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,-1,1],
[1,-1,1,-1],
[1,1,-1,-1]
]):

FI := Matrix([
[1/4,1/4,1/4,1/4],
[1/4,-1/4,-1/4,1/4],
[1/4,-1/4,1/4,-1/4],
[1/4,1/4,-1/4,-1/4]
]):

# List of polynomial parametrizations

P0 := [
a0*b0*c0+a1*b1*c1,
a0*b1*c1+a1*b0*c0,
a0*b0*c1+a1*b1*c0,
a0*b1*c0+a1*b0*c1
]:

# Substitutions based on the model

P := P0:
P := subs(a0 = 1-a1, P):
P := subs(b0 = 1-b1, P):
P := subs(c0 = 1-c1, 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([
  [0]
]):

# 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: