restart;
with(LinearAlgebra):

# Define indeterminates

QParam := [q1,q2,q3,q4]:
PParam := Matrix([[p1],[p2],[p3],[p4]]):

# Special Fourier parameterization and inverse

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

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

# List of polynomial parametrizations

P0 := [
b0^2*e0^4+b0^2*e1^4+2*b0*b1*e0^3*e1+2*b0*b1*e0*e1^3+b1^2*e0^4+b1^2*e1^4,
3*b0^2*e0^3*e1+3*b0^2*e0*e1^3+12*b0*b1*e0^2*e1^2+3*b1^2*e0^3*e1+3*b1^2*e0*e1^3,
6*b0^2*e0^2*e1^2+6*b0*b1*e0^3*e1+6*b0*b1*e0*e1^3+6*b1^2*e0^2*e1^2,
b0^2*e0^3*e1+b0^2*e0*e1^3+2*b0*b1*e0^4+2*b0*b1*e1^4+b1^2*e0^3*e1+b1^2*e0*e1^3]:

# Substitutions based on the model

P := P0:
P := subs(b0 = 1-b1, P):
P := subs(e0 = 1-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([
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: