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],
[1,1/9,1/9,-1/3,1,1/9,-1/3],
[1,-1/3,-1/3,1/3,1,-1/3,1/3],
[1,5/9,1/9,1/9,-1/3,-1/3,-1/3],
[1,1/9,-1/3,-1/9,-1/3,1/9,1/3],
[1,-1/3,5/21,1/21,-1/3,1/21,-1/7]]):

FI := Matrix([
[1/64,9/64,3/32,9/64,9/32,21/64],
[9/64,9/64,-9/32,45/64,9/32,-63/64],
[9/64,9/64,-9/32,9/64,-27/32,45/64],
[9/32,-27/32,9/16,9/32,-9/16,9/32],
[3/64,27/64,9/32,-9/64,-9/32,-21/64],
[9/32,9/32,-9/16,-27/32,9/16,9/32],
[3/32,-9/32,3/16,-9/32,9/16,-9/32]]):

# List of polynomial parametrizations

P0 := [
b0^2*e0^4+3*b0^2*e1^4+6*b0*b1*e0^3*e1+6*b0*b1*e0*e1^3+12*b0*b1*e1^4+3*b1^2*e0^4+6*b1^2*e0^3*e1+6*b1^2*e0*e1^3+21*b1^2*e1^4,
9*b0^2*e0^3*e1+9*b0^2*e0*e1^3+18*b0^2*e1^4+72*b0*b1*e0^2*e1^2+72*b0*b1*e0*e1^3+72*b0*b1*e1^4+27*b1^2*e0^3*e1+72*b1^2*e0^2*e1^2+99*b1^2*e0*e1^3+126*b1^2*e1^4,
18*b0^2*e0^2*e1^2+18*b0^2*e1^4+18*b0*b1*e0^3*e1+36*b0*b1*e0^2*e1^2+90*b0*b1*e0*e1^3+72*b0*b1*e1^4+18*b1^2*e0^3*e1+90*b1^2*e0^2*e1^2+90*b1^2*e0*e1^3+126*b1^2*e1^4,
18*b0^2*e0^2*e1^2+36*b0^2*e0*e1^3+18*b0^2*e1^4+72*b0*b1*e0^2*e1^2+288*b0*b1*e0*e1^3+72*b0*b1*e1^4+126*b1^2*e0^2*e1^2+396*b1^2*e0*e1^3+126*b1^2*e1^4,
3*b0^2*e0^3*e1+3*b0^2*e0*e1^3+6*b0^2*e1^4+6*b0*b1*e0^4+12*b0*b1*e0^3*e1+12*b0*b1*e0*e1^3+42*b0*b1*e1^4+6*b1^2*e0^4+21*b1^2*e0^3*e1+21*b1^2*e0*e1^3+60*b1^2*e1^4,
18*b0^2*e0^2*e1^2+36*b0^2*e0*e1^3+18*b0^2*e1^4+36*b0*b1*e0^3*e1+108*b0*b1*e0^2*e1^2+108*b0*b1*e0*e1^3+180*b0*b1*e1^4+36*b1^2*e0^3*e1+162*b1^2*e0^2*e1^2+216*b1^2*e0*e1^3+234*b1^2*e1^4,
24*b0^2*e0*e1^3+36*b0*b1*e0^2*e1^2+72*b0*b1*e0*e1^3+36*b0*b1*e1^4+36*b1^2*e0^2*e1^2+144*b1^2*e0*e1^3+36*b1^2*e1^4]:

# Substitutions based on the model

P := P0:
P := subs(b0 = 1-3*b1, P):
P := subs(e0 = 1-3*e1, 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([
q5^2-q4*q6,
q3*q5-q2*q6,
q3*q4-q2*q5,
q2*q4-q1*q6,
q2^3-q1*q3^2]):

# 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: