^3-4*nu*t^2*(5*nu-4)*X^2+4*t*(-8*t^3*nu+3*nu-2*nu^2+4*t^3*nu^2-1)*X-nu^2+2*nu+48*t^6*nu^2+8*t^3*nu^2-1-24*t^3*nu+16*t^3

>    eq1:=factor(discrim(pol1,X)):

>    nops(eq1);

5

>    seq(nops(op(i,eq1)),i=1..5);

1, 2, 2, 2, 115

>    seq(op(i,eq1),i=1..4);;

256, (nu-1)^6, nu^9, t^7

A first equation between Q1 and Q3

>    eq1:=collect(op(5,eq1),[Q1,Q3],factor);

eq1 := -432*nu^7*t^9*(nu-1)^2*Q1^4+8*t^5*nu^3*(-108*nu^2+1+36*nu^6*t^3-48*nu+8*nu^6-72*t^3*nu^3+36*t^3*nu^4+68*nu^3-48*nu^5+63*nu^4)*Q1^3+(-48*nu^5*t^7*(-5+32*nu^3-24*t^3*nu^3-8*nu^4+24*t^3*nu^4-38*nu-...
eq1 := -432*nu^7*t^9*(nu-1)^2*Q1^4+8*t^5*nu^3*(-108*nu^2+1+36*nu^6*t^3-48*nu+8*nu^6-72*t^3*nu^3+36*t^3*nu^4+68*nu^3-48*nu^5+63*nu^4)*Q1^3+(-48*nu^5*t^7*(-5+32*nu^3-24*t^3*nu^3-8*nu^4+24*t^3*nu^4-38*nu-...
eq1 := -432*nu^7*t^9*(nu-1)^2*Q1^4+8*t^5*nu^3*(-108*nu^2+1+36*nu^6*t^3-48*nu+8*nu^6-72*t^3*nu^3+36*t^3*nu^4+68*nu^3-48*nu^5+63*nu^4)*Q1^3+(-48*nu^5*t^7*(-5+32*nu^3-24*t^3*nu^3-8*nu^4+24*t^3*nu^4-38*nu-...
eq1 := -432*nu^7*t^9*(nu-1)^2*Q1^4+8*t^5*nu^3*(-108*nu^2+1+36*nu^6*t^3-48*nu+8*nu^6-72*t^3*nu^3+36*t^3*nu^4+68*nu^3-48*nu^5+63*nu^4)*Q1^3+(-48*nu^5*t^7*(-5+32*nu^3-24*t^3*nu^3-8*nu^4+24*t^3*nu^4-38*nu-...
eq1 := -432*nu^7*t^9*(nu-1)^2*Q1^4+8*t^5*nu^3*(-108*nu^2+1+36*nu^6*t^3-48*nu+8*nu^6-72*t^3*nu^3+36*t^3*nu^4+68*nu^3-48*nu^5+63*nu^4)*Q1^3+(-48*nu^5*t^7*(-5+32*nu^3-24*t^3*nu^3-8*nu^4+24*t^3*nu^4-38*nu-...
eq1 := -432*nu^7*t^9*(nu-1)^2*Q1^4+8*t^5*nu^3*(-108*nu^2+1+36*nu^6*t^3-48*nu+8*nu^6-72*t^3*nu^3+36*t^3*nu^4+68*nu^3-48*nu^5+63*nu^4)*Q1^3+(-48*nu^5*t^7*(-5+32*nu^3-24*t^3*nu^3-8*nu^4+24*t^3*nu^4-38*nu-...
eq1 := -432*nu^7*t^9*(nu-1)^2*Q1^4+8*t^5*nu^3*(-108*nu^2+1+36*nu^6*t^3-48*nu+8*nu^6-72*t^3*nu^3+36*t^3*nu^4+68*nu^3-48*nu^5+63*nu^4)*Q1^3+(-48*nu^5*t^7*(-5+32*nu^3-24*t^3*nu^3-8*nu^4+24*t^3*nu^4-38*nu-...
eq1 := -432*nu^7*t^9*(nu-1)^2*Q1^4+8*t^5*nu^3*(-108*nu^2+1+36*nu^6*t^3-48*nu+8*nu^6-72*t^3*nu^3+36*t^3*nu^4+68*nu^3-48*nu^5+63*nu^4)*Q1^3+(-48*nu^5*t^7*(-5+32*nu^3-24*t^3*nu^3-8*nu^4+24*t^3*nu^4-38*nu-...
eq1 := -432*nu^7*t^9*(nu-1)^2*Q1^4+8*t^5*nu^3*(-108*nu^2+1+36*nu^6*t^3-48*nu+8*nu^6-72*t^3*nu^3+36*t^3*nu^4+68*nu^3-48*nu^5+63*nu^4)*Q1^3+(-48*nu^5*t^7*(-5+32*nu^3-24*t^3*nu^3-8*nu^4+24*t^3*nu^4-38*nu-...

>    eq2:=factor(discrim(pol2,X)):

>    nops(eq2);

4

>    seq(nops(op(i,eq2)),i=1..4);

1, 2, 2, 42

>    seq(op(i,eq2),i=1..3);;

8192, t^10, nu^3

Second equation between Q1 and Q3

>    eq2:=collect(op(4,eq2),[Q1,Q3],factor);

eq2 := 256*t^8*nu^5*Q1^3+2*t^4*nu*(-4*nu^3-384*t^3*nu^3+nu^4+76*nu^2+192*t^3*nu^4-144*nu-432)*Q1^2+(-144*t^6*nu^3*(nu^2-2*nu-12)*Q3+2*nu^5*t^3+nu^3+288*t^3*nu-8+12*nu+96*t^3*nu^3-304*t^3*nu^2-6*nu^2-12...
eq2 := 256*t^8*nu^5*Q1^3+2*t^4*nu*(-4*nu^3-384*t^3*nu^3+nu^4+76*nu^2+192*t^3*nu^4-144*nu-432)*Q1^2+(-144*t^6*nu^3*(nu^2-2*nu-12)*Q3+2*nu^5*t^3+nu^3+288*t^3*nu-8+12*nu+96*t^3*nu^3-304*t^3*nu^2-6*nu^2-12...
eq2 := 256*t^8*nu^5*Q1^3+2*t^4*nu*(-4*nu^3-384*t^3*nu^3+nu^4+76*nu^2+192*t^3*nu^4-144*nu-432)*Q1^2+(-144*t^6*nu^3*(nu^2-2*nu-12)*Q3+2*nu^5*t^3+nu^3+288*t^3*nu-8+12*nu+96*t^3*nu^3-304*t^3*nu^2-6*nu^2-12...
eq2 := 256*t^8*nu^5*Q1^3+2*t^4*nu*(-4*nu^3-384*t^3*nu^3+nu^4+76*nu^2+192*t^3*nu^4-144*nu-432)*Q1^2+(-144*t^6*nu^3*(nu^2-2*nu-12)*Q3+2*nu^5*t^3+nu^3+288*t^3*nu-8+12*nu+96*t^3*nu^3-304*t^3*nu^2-6*nu^2-12...

We finally eliminate Q3

>    res:=factor(resultant(eq1,eq2,Q3)):

>    nops(res);

5

>    seq(nops(op(i,res)),i=1..5);

1, 2, 2, 107, 2

>    op(1,res); op(2,res);op(3,res);

128

nu^11

t^17

>    op(4,res);

-24*Q1-nu^5*Q1+4472*nu^6*t^6*Q1+3538944*nu^10*t^12*Q1+8081408*nu^8*t^11*Q1^3-5767168*nu^10*t^12*Q1^4-15204352*nu^9*t^12*Q1^4+9527296*nu^9*t^11*Q1^3+3538944*nu^9*t^12*Q1-7077888*nu^10*t^13*Q1^2+1003520*...
-24*Q1-nu^5*Q1+4472*nu^6*t^6*Q1+3538944*nu^10*t^12*Q1+8081408*nu^8*t^11*Q1^3-5767168*nu^10*t^12*Q1^4-15204352*nu^9*t^12*Q1^4+9527296*nu^9*t^11*Q1^3+3538944*nu^9*t^12*Q1-7077888*nu^10*t^13*Q1^2+1003520*...
-24*Q1-nu^5*Q1+4472*nu^6*t^6*Q1+3538944*nu^10*t^12*Q1+8081408*nu^8*t^11*Q1^3-5767168*nu^10*t^12*Q1^4-15204352*nu^9*t^12*Q1^4+9527296*nu^9*t^11*Q1^3+3538944*nu^9*t^12*Q1-7077888*nu^10*t^13*Q1^2+1003520*...
-24*Q1-nu^5*Q1+4472*nu^6*t^6*Q1+3538944*nu^10*t^12*Q1+8081408*nu^8*t^11*Q1^3-5767168*nu^10*t^12*Q1^4-15204352*nu^9*t^12*Q1^4+9527296*nu^9*t^11*Q1^3+3538944*nu^9*t^12*Q1-7077888*nu^10*t^13*Q1^2+1003520*...
-24*Q1-nu^5*Q1+4472*nu^6*t^6*Q1+3538944*nu^10*t^12*Q1+8081408*nu^8*t^11*Q1^3-5767168*nu^10*t^12*Q1^4-15204352*nu^9*t^12*Q1^4+9527296*nu^9*t^11*Q1^3+3538944*nu^9*t^12*Q1-7077888*nu^10*t^13*Q1^2+1003520*...
-24*Q1-nu^5*Q1+4472*nu^6*t^6*Q1+3538944*nu^10*t^12*Q1+8081408*nu^8*t^11*Q1^3-5767168*nu^10*t^12*Q1^4-15204352*nu^9*t^12*Q1^4+9527296*nu^9*t^11*Q1^3+3538944*nu^9*t^12*Q1-7077888*nu^10*t^13*Q1^2+1003520*...
-24*Q1-nu^5*Q1+4472*nu^6*t^6*Q1+3538944*nu^10*t^12*Q1+8081408*nu^8*t^11*Q1^3-5767168*nu^10*t^12*Q1^4-15204352*nu^9*t^12*Q1^4+9527296*nu^9*t^11*Q1^3+3538944*nu^9*t^12*Q1-7077888*nu^10*t^13*Q1^2+1003520*...
-24*Q1-nu^5*Q1+4472*nu^6*t^6*Q1+3538944*nu^10*t^12*Q1+8081408*nu^8*t^11*Q1^3-5767168*nu^10*t^12*Q1^4-15204352*nu^9*t^12*Q1^4+9527296*nu^9*t^11*Q1^3+3538944*nu^9*t^12*Q1-7077888*nu^10*t^13*Q1^2+1003520*...
-24*Q1-nu^5*Q1+4472*nu^6*t^6*Q1+3538944*nu^10*t^12*Q1+8081408*nu^8*t^11*Q1^3-5767168*nu^10*t^12*Q1^4-15204352*nu^9*t^12*Q1^4+9527296*nu^9*t^11*Q1^3+3538944*nu^9*t^12*Q1-7077888*nu^10*t^13*Q1^2+1003520*...
-24*Q1-nu^5*Q1+4472*nu^6*t^6*Q1+3538944*nu^10*t^12*Q1+8081408*nu^8*t^11*Q1^3-5767168*nu^10*t^12*Q1^4-15204352*nu^9*t^12*Q1^4+9527296*nu^9*t^11*Q1^3+3538944*nu^9*t^12*Q1-7077888*nu^10*t^13*Q1^2+1003520*...
-24*Q1-nu^5*Q1+4472*nu^6*t^6*Q1+3538944*nu^10*t^12*Q1+8081408*nu^8*t^11*Q1^3-5767168*nu^10*t^12*Q1^4-15204352*nu^9*t^12*Q1^4+9527296*nu^9*t^11*Q1^3+3538944*nu^9*t^12*Q1-7077888*nu^10*t^13*Q1^2+1003520*...
-24*Q1-nu^5*Q1+4472*nu^6*t^6*Q1+3538944*nu^10*t^12*Q1+8081408*nu^8*t^11*Q1^3-5767168*nu^10*t^12*Q1^4-15204352*nu^9*t^12*Q1^4+9527296*nu^9*t^11*Q1^3+3538944*nu^9*t^12*Q1-7077888*nu^10*t^13*Q1^2+1003520*...

>    op(5,res);

(95*nu^2-52*nu-288*t^3*nu^4+512*t^3*nu^3+256*t^6*nu^4+256*nu^5*t^7*Q1-320*nu^6*t^7*Q1-94*nu^3+53*nu^4-640*nu^5*t^6+512*t^4*nu^3*Q1+2*nu^6+64*nu^6*t^8*Q1^2-16*nu^5-368*t^3*nu^2+96*t^3*nu-96*Q1*t^4*nu^2+...
(95*nu^2-52*nu-288*t^3*nu^4+512*t^3*nu^3+256*t^6*nu^4+256*nu^5*t^7*Q1-320*nu^6*t^7*Q1-94*nu^3+53*nu^4-640*nu^5*t^6+512*t^4*nu^3*Q1+2*nu^6+64*nu^6*t^8*Q1^2-16*nu^5-368*t^3*nu^2+96*t^3*nu-96*Q1*t^4*nu^2+...

Two candidates for the minimal polynomial of Q1

>    alg1:=collect(op(4,res),Q1,factor);

alg1 := 8388608*nu^10*t^13*Q1^5-16*nu^6*t^9*(360448*t^3*nu^4-11180*nu^3+5972*nu+2795*nu^4+950272*t^3*nu^3+8194*nu^2-18069)*Q1^4+8*t^5*nu^2*(-255*nu^2+324+1796*t^3*nu^8+167*nu^6-162*nu-30624*nu^6*t^3+8*...
alg1 := 8388608*nu^10*t^13*Q1^5-16*nu^6*t^9*(360448*t^3*nu^4-11180*nu^3+5972*nu+2795*nu^4+950272*t^3*nu^3+8194*nu^2-18069)*Q1^4+8*t^5*nu^2*(-255*nu^2+324+1796*t^3*nu^8+167*nu^6-162*nu-30624*nu^6*t^3+8*...
alg1 := 8388608*nu^10*t^13*Q1^5-16*nu^6*t^9*(360448*t^3*nu^4-11180*nu^3+5972*nu+2795*nu^4+950272*t^3*nu^3+8194*nu^2-18069)*Q1^4+8*t^5*nu^2*(-255*nu^2+324+1796*t^3*nu^8+167*nu^6-162*nu-30624*nu^6*t^3+8*...
alg1 := 8388608*nu^10*t^13*Q1^5-16*nu^6*t^9*(360448*t^3*nu^4-11180*nu^3+5972*nu+2795*nu^4+950272*t^3*nu^3+8194*nu^2-18069)*Q1^4+8*t^5*nu^2*(-255*nu^2+324+1796*t^3*nu^8+167*nu^6-162*nu-30624*nu^6*t^3+8*...
alg1 := 8388608*nu^10*t^13*Q1^5-16*nu^6*t^9*(360448*t^3*nu^4-11180*nu^3+5972*nu+2795*nu^4+950272*t^3*nu^3+8194*nu^2-18069)*Q1^4+8*t^5*nu^2*(-255*nu^2+324+1796*t^3*nu^8+167*nu^6-162*nu-30624*nu^6*t^3+8*...
alg1 := 8388608*nu^10*t^13*Q1^5-16*nu^6*t^9*(360448*t^3*nu^4-11180*nu^3+5972*nu+2795*nu^4+950272*t^3*nu^3+8194*nu^2-18069)*Q1^4+8*t^5*nu^2*(-255*nu^2+324+1796*t^3*nu^8+167*nu^6-162*nu-30624*nu^6*t^3+8*...
alg1 := 8388608*nu^10*t^13*Q1^5-16*nu^6*t^9*(360448*t^3*nu^4-11180*nu^3+5972*nu+2795*nu^4+950272*t^3*nu^3+8194*nu^2-18069)*Q1^4+8*t^5*nu^2*(-255*nu^2+324+1796*t^3*nu^8+167*nu^6-162*nu-30624*nu^6*t^3+8*...
alg1 := 8388608*nu^10*t^13*Q1^5-16*nu^6*t^9*(360448*t^3*nu^4-11180*nu^3+5972*nu+2795*nu^4+950272*t^3*nu^3+8194*nu^2-18069)*Q1^4+8*t^5*nu^2*(-255*nu^2+324+1796*t^3*nu^8+167*nu^6-162*nu-30624*nu^6*t^3+8*...

>    degree(alg1,Q1);

5

The second candidate

>    alg2:=collect(op(1,op(5,res)),Q1,factor);

alg2 := 64*nu^6*t^8*Q1^2-32*t^4*nu^2*(-16*nu+3+28*nu^2+10*t^3*nu^4-8*t^3*nu^3+5*nu^4-20*nu^3)*Q1+95*nu^2-52*nu-288*t^3*nu^4+512*t^3*nu^3+256*t^6*nu^4-94*nu^3+53*nu^4-640*nu^5*t^6+2*nu^6-16*nu^5-368*t^3...
alg2 := 64*nu^6*t^8*Q1^2-32*t^4*nu^2*(-16*nu+3+28*nu^2+10*t^3*nu^4-8*t^3*nu^3+5*nu^4-20*nu^3)*Q1+95*nu^2-52*nu-288*t^3*nu^4+512*t^3*nu^3+256*t^6*nu^4-94*nu^3+53*nu^4-640*nu^5*t^6+2*nu^6-16*nu^5-368*t^3...

>    degree(alg2,Q1);

2

Let us decide

>    n:=2:normal(series(subs( Q1=normal(collect(coeff(subs(x=0,w=1,z=1,q=2,Qser(3*n)),y,1),t)),alg2),t,n+1));

series((2*nu^6-16*nu^5+53*nu^4-94*nu^3+95*nu^2-52*nu+12)+O(t^3),t,3)

>    n:=4:normal(series(subs( Q1=normal(collect(coeff(subs(x=0,w=1,z=1,q=2,Qser(3*n)),y,1),t)),alg1),t,n+2));

series(O(t^8),t,8)

Hence the minimal polynomial of Q1 has degree 5. Let us check

>    with(gfun):

>    factor(algeqtoseries(alg1,t,Q1,10,true));

[series((nu+1)*nu*t^2+4*nu^2*(nu+1)*(nu^2+1)*t^5+4*nu^3*(nu+1)*(8*nu^4+2*nu^3+9*nu^2+4*nu+9)*t^8+16*nu^4*(nu+1)*(21*nu^6+11*nu^5+28*nu^4+21*nu^3+36*nu^2+24*nu+27)*t^11+O(t^12),t,12)]
[series((nu+1)*nu*t^2+4*nu^2*(nu+1)*(nu^2+1)*t^5+4*nu^3*(nu+1)*(8*nu^4+2*nu^3+9*nu^2+4*nu+9)*t^8+16*nu^4*(nu+1)*(21*nu^6+11*nu^5+28*nu^4+21*nu^3+36*nu^2+24*nu+27)*t^11+O(t^12),t,12)]

>    n:=4:map(factor,collect(coeff(subs(x=0,q=2,w=1,z=1,Qser(3*n)),y,1),t));

(nu+1)*nu*t^2+4*nu^2*(nu+1)*(nu^2+1)*t^5+4*nu^3*(nu+1)*(8*nu^4+2*nu^3+9*nu^2+4*nu+9)*t^8+16*nu^4*(nu+1)*(21*nu^6+11*nu^5+28*nu^4+21*nu^3+36*nu^2+24*nu+27)*t^11

In the series t*Q1, denoted Q1t below, all exponents of t are multiples of  3. We denote t^3=w.

>    alg1t:=collect(subs(t=w^(1/3),factor(numer(subs(Q1=Q1t/t,alg1)))),Q1t,factor);

alg1t := 8388608*nu^10*w^3*Q1t^5-16*w^2*nu^6*(950272*w*nu^3-18069+2795*nu^4+5972*nu-11180*nu^3+360448*w*nu^4+8194*nu^2)*Q1t^4+8*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^7+125440*w^2...
alg1t := 8388608*nu^10*w^3*Q1t^5-16*w^2*nu^6*(950272*w*nu^3-18069+2795*nu^4+5972*nu-11180*nu^3+360448*w*nu^4+8194*nu^2)*Q1t^4+8*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^7+125440*w^2...
alg1t := 8388608*nu^10*w^3*Q1t^5-16*w^2*nu^6*(950272*w*nu^3-18069+2795*nu^4+5972*nu-11180*nu^3+360448*w*nu^4+8194*nu^2)*Q1t^4+8*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^7+125440*w^2...
alg1t := 8388608*nu^10*w^3*Q1t^5-16*w^2*nu^6*(950272*w*nu^3-18069+2795*nu^4+5972*nu-11180*nu^3+360448*w*nu^4+8194*nu^2)*Q1t^4+8*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^7+125440*w^2...
alg1t := 8388608*nu^10*w^3*Q1t^5-16*w^2*nu^6*(950272*w*nu^3-18069+2795*nu^4+5972*nu-11180*nu^3+360448*w*nu^4+8194*nu^2)*Q1t^4+8*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^7+125440*w^2...
alg1t := 8388608*nu^10*w^3*Q1t^5-16*w^2*nu^6*(950272*w*nu^3-18069+2795*nu^4+5972*nu-11180*nu^3+360448*w*nu^4+8194*nu^2)*Q1t^4+8*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^7+125440*w^2...
alg1t := 8388608*nu^10*w^3*Q1t^5-16*w^2*nu^6*(950272*w*nu^3-18069+2795*nu^4+5972*nu-11180*nu^3+360448*w*nu^4+8194*nu^2)*Q1t^4+8*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^7+125440*w^2...
alg1t := 8388608*nu^10*w^3*Q1t^5-16*w^2*nu^6*(950272*w*nu^3-18069+2795*nu^4+5972*nu-11180*nu^3+360448*w*nu^4+8194*nu^2)*Q1t^4+8*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^7+125440*w^2...

>   

>   

>    vnu:=v=(nu+1)/(nu-1);

vnu := v = (nu+1)/(nu-1)

>    nuv:=isolate(vnu,nu);

nuv := nu = (1+v)/(v-1)

The parametrization of the paper (with v=(nu+1)/(nu-1))

>    wS := 1/64*(-2*v-2*S+4*S^3-S^2+v^2)*(-S+v)*(S-2+v)/S^2/(1+v)^3;

wS := 1/64*(-2*v-2*S+4*S^3-S^2+v^2)*(-S+v)*(S-2+v)/S^2/(1+v)^3

let us check that it indeed factors the minimal polynomial of Q1

>    res:=numer(factor(subs(w=wS,nuv,alg1t))):

>    nops(res);

3

>    op(1,res);

-1

>    op(2,res);

-2*v*S-4*S^3*v-S^2*v+2*v^2*S+S^2*v^2+2*v^2+3*S^4-v^3+8*S^3*Q1t-2*Q1t*S^2-2*v^2*Q1t+2*Q1t*v^3-4*v*Q1t-4*Q1t*S-4*Q1t*v*S+8*Q1t*S^3*v-2*Q1t*S^2*v

>    degree(op(3,res),Q1t);

4

The right factor is not op(3,res), but the linear one

>    factor(subs(S=v,Q1t=0,op(3,res)));

128*v^8*(v-2)*(1+v)^2*(v-3)^3*(v-1)^4

>    Q1tsol:=factor(solve(op(2,res),Q1t));

Q1tsol := 1/2*(-S+v)*(v^2-v*S-S^2*v-2*v+3*S^3)/(-2*v-2*S+4*S^3-S^2+v^2)/(1+v)

Expression of  Q3? We get back to one of the equation we had between Q1 and Q3, and introduce Q3t:= t^3 Q3

>    eq2;

256*t^8*nu^5*Q1^3+2*t^4*nu*(-4*nu^3-384*t^3*nu^3+nu^4+76*nu^2+192*t^3*nu^4-144*nu-432)*Q1^2+(-144*t^6*nu^3*(nu^2-2*nu-12)*Q3+2*nu^5*t^3+nu^3+288*t^3*nu-8+12*nu+96*t^3*nu^3-304*t^3*nu^2-6*nu^2-12*t^3*nu...
256*t^8*nu^5*Q1^3+2*t^4*nu*(-4*nu^3-384*t^3*nu^3+nu^4+76*nu^2+192*t^3*nu^4-144*nu-432)*Q1^2+(-144*t^6*nu^3*(nu^2-2*nu-12)*Q3+2*nu^5*t^3+nu^3+288*t^3*nu-8+12*nu+96*t^3*nu^3-304*t^3*nu^2-6*nu^2-12*t^3*nu...
256*t^8*nu^5*Q1^3+2*t^4*nu*(-4*nu^3-384*t^3*nu^3+nu^4+76*nu^2+192*t^3*nu^4-144*nu-432)*Q1^2+(-144*t^6*nu^3*(nu^2-2*nu-12)*Q3+2*nu^5*t^3+nu^3+288*t^3*nu-8+12*nu+96*t^3*nu^3-304*t^3*nu^2-6*nu^2-12*t^3*nu...
256*t^8*nu^5*Q1^3+2*t^4*nu*(-4*nu^3-384*t^3*nu^3+nu^4+76*nu^2+192*t^3*nu^4-144*nu-432)*Q1^2+(-144*t^6*nu^3*(nu^2-2*nu-12)*Q3+2*nu^5*t^3+nu^3+288*t^3*nu-8+12*nu+96*t^3*nu^3-304*t^3*nu^2-6*nu^2-12*t^3*nu...

>    subs(t=w^(1/3),numer(factor(subs(Q1=Q1t/t,Q3=Q3t/t^3,eq2))));

8*w*nu-256*w^2*nu^2-4*w*nu^2-6*w*nu^3-48*w^2*nu^3+5*w*nu^4+240*w^2*nu^4-w*nu^5-76*w^2*nu^5-8*Q1t+8*nu^2*Q3t+Q1t*nu^3+12*Q1t*nu-6*Q1t*nu^2+6*nu^4*Q3t-12*nu^3*Q3t-nu^5*Q3t-144*Q1t*w*nu^5*Q3t+288*Q1t*w*nu...
8*w*nu-256*w^2*nu^2-4*w*nu^2-6*w*nu^3-48*w^2*nu^3+5*w*nu^4+240*w^2*nu^4-w*nu^5-76*w^2*nu^5-8*Q1t+8*nu^2*Q3t+Q1t*nu^3+12*Q1t*nu-6*Q1t*nu^2+6*nu^4*Q3t-12*nu^3*Q3t-nu^5*Q3t-144*Q1t*w*nu^5*Q3t+288*Q1t*w*nu...
8*w*nu-256*w^2*nu^2-4*w*nu^2-6*w*nu^3-48*w^2*nu^3+5*w*nu^4+240*w^2*nu^4-w*nu^5-76*w^2*nu^5-8*Q1t+8*nu^2*Q3t+Q1t*nu^3+12*Q1t*nu-6*Q1t*nu^2+6*nu^4*Q3t-12*nu^3*Q3t-nu^5*Q3t-144*Q1t*w*nu^5*Q3t+288*Q1t*w*nu...
8*w*nu-256*w^2*nu^2-4*w*nu^2-6*w*nu^3-48*w^2*nu^3+5*w*nu^4+240*w^2*nu^4-w*nu^5-76*w^2*nu^5-8*Q1t+8*nu^2*Q3t+Q1t*nu^3+12*Q1t*nu-6*Q1t*nu^2+6*nu^4*Q3t-12*nu^3*Q3t-nu^5*Q3t-144*Q1t*w*nu^5*Q3t+288*Q1t*w*nu...

>    res:=numer(factor(subs(nuv,Q1t=Q1tsol,w=wS,%)));

res := -(2592*S^4*v+25920*S^3*v^2+2592*S^2*v^2+3456*v^3*S-432*S^4+1296*v^4-2592*v^5+1944*v^6+17496*S^2*v^4+18144*S^5*v-4752*S^5*v^2-25056*S^5*v^3+16200*S^4*v^2-24664*S^4*v^3-59904*S^3*v^3+57456*S^3*v^4...
res := -(2592*S^4*v+25920*S^3*v^2+2592*S^2*v^2+3456*v^3*S-432*S^4+1296*v^4-2592*v^5+1944*v^6+17496*S^2*v^4+18144*S^5*v-4752*S^5*v^2-25056*S^5*v^3+16200*S^4*v^2-24664*S^4*v^3-59904*S^3*v^3+57456*S^3*v^4...
res := -(2592*S^4*v+25920*S^3*v^2+2592*S^2*v^2+3456*v^3*S-432*S^4+1296*v^4-2592*v^5+1944*v^6+17496*S^2*v^4+18144*S^5*v-4752*S^5*v^2-25056*S^5*v^3+16200*S^4*v^2-24664*S^4*v^3-59904*S^3*v^3+57456*S^3*v^4...
res := -(2592*S^4*v+25920*S^3*v^2+2592*S^2*v^2+3456*v^3*S-432*S^4+1296*v^4-2592*v^5+1944*v^6+17496*S^2*v^4+18144*S^5*v-4752*S^5*v^2-25056*S^5*v^3+16200*S^4*v^2-24664*S^4*v^3-59904*S^3*v^3+57456*S^3*v^4...
res := -(2592*S^4*v+25920*S^3*v^2+2592*S^2*v^2+3456*v^3*S-432*S^4+1296*v^4-2592*v^5+1944*v^6+17496*S^2*v^4+18144*S^5*v-4752*S^5*v^2-25056*S^5*v^3+16200*S^4*v^2-24664*S^4*v^3-59904*S^3*v^3+57456*S^3*v^4...
res := -(2592*S^4*v+25920*S^3*v^2+2592*S^2*v^2+3456*v^3*S-432*S^4+1296*v^4-2592*v^5+1944*v^6+17496*S^2*v^4+18144*S^5*v-4752*S^5*v^2-25056*S^5*v^3+16200*S^4*v^2-24664*S^4*v^3-59904*S^3*v^3+57456*S^3*v^4...
res := -(2592*S^4*v+25920*S^3*v^2+2592*S^2*v^2+3456*v^3*S-432*S^4+1296*v^4-2592*v^5+1944*v^6+17496*S^2*v^4+18144*S^5*v-4752*S^5*v^2-25056*S^5*v^3+16200*S^4*v^2-24664*S^4*v^3-59904*S^3*v^3+57456*S^3*v^4...
res := -(2592*S^4*v+25920*S^3*v^2+2592*S^2*v^2+3456*v^3*S-432*S^4+1296*v^4-2592*v^5+1944*v^6+17496*S^2*v^4+18144*S^5*v-4752*S^5*v^2-25056*S^5*v^3+16200*S^4*v^2-24664*S^4*v^3-59904*S^3*v^3+57456*S^3*v^4...
res := -(2592*S^4*v+25920*S^3*v^2+2592*S^2*v^2+3456*v^3*S-432*S^4+1296*v^4-2592*v^5+1944*v^6+17496*S^2*v^4+18144*S^5*v-4752*S^5*v^2-25056*S^5*v^3+16200*S^4*v^2-24664*S^4*v^3-59904*S^3*v^3+57456*S^3*v^4...
res := -(2592*S^4*v+25920*S^3*v^2+2592*S^2*v^2+3456*v^3*S-432*S^4+1296*v^4-2592*v^5+1944*v^6+17496*S^2*v^4+18144*S^5*v-4752*S^5*v^2-25056*S^5*v^3+16200*S^4*v^2-24664*S^4*v^3-59904*S^3*v^3+57456*S^3*v^4...
res := -(2592*S^4*v+25920*S^3*v^2+2592*S^2*v^2+3456*v^3*S-432*S^4+1296*v^4-2592*v^5+1944*v^6+17496*S^2*v^4+18144*S^5*v-4752*S^5*v^2-25056*S^5*v^3+16200*S^4*v^2-24664*S^4*v^3-59904*S^3*v^3+57456*S^3*v^4...

>    nops(res);

3

>    degree(op(3,res),Q3t);degree(op(2,res),Q3t);

1

1

The right factor is op(3,res):

>    factor(subs(S=v,Q3t=0, op(3,res)));factor(subs(S=v,Q3t=0, op(2,res)));

0

-256*v^4*(1+v)^3*(v-3)^3

>    Q3tsol:=factor(solve(op(3,res),Q3t));

Q3tsol := -1/128*(32*v*S-102*S^3*v-190*S^2*v-26*v^2*S-48*S^4*v+40*S^3*v^2+224*S^2*v^2+6*v^3*S+106*S^3-4*S^2+36*v^2-67*S^4-18*v^3+232*S^5+3*v^4-128*S^5*v+64*S^4*v^2-64*S^6-24*v-8*S-64*S^2*v^3)*(-S+v)^2/...
Q3tsol := -1/128*(32*v*S-102*S^3*v-190*S^2*v-26*v^2*S-48*S^4*v+40*S^3*v^2+224*S^2*v^2+6*v^3*S+106*S^3-4*S^2+36*v^2-67*S^4-18*v^3+232*S^5+3*v^4-128*S^5*v+64*S^4*v^2-64*S^6-24*v-8*S-64*S^2*v^3)*(-S+v)^2/...

>    factor(Q3tsol*wS);

-1/8192*(32*v*S-102*S^3*v-190*S^2*v-26*v^2*S-48*S^4*v+40*S^3*v^2+224*S^2*v^2+6*v^3*S+106*S^3-4*S^2+36*v^2-67*S^4-18*v^3+232*S^5+3*v^4-128*S^5*v+64*S^4*v^2-64*S^6-24*v-8*S-64*S^2*v^3)*(-S+v)^3/S^4/(1+v)...
-1/8192*(32*v*S-102*S^3*v-190*S^2*v-26*v^2*S-48*S^4*v+40*S^3*v^2+224*S^2*v^2+6*v^3*S+106*S^3-4*S^2+36*v^2-67*S^4-18*v^3+232*S^5+3*v^4-128*S^5*v+64*S^4*v^2-64*S^6-24*v-8*S-64*S^2*v^3)*(-S+v)^3/S^4/(1+v)...

>   

Asymptotics

>   

>   

>