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,-1,1],
[1,0,-1]]):

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

# List of polynomial parametrizations

P0 := [
b0^2*d0^3+b0^2*d1^3+2*b0*b1*d0^2*d1+2*b0*b1*d0*d1^2+b1^2*d0^3+b1^2*d1^3,
2*b0^2*d0^2*d1+2*b0^2*d0*d1^2+4*b0*b1*d0^2*d1+4*b0*b1*d0*d1^2+2*b1^2*d0^2*d1+2*b1^2*d0*d1^2,
b0^2*d0^2*d1+b0^2*d0*d1^2+2*b0*b1*d0^3+2*b0*b1*d1^3+b1^2*d0^2*d1+b1^2*d0*d1^2]:

# Substitutions based on the model

P := P0:
P := subs(b0 = 1-b1, P):
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([
]):

# 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: