Le cas nu=0 (triangulations bien 3-coloriées, donc eulériennes)

When  nu=0

>    subs(nu=0,eqcatQ);

2520*t*D(Q)(0)*y^7

Let us first express  Q1 and Q3 in terms of Q2 and Q4, using the following expressions of the second and fourth derivative:

>    D2Qsol;

2*D(Q)(0)/nu/t

>    D4Qsol;

(-144-24/nu^2/t^3)*D(Q)(0)+4/nu*(1+nu)/t*`@@`(D,3)(Q)(0)+24/nu*(2+nu)/t

>    isolate(D4Qsol-`@@`(D,4)(Q)(0), `@@`(D,3)(Q)(0));

`@@`(D,3)(Q)(0) = 1/4*(-(-144-24/nu^2/t^3)*D(Q)(0)-24/nu*(2+nu)/t+`@@`(D,4)(Q)(0))*nu/(1+nu)*t

>    eqcatmixte:=collect(factor(subs(%,D(Q)(0)= nu*t*`@@`(D,2)(Q)(0)/2,eqcatQ/nu)),Q,factor);

eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...
eqcatmixte := (-378*t^7*y^8*nu^4*Q(y)+7*t^5*y^5*nu^2*(-18*t^2*nu^2+20*y^2+59*nu*y^2+29*nu^2*y^2+18*nu^2*y*t))*`@@`(D,5)(Q)(0)+(-34020*nu^6*t^9*y^8*Q(y)-315*t^7*y^5*nu^4*(648*nu^3*t^3*y^2-163*nu^2*y^2+3...

>    eqcatnu0:=collect(op(3,factor(subs(nu=0,eqcatmixte))),Q,factor);

eqcatnu0 := y^2*t^2*`@@`(D,2)(Q)(0)+4*y^5*t^3*Q(y)^3-y^2*t*(-t^2+8*y^2+10*t*y)*Q(y)^2+(4*y^2+6*t^2*y^3-2*t^2+2*t*y)*Q(y)-2*t*y+2*t^2-4*y^2-t^3*y^2

We apply the quadratic method : the discriminant of the above equation (in Q(y)) has a double root (in y)

>    e1:=op(4,factor(discrim(subs(Q(y)=Qy,eqcatnu0),Qy)));

e1 := -120*t^7*y^6*`@@`(D,2)(Q)(0)-216*y^10*t^9*`@@`(D,2)(Q)(0)+6*y^5*t^8*`@@`(D,2)(Q)(0)+108*y^10*t^8*`@@`(D,2)(Q)(0)^2-1344*y^9*t^4*`@@`(D,2)(Q)(0)+648*y^9*t^7*`@@`(D,2)(Q)(0)+324*y^8*t^8*`@@`(D,2)(Q...
e1 := -120*t^7*y^6*`@@`(D,2)(Q)(0)-216*y^10*t^9*`@@`(D,2)(Q)(0)+6*y^5*t^8*`@@`(D,2)(Q)(0)+108*y^10*t^8*`@@`(D,2)(Q)(0)^2-1344*y^9*t^4*`@@`(D,2)(Q)(0)+648*y^9*t^7*`@@`(D,2)(Q)(0)+324*y^8*t^8*`@@`(D,2)(Q...
e1 := -120*t^7*y^6*`@@`(D,2)(Q)(0)-216*y^10*t^9*`@@`(D,2)(Q)(0)+6*y^5*t^8*`@@`(D,2)(Q)(0)+108*y^10*t^8*`@@`(D,2)(Q)(0)^2-1344*y^9*t^4*`@@`(D,2)(Q)(0)+648*y^9*t^7*`@@`(D,2)(Q)(0)+324*y^8*t^8*`@@`(D,2)(Q...
e1 := -120*t^7*y^6*`@@`(D,2)(Q)(0)-216*y^10*t^9*`@@`(D,2)(Q)(0)+6*y^5*t^8*`@@`(D,2)(Q)(0)+108*y^10*t^8*`@@`(D,2)(Q)(0)^2-1344*y^9*t^4*`@@`(D,2)(Q)(0)+648*y^9*t^7*`@@`(D,2)(Q)(0)+324*y^8*t^8*`@@`(D,2)(Q...
e1 := -120*t^7*y^6*`@@`(D,2)(Q)(0)-216*y^10*t^9*`@@`(D,2)(Q)(0)+6*y^5*t^8*`@@`(D,2)(Q)(0)+108*y^10*t^8*`@@`(D,2)(Q)(0)^2-1344*y^9*t^4*`@@`(D,2)(Q)(0)+648*y^9*t^7*`@@`(D,2)(Q)(0)+324*y^8*t^8*`@@`(D,2)(Q...

We now take the discriminant wrt y

>    e2:=factor(discrim(subs(`@@`(D,2)(Q)(0)=2*Q2,e1),y));

e2 := 87747802561511424*(-2+t^2*Q2)^2*(-2*t+22*t^4+2*t^7+Q2-12*t^3*Q2-8*Q2*t^6+8*Q2^2*t^5)^3*(t^12-4*t^11*Q2+4*t^10*Q2^2-14*t^9-456*t^8*Q2+1040*t^6-4864*t^5*Q2+9600*t^3-1024)^6*t^75
e2 := 87747802561511424*(-2+t^2*Q2)^2*(-2*t+22*t^4+2*t^7+Q2-12*t^3*Q2-8*Q2*t^6+8*Q2^2*t^5)^3*(t^12-4*t^11*Q2+4*t^10*Q2^2-14*t^9-456*t^8*Q2+1040*t^6-4864*t^5*Q2+9600*t^3-1024)^6*t^75

>    alg1:=collect(op(1,op(3,e2)),Q2,factor); alg2:=collect(op(1,op(4,e2)),Q2,factor);

alg1 := 8*Q2^2*t^5+(1-12*t^3-8*t^6)*Q2+2*t*(-1+11*t^3+t^6)

alg2 := 4*t^10*Q2^2-4*t^5*(1216+t^6+114*t^3)*Q2+(2+t)*(t^9-22*t^6+1216*t^3-128)*(t^2-2*t+4)

>    n:=6: factor(series(subs(Q2=coeff(subs(q=3,w=1,z=1,x=0,nu=0,Qser(n)),y,2),alg1),t,n+1));

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

>    n:=3: factor(series(subs(Q2=coeff(subs(q=3,w=1,z=1,x=0,nu=0,Qser(n)),y,2),alg2),t,n+1));

series(-1024+9600*t^3+O(t^4),t,4)

C'est donc alg1

>    factor([solve(alg1,Q2)]);

[1/16/t^5*(-1+12*t^3+8*t^6+(-(2*t-1)^3*(1+2*t+4*t^2)^3)^(1/2)), 1/16/t^5*(-1+12*t^3+8*t^6-(-(2*t-1)^3*(1+2*t+4*t^2)^3)^(1/2))]

>    series(op(1,%)/2,t,32);

series(1*t+1*t^4+3*t^7+12*t^10+56*t^13+288*t^16+1584*t^19+9152*t^22+54912*t^25+O(t^28),t,28)

C'est bien les cartes bicubiques

>