restart;
with(LinearAlgebra):

# Define indeterminates

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

# Special Fourier parameterization and inverse

F := Matrix([
[1,1,1],
[1,0,-1/3],
[1,-1,1]]):

FI := Matrix([
[1/8,3/4,1/8],
[1/2,0,-1/2],
[3/8,-3/4,3/8]]):

# List of polynomial parametrizations

P0 := [
d0^4+d1^4,
4*d0^3*d1+4*d0*d1^3,
6*d0^2*d1^2]:

# Substitutions based on the model

P := P0:
P := subs(d0 = 1-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([
q2^2-q1*q3]):

# 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: