restart;
with(LinearAlgebra):

# Define indeterminates

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

# Special Fourier parameterization and inverse

F := Matrix([[1,1,1,1,1],
[1,-1/3,-1/3,1,1],
[1,3/7,-1/7,-5/7,-1/7],
[1,-3/5,1/5,1/5,-3/5]]):

FI := Matrix([
[1/16,3/16,7/16,5/16],
[3/16,-3/16,9/16,-9/16],
[9/16,-9/16,-9/16,9/16],
[1/16,3/16,-5/16,1/16],
[1/8,3/8,-1/8,-3/8]
]):

# List of polynomial parametrizations

P0 := [
c0^3*f0^5+c0^3*f1^5+2*c0^2*c1*f0^4*f1+c0^2*c1*f0^3*f1^2+c0^2*c1*f0^2*f1^3+2*c0^2*c1*f0*f1^4+2*c0*c1^2*f0^4*f1+c0*c1^2*f0^3*f1^2+c0*c1^2*f0^2*f1^3+2*c0*c1^2*f0*f1^4+c1^3*f0^5+c1^3*f1^5,
3*c0^3*f0^4*f1+3*c0^3*f0*f1^4+9*c0^2*c1*f0^3*f1^2+9*c0^2*c1*f0^2*f1^3+9*c0*c1^2*f0^3*f1^2+9*c0*c1^2*f0^2*f1^3+3*c1^3*f0^4*f1+3*c1^3*f0*f1^4,
9*c0^3*f0^3*f1^2+9*c0^3*f0^2*f1^3+9*c0^2*c1*f0^4*f1+18*c0^2*c1*f0^3*f1^2+18*c0^2*c1*f0^2*f1^3+9*c0^2*c1*f0*f1^4+9*c0*c1^2*f0^4*f1+18*c0*c1^2*f0^3*f1^2+18*c0*c1^2*f0^2*f1^3+9*c0*c1^2*f0*f1^4+9*c1^3*f0^3*f1^2+9*c1^3*f0^2*f1^3,
c0^3*f0^3*f1^2+c0^3*f0^2*f1^3+c0^2*c1*f0^5+2*c0^2*c1*f0^4*f1+2*c0^2*c1*f0*f1^4+c0^2*c1*f1^5+c0*c1^2*f0^5+2*c0*c1^2*f0^4*f1+2*c0*c1^2*f0*f1^4+c0*c1^2*f1^5+c1^3*f0^3*f1^2+c1^3*f0^2*f1^3,
2*c0^3*f0^4*f1+2*c0^3*f0*f1^4+2*c0^2*c1*f0^5+2*c0^2*c1*f0^4*f1+2*c0^2*c1*f0^3*f1^2+2*c0^2*c1*f0^2*f1^3+2*c0^2*c1*f0*f1^4+2*c0^2*c1*f1^5+2*c0*c1^2*f0^5+2*c0*c1^2*f0^4*f1+2*c0*c1^2*f0^3*f1^2+2*c0*c1^2*f0^2*f1^3+2*c0*c1^2*f0*f1^4+2*c0*c1^2*f1^5+2*c1^3*f0^4*f1+2*c1^3*f0*f1^4]:

# Substitutions based on the model

P := P0:
P := subs(c0 = 1-c1, P):
P := subs(f0 = 1-f1, 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: