restart;
with(LinearAlgebra):

# Define indeterminates

QParam := [q1,q2,q3,q4,q5]:
PParam := Matrix([[p1],[p2],[p3],[p4],[p5],[p6]]):

# Special Fourier parameterization and inverse

F := Matrix([[1,1,1,1,1,1],
[1,-1,1,1,-1,1],
[1,1/3,-1/3,1/9,-1/9,-1/3],
[1,-1,1,-1/3,1/3,-1/3],
[1,0,-1,-1,0,1]]):

FI := Matrix([
[1/16,1/16,9/16,3/16,1/8],
[1/8,-1/8,3/8,-3/8,0],
[1/16,1/16,-3/16,3/16,-1/8],
[3/16,3/16,3/16,-3/16,-3/8],
[3/8,-3/8,-3/8,3/8,0],
[3/16,3/16,-9/16,-3/16,3/8]
]):

# List of polynomial parametrizations

P0 := [
d0^4*f0^5+d0^4*f1^5+3*d0^3*d1*f0^4*f1+d0^3*d1*f0^3*f1^2+d0^3*d1*f0^2*f1^3+3*d0^3*d1*f0*f1^4+6*d0^2*d1^2*f0^3*f1^2+6*d0^2*d1^2*f0^2*f1^3+3*d0*d1^3*f0^4*f1+d0*d1^3*f0^3*f1^2+d0*d1^3*f0^2*f1^3+3*d0*d1^3*f0*f1^4+d1^4*f0^5+d1^4*f1^5,
2*d0^4*f0^4*f1+2*d0^4*f0*f1^4+2*d0^3*d1*f0^4*f1+6*d0^3*d1*f0^3*f1^2+6*d0^3*d1*f0^2*f1^3+2*d0^3*d1*f0*f1^4+12*d0^2*d1^2*f0^3*f1^2+12*d0^2*d1^2*f0^2*f1^3+2*d0*d1^3*f0^4*f1+6*d0*d1^3*f0^3*f1^2+6*d0*d1^3*f0^2*f1^3+2*d0*d1^3*f0*f1^4+2*d1^4*f0^4*f1+2*d1^4*f0*f1^4,
d0^4*f0^3*f1^2+d0^4*f0^2*f1^3+d0^3*d1*f0^5+3*d0^3*d1*f0^3*f1^2+3*d0^3*d1*f0^2*f1^3+d0^3*d1*f1^5+6*d0^2*d1^2*f0^4*f1+6*d0^2*d1^2*f0*f1^4+d0*d1^3*f0^5+3*d0*d1^3*f0^3*f1^2+3*d0*d1^3*f0^2*f1^3+d0*d1^3*f1^5+d1^4*f0^3*f1^2+d1^4*f0^2*f1^3,
3*d0^4*f0^4*f1+3*d0^4*f0*f1^4+3*d0^3*d1*f0^5+9*d0^3*d1*f0^3*f1^2+9*d0^3*d1*f0^2*f1^3+3*d0^3*d1*f1^5+12*d0^2*d1^2*f0^4*f1+6*d0^2*d1^2*f0^3*f1^2+6*d0^2*d1^2*f0^2*f1^3+12*d0^2*d1^2*f0*f1^4+3*d0*d1^3*f0^5+9*d0*d1^3*f0^3*f1^2+9*d0*d1^3*f0^2*f1^3+3*d0*d1^3*f1^5+3*d1^4*f0^4*f1+3*d1^4*f0*f1^4,
6*d0^4*f0^3*f1^2+6*d0^4*f0^2*f1^3+6*d0^3*d1*f0^4*f1+18*d0^3*d1*f0^3*f1^2+18*d0^3*d1*f0^2*f1^3+6*d0^3*d1*f0*f1^4+12*d0^2*d1^2*f0^4*f1+24*d0^2*d1^2*f0^3*f1^2+24*d0^2*d1^2*f0^2*f1^3+12*d0^2*d1^2*f0*f1^4+6*d0*d1^3*f0^4*f1+18*d0*d1^3*f0^3*f1^2+18*d0*d1^3*f0^2*f1^3+6*d0*d1^3*f0*f1^4+6*d1^4*f0^3*f1^2+6*d1^4*f0^2*f1^3,
3*d0^4*f0^3*f1^2+3*d0^4*f0^2*f1^3+9*d0^3*d1*f0^4*f1+3*d0^3*d1*f0^3*f1^2+3*d0^3*d1*f0^2*f1^3+9*d0^3*d1*f0*f1^4+6*d0^2*d1^2*f0^5+12*d0^2*d1^2*f0^3*f1^2+12*d0^2*d1^2*f0^2*f1^3+6*d0^2*d1^2*f1^5+9*d0*d1^3*f0^4*f1+3*d0*d1^3*f0^3*f1^2+3*d0*d1^3*f0^2*f1^3+9*d0*d1^3*f0*f1^4+3*d1^4*f0^3*f1^2+3*d1^4*f0^2*f1^3]:

# Substitutions based on the model

P := P0:
P := subs(d0 = 1-d1, 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([
q3*q4-q2*q5,
q3^2-q1*q5,
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: