Asymptotics

Recal that w=t^3. Expression of w in terms of the series S

>    wS;

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

>    algSv:=collect(numer(w-wS),w,factor);

algSv := 64*w*S^2*(1+v)^3-(-2*v-2*S+4*S^3-S^2+v^2)*(-S+v)*(S-2+v)

>   

Expansion of S in terms of w

>    factor(op(2,algeqtoseries(subs(vnu,algS),w,S,5,true)));

series((nu+1)/(nu-1)+(-8*nu^2*(nu+1)/(nu-1))*w+(-48*(nu^2+1)*(nu+1)*nu^3/(nu-1))*w^2+(-64*(9+9*nu^2+4*nu+2*nu^3+8*nu^4)*(nu+1)*nu^4/(nu-1))*w^3+(-320*(27+36*nu^2+24*nu+21*nu^3+28*nu^4+21*nu^6+11*nu^5)*...
series((nu+1)/(nu-1)+(-8*nu^2*(nu+1)/(nu-1))*w+(-48*(nu^2+1)*(nu+1)*nu^3/(nu-1))*w^2+(-64*(9+9*nu^2+4*nu+2*nu^3+8*nu^4)*(nu+1)*nu^4/(nu-1))*w^3+(-320*(27+36*nu^2+24*nu+21*nu^3+28*nu^4+21*nu^6+11*nu^5)*...

A better series, for the analysis, is U such that S=v(1-2U). Indeed, its coefficients are non negative

>    algUv:=collect(op(3,factor(subs(S=v*(1-2*U),algSv))),U,factor);

algUv := -8*v^3*U^5+v^2*(-9+20*v)*U^4-2*v^2*(9*v-7)*U^3+(1+16*w*v^3-v-7*v^2+7*v^3+48*w*v+16*w+48*w*v^2)*U^2-(1+v)*(v^2+16*w*v^2+32*w*v-2*v+1+16*w)*U+4*w*(1+v)^3
algUv := -8*v^3*U^5+v^2*(-9+20*v)*U^4-2*v^2*(9*v-7)*U^3+(1+16*w*v^3-v-7*v^2+7*v^3+48*w*v+16*w+48*w*v^2)*U^2-(1+v)*(v^2+16*w*v^2+32*w*v-2*v+1+16*w)*U+4*w*(1+v)^3

>    algU:=collect(numer(factor(subs(vnu,algUv))),U,factor);

algU := -8*(nu+1)^3*U^5+(11*nu+29)*(nu+1)^2*U^4-4*(nu+8)*(nu+1)^2*U^3+(128*w*nu^3+12+32*nu+12*nu^2)*U^2-8*nu*(16*w*nu^2+1)*U+32*w*nu^3

>    op(3,algeqtoseries(algU,w,U,5,true));

series(4*nu^2*w+24*nu^3*(nu^2+1)*w^2+32*nu^4*(9+9*nu^2+4*nu+2*nu^3+8*nu^4)*w^3+160*nu^5*(27+36*nu^2+24*nu+21*nu^3+28*nu^4+21*nu^6+11*nu^5)*w^4+384*nu^6*(189+356*nu^2+252*nu+297*nu^3+311*nu^4+208*nu^6+2...
series(4*nu^2*w+24*nu^3*(nu^2+1)*w^2+32*nu^4*(9+9*nu^2+4*nu+2*nu^3+8*nu^4)*w^3+160*nu^5*(27+36*nu^2+24*nu+21*nu^3+28*nu^4+21*nu^6+11*nu^5)*w^4+384*nu^6*(189+356*nu^2+252*nu+297*nu^3+311*nu^4+208*nu^6+2...

>    factor(discrim(algU,U));

-4096*(nu-1)*(nu-3)^2*(nu+1)^6*(-36+75*nu^2-20*nu+864*w*nu+864*w*nu^2-42*nu^3-2592*w*nu^3+7*nu^4+864*w*nu^4+27648*w^2*nu^4)*(23-12*nu^2-15*nu+7104*w^2*nu^7+4*nu^3-48*w*nu^3-1728*w^2*nu^9+96*w*nu^4-1056...
-4096*(nu-1)*(nu-3)^2*(nu+1)^6*(-36+75*nu^2-20*nu+864*w*nu+864*w*nu^2-42*nu^3-2592*w*nu^3+7*nu^4+864*w*nu^4+27648*w^2*nu^4)*(23-12*nu^2-15*nu+7104*w^2*nu^7+4*nu^3-48*w*nu^3-1728*w^2*nu^9+96*w*nu^4-1056...

>    algr2:=collect(op(5,%),w,factor);

algr2 := 27648*w^2*nu^4+864*nu*(nu-1)*(-1+nu^2-2*nu)*w+(7*nu^2-14*nu-9)*(nu-2)^2

>    algr1:=collect(op(6,%%),w,factor);

algr1 := 131072*nu^9*w^3-192*nu^6*(3*nu+5)*(nu-1)*(3*nu-11)*w^2-48*nu^3*(nu-1)^2*w+(nu-1)*(4*nu^2-8*nu-23)

The radius is found among the roots of these two polynomials

>    coeff(algU,U,5);

-8*(nu+1)^3

>    with(plots):

The general picture is similar to the case of general planar maps

>    implicitplot([algr1,algr2],nu=0..4,w=0..0.1,numpoints=50000);

[Maple Plot]

Let us zoom between 2.5 and 4

>    implicitplot([algr1,algr2],nu=2.5..4,w=-0.02..0.02,numpoints=100000);

[Maple Plot]

Where do the two curves meet?

>    factor(resultant(algr1,algr2,w));

4194304*nu^12*(67482-31038*nu+69811*nu^2-54292*nu^3+13573*nu^4)*(6-14*nu+7*nu^2)^3*(nu+1)^4*(nu-3)^4

We are only interested in positive values of nu

>    res1:=(67482-31038*nu+69811*nu^2-54292*nu^3+13573*nu^4); res2:= (6-14*nu+7*nu^2);

res1 := 67482-31038*nu+69811*nu^2-54292*nu^3+13573*nu^4

res2 := 6-14*nu+7*nu^2

>    {solve(res1)}: evalf(%);

{-.145791810+.9404920565*I, 2.145791810-.9404920565*I, -.145791810-.9404920565*I, 2.145791810+.9404920565*I}

>    plot(res1,nu=0..3);

[Maple Plot]

>    plot(res2,nu=0..4);

[Maple Plot]

>    {solve(res2)};evalf(%);

{1-1/7*7^(1/2), 1+1/7*7^(1/2)}

{.6220355269, 1.377964473}

When the sign is - , the common root of algr1 and algr2 is negative and irrelevant for us

>    factor(subs(nu=1-1/7*7^(1/2),algr1));factor(subs(nu=1-1/7*7^(1/2),algr2));

-1/661624362*(-9415+3553*7^(1/2))*(95551488*w^2-12340944*w-5609520*w*7^(1/2)+1878268+698005*7^(1/2))*(864*w+55+25*7^(1/2))

-1/1323*(-23+8*7^(1/2))*(3456*w-77-35*7^(1/2))*(864*w+55+25*7^(1/2))

>    factor(subs(nu=1+1/7*7^(1/2),algr1));factor(subs(nu=1+1/7*7^(1/2),algr2));

1/661624362*(9415+3553*7^(1/2))*(95551488*w^2-12340944*w+5609520*w*7^(1/2)+1878268-698005*7^(1/2))*(864*w+55-25*7^(1/2))

1/1323*(23+8*7^(1/2))*(3456*w-77+35*7^(1/2))*(864*w+55-25*7^(1/2))

>    nuc:=1+1/7*7^(1/2);evalf(nuc);

nuc := 1+1/7*7^(1/2)

1.377964473

>    rhoc:=solve((-864*w-55+25*7^(1/2)));

rhoc := -55/864+25/864*7^(1/2)

>   

Expansions of the roots of algr1 and algr2  before  nu_c

>    R1ser:=map(factor,map(expand,map(rationalize,op(algeqtoseries(subs(nu=nuc-x,w=rhoc+x*s,algr1),x,s,3,true)))));

R1ser := series((-35/96+7/48*7^(1/2))+(-9359/7776+112/243*7^(1/2))*x+(788263/629856*7^(1/2)-2092447/629856)*x^2+O(x^3),x,3)

>    R2ser:=map(factor,map(expand,map(rationalize,op(algeqtoseries(subs(nu=nuc-x,w=rhoc+x*s,algr2),x,s,3,true)))));

R2ser := series((-35/96+7/48*7^(1/2))+(-9359/7776+112/243*7^(1/2))*x+(-502103/157464+190463/157464*7^(1/2))*x^2+O(x^3),x,3)

>    factor(series(R2ser-R1ser,x));

series((84035/629856-26411/629856*7^(1/2))*x^2+O(x^3),x,3)

>    evalf(%);

series(.224782523e-1*x^2+O(x^3),x,3)

Hence before  nu_c, rho_2 dominates  rho_1  (we denote by rho_i the root of P_i or algr_i)

After nu_c

>    R1ser:=map(factor,map(expand,map(rationalize,op(algeqtoseries(subs(nu=nuc+x,w=rhoc+x*s,algr1),x,s,3,true)))));

R1ser := series((35/96-7/48*7^(1/2))+(-9359/7776+112/243*7^(1/2))*x+(2092447/629856-788263/629856*7^(1/2))*x^2+O(x^3),x,3)

>    R2ser:=map(factor,map(expand,map(rationalize,op(algeqtoseries(subs(nu=nuc+x,w=rhoc+x*s,algr2),x,s,3,true)))));

R2ser := series((35/96-7/48*7^(1/2))+(-9359/7776+112/243*7^(1/2))*x+(502103/157464-190463/157464*7^(1/2))*x^2+O(x^3),x,3)

>    factor(series(R2ser-R1ser,x));

series((26411/629856*7^(1/2)-84035/629856)*x^2+O(x^3),x,3)

>    evalf(%);

series(-.224782523e-1*x^2+O(x^3),x,3)

The two curves have crossed and rho_2 < rho_1

Some plots of U

>    implicitplot(subs(nu=0.8,algU),w=0..0.05,U=0..0.4);

[Maple Plot]

The radius is  rho_2

>    fsolve(subs(nu=0.8,algr1));fsolve(subs(nu=0.8,algr2));

-.5734604134e-1

-.5824476742e-1, .3431898617e-1

>    implicitplot(subs(nu=1.2,algU),w=0..0.03,U=0..0.4);

[Maple Plot]

The radius is again  rho_2

>    fsolve(subs(nu=1.2,algr1));fsolve(subs(nu=1.2,algr2));

.1699141966e-1

-.1016858849e-1, .1725770886e-1

>    implicitplot(subs(nu=nuc,algU),w=0..0.04,U=0..0.5);

[Maple Plot]

The two curves meet

>    fsolve(subs(nu=nuc,algr1));fsolve(subs(nu=nuc,algr2));

.1289789673e-1

-.4514263855e-2, .1289789673e-1

Beyond nu_c

>    implicitplot(subs(nu=2,algU),w=0..0.02,U=0..0.4);

[Maple Plot]

The radius is now  rho_1

>    fsolve(subs(nu=2,algr1));fsolve(subs(nu=2,algr2));

.4967886142e-2

0., .3906250000e-2

The type of singularity of U

Before  nu_c

>    subs(nu=4/5,algr2);

-14148/625+169344/625*w+7077888/625*w^2

A square root singularity

>    op(2,algeqtoseries(subs(nu=4/5,w=RootOf(-14148/625+7077888/625*w^2+169344/625*w)*(1-x),algU),x,U,2,true));

124/297-2048/297*RootOf(-131+1568*_Z+65536*_Z^2)+RootOf(37888*RootOf(-131+1568*_Z+65536*_Z^2)+24057*_Z^2-2096)*x^(1/2)+O(x)

>    evalf(%);

.1808576307+.1818695605*x^(1/2)+O(x)

At the critical point, singularity in 1/3

>    op(2,algeqtoseries(subs(nu=nuc,w=rhoc*(1-x),algU),x,U,2,true));

5/9-1/9*7^(1/2)+RootOf(310*7^(1/2)-425+39366*_Z^3)*x^(1/3)+O(x^(2/3))

Beyond nu_c, a square root again

>    subs(nu=4,algr1);

34359738368*w^3-40108032*w^2-27648*w+27

>    op(2,algeqtoseries(subs(nu=4,w=RootOf(-40108032*w^2-27648*w+27+34359738368*w^3)*(1-x),algU),x,U,2,true));

-3/50-256/3*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)+33554432/75*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)^2+RootOf(-130560*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)+1543503...
-3/50-256/3*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)+33554432/75*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)^2+RootOf(-130560*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)+1543503...
-3/50-256/3*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)+33554432/75*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)^2+RootOf(-130560*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)+1543503...

>    evalf(%);

.959370638e-1+.1049120020*x^(1/2)+O(x)

>   

The form of Q1 in terms of S shows that it will inherit the singularities of S

Equation for Q1t=t*Q1

>    algQ1t:=collect(op(5,factor(resultant(numer(Q1t-Q1tsol),algS,S))),Q1t,factor);

algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...
algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...
algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...
algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...
algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...
algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...
algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...
algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...
algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...
algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...
algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...
algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...
algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...
algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...
algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...
algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...
algQ1t := -2097152*w^3*(1+v)^10*Q1t^5+64*w^2*(1+v)^6*(-768*v^4+81920*w*v^4+208896*w*v^3+3072*v^3-6752*v^2+135168*w*v^2+7360*v-28672*w*v-36864*w-117)*Q1t^4-16*w*(1+v)^2*(1749*v^2-3076*v^3+2823*v^4-1770*...

>    algQ1tnu:=collect(numer(factor(subs(vnu,algQ1t))),Q1t,factor);

algQ1tnu := -2147483648*nu^10*w^3*Q1t^5+4096*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^4-2048*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^...
algQ1tnu := -2147483648*nu^10*w^3*Q1t^5+4096*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^4-2048*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^...
algQ1tnu := -2147483648*nu^10*w^3*Q1t^5+4096*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^4-2048*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^...
algQ1tnu := -2147483648*nu^10*w^3*Q1t^5+4096*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^4-2048*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^...
algQ1tnu := -2147483648*nu^10*w^3*Q1t^5+4096*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^4-2048*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^...
algQ1tnu := -2147483648*nu^10*w^3*Q1t^5+4096*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^4-2048*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^...
algQ1tnu := -2147483648*nu^10*w^3*Q1t^5+4096*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^4-2048*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^...
algQ1tnu := -2147483648*nu^10*w^3*Q1t^5+4096*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^4-2048*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^...
algQ1tnu := -2147483648*nu^10*w^3*Q1t^5+4096*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^4-2048*w*nu^2*(-30624*w*nu^6-49728*w*nu^3+796*w*nu^4-256*nu^4-64*nu^...

Quel est le type de singularité de Q1 ?

Before nu_c

>    subs(nu=4/5,algr2);

-14148/625+169344/625*w+7077888/625*w^2

singularité en 3/2

>    op(2,algeqtoseries(subs(nu=4/5,w=RootOf(-14148/625+7077888/625*w^2+169344/625*w)*(1-x),algQ1tnu),x,Q1t,3,true));

123/262-1536/131*RootOf(-131+1568*_Z+65536*_Z^2)+(-17152/4323*RootOf(-131+1568*_Z+65536*_Z^2)-1/4323)*x+RootOf(2673*_Z^2+2368*RootOf(-131+1568*_Z+65536*_Z^2)-131)*x^(3/2)+O(x^(7/4))
123/262-1536/131*RootOf(-131+1568*_Z+65536*_Z^2)+(-17152/4323*RootOf(-131+1568*_Z+65536*_Z^2)-1/4323)*x+RootOf(2673*_Z^2+2368*RootOf(-131+1568*_Z+65536*_Z^2)-131)*x^(3/2)+O(x^(7/4))

>    evalf(%);

.670689866e-1-.1363958479*x+.1364021704*x^(3/2)+O(x^(7/4))

Au point critique, singularité en 4/3

>    op(2,algeqtoseries(subs(nu=nuc,w=rhoc*(1-x),algQ1tnu),x,Q1t,3,true));

23/30-4/15*7^(1/2)+(1/10-1/10*7^(1/2))*x+RootOf(4096*_Z^3-25*7^(1/2)+55)*x^(4/3)+O(x^(13/9))

Au dela, on retrouve une racine

>    subs(nu=4,algr1);

34359738368*w^3-40108032*w^2-27648*w+27

>    op(2,algeqtoseries(subs(nu=4,w=RootOf(-40108032*w^2-27648*w+27+34359738368*w^3)*(1-x),algQ1tnu),x,Q1t,3,true));

-1/4-128/3*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)+16777216/27*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)^2+(32/3*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)+46137344/135*Root...
-1/4-128/3*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)+16777216/27*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)^2+(32/3*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)+46137344/135*Root...
-1/4-128/3*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)+16777216/27*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)^2+(32/3*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)+46137344/135*Root...
-1/4-128/3*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)+16777216/27*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)^2+(32/3*RootOf(34359738368*_Z^3-40108032*_Z^2-27648*_Z+27)+46137344/135*Root...

>    evalf(%);

.191748338e-1-.407858311e-1*x+.4371333419e-1*x^(3/2)+O(x^(7/4))

>   

>   

>