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,-1/3],
[1,-1/3,1,-1/3],
[1,-1/3,-1/3,1/3]]):

FI := Matrix([
[1/16,3/8,3/16,3/8],
[3/8,3/4,-3/8,-3/4],
[3/16,-3/8,9/16,-3/8],
[3/8,-3/4,-3/8,3/4]]):

# List of polynomial parametrizations

P0 := [
c0^3+2*c1^3+c2^3,
6*c0^2*c1+6*c0*c1^2+6*c1^2*c2+6*c1*c2^2,
3*c0^2*c2+3*c0*c2^2+6*c1^3,
6*c0*c1^2+12*c0*c1*c2+6*c1^2*c2]:

# Substitutions based on the model

P := P0:
P := subs(c0 = 1-2*c1-c2, 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([
q2^2*q3-q1*q4^2
]):

# 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: