restart;
with(LinearAlgebra):

# Define indeterminates

QParam := [q1,q2,q3,q4,q5,q6]:
PParam := Matrix([[p1],[p2],[p3],[p4],[p5],[p6],[p7]]):

# Special Fourier parameterization and inverse

F := Matrix([[1,1,1,1,1,1,1],
[1,-1/3,-1/3,1,1,-1/3,1],
[1,1/3,-1/3,-1,-1,1/3,1],
[1,-1,1,-1,-1,-1,1],
[1,1/2,0,-1/2,1/2,-1/2,-1],
[1,-1/2,0,1/2,-1/2,1/2,-1]]):

FI := Matrix([
[1/16,3/16,3/16,1/16,1/4,1/4],
[3/16,-3/16,3/16,-3/16,3/8,-3/8],
[3/8,-3/8,-3/8,3/8,0,0],
[1/16,3/16,-3/16,-1/16,-1/8,1/8],
[1/16,3/16,-3/16,-1/16,1/8,-1/8],
[3/16,-3/16,3/16,-3/16,-3/8,3/8],
[1/16,3/16,3/16,1/16,-1/4,-1/4]
]):

# List of polynomial parametrizations

P0 := [
b0^2*d0^3*g0^5+b0^2*d0^3*g1^5+2*b0^2*d0^2*d1*g0^4*g1+b0^2*d0^2*d1*g0^3*g1^2+b0^2*d0^2*d1*g0^2*g1^3+2*b0^2*d0^2*d1*g0*g1^4+2*b0^2*d0*d1^2*g0^4*g1+b0^2*d0*d1^2*g0^3*g1^2+b0^2*d0*d1^2*g0^2*g1^3+2*b0^2*d0*d1^2*g0*g1^4+b0^2*d1^3*g0^5+b0^2*d1^3*g1^5+2*b0*b1*d0^3*g0^4*g1+2*b0*b1*d0^3*g0*g1^4+2*b0*b1*d0^2*d1*g0^5+2*b0*b1*d0^2*d1*g0^4*g1+2*b0*b1*d0^2*d1*g0^3*g1^2+2*b0*b1*d0^2*d1*g0^2*g1^3+2*b0*b1*d0^2*d1*g0*g1^4+2*b0*b1*d0^2*d1*g1^5+2*b0*b1*d0*d1^2*g0^5+2*b0*b1*d0*d1^2*g0^4*g1+2*b0*b1*d0*d1^2*g0^3*g1^2+2*b0*b1*d0*d1^2*g0^2*g1^3+2*b0*b1*d0*d1^2*g0*g1^4+2*b0*b1*d0*d1^2*g1^5+2*b0*b1*d1^3*g0^4*g1+2*b0*b1*d1^3*g0*g1^4+b1^2*d0^3*g0^5+b1^2*d0^3*g1^5+2*b1^2*d0^2*d1*g0^4*g1+b1^2*d0^2*d1*g0^3*g1^2+b1^2*d0^2*d1*g0^2*g1^3+2*b1^2*d0^2*d1*g0*g1^4+2*b1^2*d0*d1^2*g0^4*g1+b1^2*d0*d1^2*g0^3*g1^2+b1^2*d0*d1^2*g0^2*g1^3+2*b1^2*d0*d1^2*g0*g1^4+b1^2*d1^3*g0^5+b1^2*d1^3*g1^5,
3*b0^2*d0^3*g0^4*g1+3*b0^2*d0^3*g0*g1^4+9*b0^2*d0^2*d1*g0^3*g1^2+9*b0^2*d0^2*d1*g0^2*g1^3+9*b0^2*d0*d1^2*g0^3*g1^2+9*b0^2*d0*d1^2*g0^2*g1^3+3*b0^2*d1^3*g0^4*g1+3*b0^2*d1^3*g0*g1^4+6*b0*b1*d0^3*g0^3*g1^2+6*b0*b1*d0^3*g0^2*g1^3+6*b0*b1*d0^2*d1*g0^4*g1+12*b0*b1*d0^2*d1*g0^3*g1^2+12*b0*b1*d0^2*d1*g0^2*g1^3+6*b0*b1*d0^2*d1*g0*g1^4+6*b0*b1*d0*d1^2*g0^4*g1+12*b0*b1*d0*d1^2*g0^3*g1^2+12*b0*b1*d0*d1^2*g0^2*g1^3+6*b0*b1*d0*d1^2*g0*g1^4+6*b0*b1*d1^3*g0^3*g1^2+6*b0*b1*d1^3*g0^2*g1^3+3*b1^2*d0^3*g0^4*g1+3*b1^2*d0^3*g0*g1^4+9*b1^2*d0^2*d1*g0^3*g1^2+9*b1^2*d0^2*d1*g0^2*g1^3+9*b1^2*d0*d1^2*g0^3*g1^2+9*b1^2*d0*d1^2*g0^2*g1^3+3*b1^2*d1^3*g0^4*g1+3*b1^2*d1^3*g0*g1^4,
6*b0^2*d0^3*g0^3*g1^2+6*b0^2*d0^3*g0^2*g1^3+6*b0^2*d0^2*d1*g0^4*g1+12*b0^2*d0^2*d1*g0^3*g1^2+12*b0^2*d0^2*d1*g0^2*g1^3+6*b0^2*d0^2*d1*g0*g1^4+6*b0^2*d0*d1^2*g0^4*g1+12*b0^2*d0*d1^2*g0^3*g1^2+12*b0^2*d0*d1^2*g0^2*g1^3+6*b0^2*d0*d1^2*g0*g1^4+6*b0^2*d1^3*g0^3*g1^2+6*b0^2*d1^3*g0^2*g1^3+12*b0*b1*d0^3*g0^3*g1^2+12*b0*b1*d0^3*g0^2*g1^3+12*b0*b1*d0^2*d1*g0^4*g1+24*b0*b1*d0^2*d1*g0^3*g1^2+24*b0*b1*d0^2*d1*g0^2*g1^3+12*b0*b1*d0^2*d1*g0*g1^4+12*b0*b1*d0*d1^2*g0^4*g1+24*b0*b1*d0*d1^2*g0^3*g1^2+24*b0*b1*d0*d1^2*g0^2*g1^3+12*b0*b1*d0*d1^2*g0*g1^4+12*b0*b1*d1^3*g0^3*g1^2+12*b0*b1*d1^3*g0^2*g1^3+6*b1^2*d0^3*g0^3*g1^2+6*b1^2*d0^3*g0^2*g1^3+6*b1^2*d0^2*d1*g0^4*g1+12*b1^2*d0^2*d1*g0^3*g1^2+12*b1^2*d0^2*d1*g0^2*g1^3+6*b1^2*d0^2*d1*g0*g1^4+6*b1^2*d0*d1^2*g0^4*g1+12*b1^2*d0*d1^2*g0^3*g1^2+12*b1^2*d0*d1^2*g0^2*g1^3+6*b1^2*d0*d1^2*g0*g1^4+6*b1^2*d1^3*g0^3*g1^2+6*b1^2*d1^3*g0^2*g1^3,
b0^2*d0^3*g0^3*g1^2+b0^2*d0^3*g0^2*g1^3+b0^2*d0^2*d1*g0^5+2*b0^2*d0^2*d1*g0^4*g1+2*b0^2*d0^2*d1*g0*g1^4+b0^2*d0^2*d1*g1^5+b0^2*d0*d1^2*g0^5+2*b0^2*d0*d1^2*g0^4*g1+2*b0^2*d0*d1^2*g0*g1^4+b0^2*d0*d1^2*g1^5+b0^2*d1^3*g0^3*g1^2+b0^2*d1^3*g0^2*g1^3+2*b0*b1*d0^3*g0^4*g1+2*b0*b1*d0^3*g0*g1^4+2*b0*b1*d0^2*d1*g0^5+2*b0*b1*d0^2*d1*g0^4*g1+2*b0*b1*d0^2*d1*g0^3*g1^2+2*b0*b1*d0^2*d1*g0^2*g1^3+2*b0*b1*d0^2*d1*g0*g1^4+2*b0*b1*d0^2*d1*g1^5+2*b0*b1*d0*d1^2*g0^5+2*b0*b1*d0*d1^2*g0^4*g1+2*b0*b1*d0*d1^2*g0^3*g1^2+2*b0*b1*d0*d1^2*g0^2*g1^3+2*b0*b1*d0*d1^2*g0*g1^4+2*b0*b1*d0*d1^2*g1^5+2*b0*b1*d1^3*g0^4*g1+2*b0*b1*d1^3*g0*g1^4+b1^2*d0^3*g0^3*g1^2+b1^2*d0^3*g0^2*g1^3+b1^2*d0^2*d1*g0^5+2*b1^2*d0^2*d1*g0^4*g1+2*b1^2*d0^2*d1*g0*g1^4+b1^2*d0^2*d1*g1^5+b1^2*d0*d1^2*g0^5+2*b1^2*d0*d1^2*g0^4*g1+2*b1^2*d0*d1^2*g0*g1^4+b1^2*d0*d1^2*g1^5+b1^2*d1^3*g0^3*g1^2+b1^2*d1^3*g0^2*g1^3,
b0^2*d0^3*g0^4*g1+b0^2*d0^3*g0*g1^4+b0^2*d0^2*d1*g0^5+b0^2*d0^2*d1*g0^4*g1+b0^2*d0^2*d1*g0^3*g1^2+b0^2*d0^2*d1*g0^2*g1^3+b0^2*d0^2*d1*g0*g1^4+b0^2*d0^2*d1*g1^5+b0^2*d0*d1^2*g0^5+b0^2*d0*d1^2*g0^4*g1+b0^2*d0*d1^2*g0^3*g1^2+b0^2*d0*d1^2*g0^2*g1^3+b0^2*d0*d1^2*g0*g1^4+b0^2*d0*d1^2*g1^5+b0^2*d1^3*g0^4*g1+b0^2*d1^3*g0*g1^4+2*b0*b1*d0^3*g0^3*g1^2+2*b0*b1*d0^3*g0^2*g1^3+2*b0*b1*d0^2*d1*g0^5+4*b0*b1*d0^2*d1*g0^4*g1+4*b0*b1*d0^2*d1*g0*g1^4+2*b0*b1*d0^2*d1*g1^5+2*b0*b1*d0*d1^2*g0^5+4*b0*b1*d0*d1^2*g0^4*g1+4*b0*b1*d0*d1^2*g0*g1^4+2*b0*b1*d0*d1^2*g1^5+2*b0*b1*d1^3*g0^3*g1^2+2*b0*b1*d1^3*g0^2*g1^3+b1^2*d0^3*g0^4*g1+b1^2*d0^3*g0*g1^4+b1^2*d0^2*d1*g0^5+b1^2*d0^2*d1*g0^4*g1+b1^2*d0^2*d1*g0^3*g1^2+b1^2*d0^2*d1*g0^2*g1^3+b1^2*d0^2*d1*g0*g1^4+b1^2*d0^2*d1*g1^5+b1^2*d0*d1^2*g0^5+b1^2*d0*d1^2*g0^4*g1+b1^2*d0*d1^2*g0^3*g1^2+b1^2*d0*d1^2*g0^2*g1^3+b1^2*d0*d1^2*g0*g1^4+b1^2*d0*d1^2*g1^5+b1^2*d1^3*g0^4*g1+b1^2*d1^3*g0*g1^4,
3*b0^2*d0^3*g0^3*g1^2+3*b0^2*d0^3*g0^2*g1^3+3*b0^2*d0^2*d1*g0^4*g1+6*b0^2*d0^2*d1*g0^3*g1^2+6*b0^2*d0^2*d1*g0^2*g1^3+3*b0^2*d0^2*d1*g0*g1^4+3*b0^2*d0*d1^2*g0^4*g1+6*b0^2*d0*d1^2*g0^3*g1^2+6*b0^2*d0*d1^2*g0^2*g1^3+3*b0^2*d0*d1^2*g0*g1^4+3*b0^2*d1^3*g0^3*g1^2+3*b0^2*d1^3*g0^2*g1^3+6*b0*b1*d0^3*g0^4*g1+6*b0*b1*d0^3*g0*g1^4+18*b0*b1*d0^2*d1*g0^3*g1^2+18*b0*b1*d0^2*d1*g0^2*g1^3+18*b0*b1*d0*d1^2*g0^3*g1^2+18*b0*b1*d0*d1^2*g0^2*g1^3+6*b0*b1*d1^3*g0^4*g1+6*b0*b1*d1^3*g0*g1^4+3*b1^2*d0^3*g0^3*g1^2+3*b1^2*d0^3*g0^2*g1^3+3*b1^2*d0^2*d1*g0^4*g1+6*b1^2*d0^2*d1*g0^3*g1^2+6*b1^2*d0^2*d1*g0^2*g1^3+3*b1^2*d0^2*d1*g0*g1^4+3*b1^2*d0*d1^2*g0^4*g1+6*b1^2*d0*d1^2*g0^3*g1^2+6*b1^2*d0*d1^2*g0^2*g1^3+3*b1^2*d0*d1^2*g0*g1^4+3*b1^2*d1^3*g0^3*g1^2+3*b1^2*d1^3*g0^2*g1^3,
b0^2*d0^3*g0^4*g1+b0^2*d0^3*g0*g1^4+b0^2*d0^2*d1*g0^5+b0^2*d0^2*d1*g0^4*g1+b0^2*d0^2*d1*g0^3*g1^2+b0^2*d0^2*d1*g0^2*g1^3+b0^2*d0^2*d1*g0*g1^4+b0^2*d0^2*d1*g1^5+b0^2*d0*d1^2*g0^5+b0^2*d0*d1^2*g0^4*g1+b0^2*d0*d1^2*g0^3*g1^2+b0^2*d0*d1^2*g0^2*g1^3+b0^2*d0*d1^2*g0*g1^4+b0^2*d0*d1^2*g1^5+b0^2*d1^3*g0^4*g1+b0^2*d1^3*g0*g1^4+2*b0*b1*d0^3*g0^5+2*b0*b1*d0^3*g1^5+4*b0*b1*d0^2*d1*g0^4*g1+2*b0*b1*d0^2*d1*g0^3*g1^2+2*b0*b1*d0^2*d1*g0^2*g1^3+4*b0*b1*d0^2*d1*g0*g1^4+4*b0*b1*d0*d1^2*g0^4*g1+2*b0*b1*d0*d1^2*g0^3*g1^2+2*b0*b1*d0*d1^2*g0^2*g1^3+4*b0*b1*d0*d1^2*g0*g1^4+2*b0*b1*d1^3*g0^5+2*b0*b1*d1^3*g1^5+b1^2*d0^3*g0^4*g1+b1^2*d0^3*g0*g1^4+b1^2*d0^2*d1*g0^5+b1^2*d0^2*d1*g0^4*g1+b1^2*d0^2*d1*g0^3*g1^2+b1^2*d0^2*d1*g0^2*g1^3+b1^2*d0^2*d1*g0*g1^4+b1^2*d0^2*d1*g1^5+b1^2*d0*d1^2*g0^5+b1^2*d0*d1^2*g0^4*g1+b1^2*d0*d1^2*g0^3*g1^2+b1^2*d0*d1^2*g0^2*g1^3+b1^2*d0*d1^2*g0*g1^4+b1^2*d0*d1^2*g1^5+b1^2*d1^3*g0^4*g1+b1^2*d1^3*g0*g1^4]:

# Substitutions based on the model

P := P0:
P := subs(b0 = 1-b1, P):
P := subs(d0 = 1-d1, P):
P := subs(g0 = 1-g1, P):

# Check that the polynomial parametrization lies in the probability simplex

suma := 0:
for i from 1 to nops(P) do
suma := suma + P[i]:
od:
normal(expand(suma));

# Ideal of Invariants in Fourier coordinates

Invariants := Matrix([
q4*q5-q3*q6,
q2*q5-q1*q6,
q2*q3-q1*q4]):

# Ideal of Invariants in probability coordinates

Fourier := MatrixMatrixMultiply(F,PParam):

PInvariants := Invariants:
for i from 1 to nops(QParam) do
PInvariants := subs(QParam[i] = Fourier[i, 1], PInvariants):
od:

# Evaluation of Invariants at the polynomial/rational parametrization

num := op(PInvariants[1,1..-1])[1]:
for j from 1 to num do
coordpoly := PInvariants[1, j]:
for i from 1 to op(PParam[1..-1,1])[1] do
coordpoly := subs(PParam[i, 1] = P0[i], coordpoly):
od:
coordpoly :=expand(coordpoly):
lprint(j,coordpoly);
od: