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],
[1,1/3,1/9,-1/9,-1/3],
[1,0,-1/3,0,1/3],
[1,-1/3,5/21,1/21,-1/7]]):

FI := Matrix([
[1/64,9/32,3/8,21/64],
[3/16,9/8,0,-21/16],
[9/64,9/32,-9/8,45/64],
[9/16,-9/8,0,9/16],
[3/32,-9/16,3/4,-9/32]]):

# List of polynomial parametrizations

P0 := [
d0^4+3*d1^4,
12*d0^3*d1+12*d0*d1^3+24*d1^4,
18*d0^2*d1^2+18*d1^4,
36*d0^2*d1^2+72*d0*d1^3+36*d1^4,
24*d0*d1^3]:

# Substitutions based on the model

P := P0:
P := subs(d0 = 1-3*d1, 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([
q3^2-q2*q4,
q2^2-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: