restart;
with(LinearAlgebra):
with(Groebner):

# Define indeterminates

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

# Special Fourier parameterization and inverse

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

FI := Matrix([
[1/8,1/8,1/2,1/8,1/8],
[1/4,-1/4,0,1/4,-1/4],
[1/8,1/8,-1/2,1/8,1/8],
[1/4,1/4,0,-1/4,-1/4],
[1/4,-1/4,0,-1/4,1/4]
]):
  
# List of polynomial parametrizations

P0 := [
1/(d0^2-d1^2)*b0^2*d0^3*f0^3+1/(d0^2-d1^2)*b0^2*d0^3*f1^3-1/(d0^2-d1^2)*b0^2*d0^2*d1*f0^2*f1-1/(d0^2-d1^2)*b0^2*d0^2*d1*f0*f1^2+1/(d0^2-d1^2)*b0^2*d0*d1^2*f0^2*f1+1/(d0^2-d1^2)*b0^2*d0*d1^2*f0*f1^2-1/(d0^2-d1^2)*b0^2*d1^3*f0^3-1/(d0^2-d1^2)*b0^2*d1^3*f1^3+2/(d0^2-d1^2)*b0*b1*d0^3*f0^2*f1+2/(d0^2-d1^2)*b0*b1*d0^3*f0*f1^2-2/(d0^2-d1^2)*b0*b1*d0^2*d1*f0^3-2/(d0^2-d1^2)*b0*b1*d0^2*d1*f1^3+2/(d0^2-d1^2)*b0*b1*d0*d1^2*f0^3+2/(d0^2-d1^2)*b0*b1*d0*d1^2*f1^3-2/(d0^2-d1^2)*b0*b1*d1^3*f0^2*f1-2/(d0^2-d1^2)*b0*b1*d1^3*f0*f1^2+1/(d0^2-d1^2)*b1^2*d0^3*f0^3+1/(d0^2-d1^2)*b1^2*d0^3*f1^3-1/(d0^2-d1^2)*b1^2*d0^2*d1*f0^2*f1-1/(d0^2-d1^2)*b1^2*d0^2*d1*f0*f1^2+1/(d0^2-d1^2)*b1^2*d0*d1^2*f0^2*f1+1/(d0^2-d1^2)*b1^2*d0*d1^2*f0*f1^2-1/(d0^2-d1^2)*b1^2*d1^3*f0^3-1/(d0^2-d1^2)*b1^2*d1^3*f1^3,
2/(d0^2-d1^2)*b0^2*d0^3*f0^2*f1+2/(d0^2-d1^2)*b0^2*d0^3*f0*f1^2-2/(d0^2-d1^2)*b0^2*d0^2*d1*f0^2*f1-2/(d0^2-d1^2)*b0^2*d0^2*d1*f0*f1^2+2/(d0^2-d1^2)*b0^2*d0*d1^2*f0^2*f1+2/(d0^2-d1^2)*b0^2*d0*d1^2*f0*f1^2-2/(d0^2-d1^2)*b0^2*d1^3*f0^2*f1-2/(d0^2-d1^2)*b0^2*d1^3*f0*f1^2+4/(d0^2-d1^2)*b0*b1*d0^3*f0^2*f1+4/(d0^2-d1^2)*b0*b1*d0^3*f0*f1^2-4/(d0^2-d1^2)*b0*b1*d0^2*d1*f0^2*f1-4/(d0^2-d1^2)*b0*b1*d0^2*d1*f0*f1^2+4/(d0^2-d1^2)*b0*b1*d0*d1^2*f0^2*f1+4/(d0^2-d1^2)*b0*b1*d0*d1^2*f0*f1^2-4/(d0^2-d1^2)*b0*b1*d1^3*f0^2*f1-4/(d0^2-d1^2)*b0*b1*d1^3*f0*f1^2+2/(d0^2-d1^2)*b1^2*d0^3*f0^2*f1+2/(d0^2-d1^2)*b1^2*d0^3*f0*f1^2-2/(d0^2-d1^2)*b1^2*d0^2*d1*f0^2*f1-2/(d0^2-d1^2)*b1^2*d0^2*d1*f0*f1^2+2/(d0^2-d1^2)*b1^2*d0*d1^2*f0^2*f1+2/(d0^2-d1^2)*b1^2*d0*d1^2*f0*f1^2-2/(d0^2-d1^2)*b1^2*d1^3*f0^2*f1-2/(d0^2-d1^2)*b1^2*d1^3*f0*f1^2,
1/(d0^2-d1^2)*b0^2*d0^3*f0^2*f1+1/(d0^2-d1^2)*b0^2*d0^3*f0*f1^2-1/(d0^2-d1^2)*b0^2*d0^2*d1*f0^3-1/(d0^2-d1^2)*b0^2*d0^2*d1*f1^3+1/(d0^2-d1^2)*b0^2*d0*d1^2*f0^3+1/(d0^2-d1^2)*b0^2*d0*d1^2*f1^3-1/(d0^2-d1^2)*b0^2*d1^3*f0^2*f1-1/(d0^2-d1^2)*b0^2*d1^3*f0*f1^2+2/(d0^2-d1^2)*b0*b1*d0^3*f0^3+2/(d0^2-d1^2)*b0*b1*d0^3*f1^3-2/(d0^2-d1^2)*b0*b1*d0^2*d1*f0^2*f1-2/(d0^2-d1^2)*b0*b1*d0^2*d1*f0*f1^2+2/(d0^2-d1^2)*b0*b1*d0*d1^2*f0^2*f1+2/(d0^2-d1^2)*b0*b1*d0*d1^2*f0*f1^2-2/(d0^2-d1^2)*b0*b1*d1^3*f0^3-2/(d0^2-d1^2)*b0*b1*d1^3*f1^3+1/(d0^2-d1^2)*b1^2*d0^3*f0^2*f1+1/(d0^2-d1^2)*b1^2*d0^3*f0*f1^2-1/(d0^2-d1^2)*b1^2*d0^2*d1*f0^3-1/(d0^2-d1^2)*b1^2*d0^2*d1*f1^3+1/(d0^2-d1^2)*b1^2*d0*d1^2*f0^3+1/(d0^2-d1^2)*b1^2*d0*d1^2*f1^3-1/(d0^2-d1^2)*b1^2*d1^3*f0^2*f1-1/(d0^2-d1^2)*b1^2*d1^3*f0*f1^2,
2/(d0^2-d1^2)*b0^2*d0^2*d1*f0^3+2/(d0^2-d1^2)*b0^2*d0^2*d1*f0^2*f1+2/(d0^2-d1^2)*b0^2*d0^2*d1*f0*f1^2+2/(d0^2-d1^2)*b0^2*d0^2*d1*f1^3-2/(d0^2-d1^2)*b0^2*d0*d1^2*f0^3-2/(d0^2-d1^2)*b0^2*d0*d1^2*f0^2*f1-2/(d0^2-d1^2)*b0^2*d0*d1^2*f0*f1^2-2/(d0^2-d1^2)*b0^2*d0*d1^2*f1^3+4/(d0^2-d1^2)*b0*b1*d0^2*d1*f0^3+4/(d0^2-d1^2)*b0*b1*d0^2*d1*f0^2*f1+4/(d0^2-d1^2)*b0*b1*d0^2*d1*f0*f1^2+4/(d0^2-d1^2)*b0*b1*d0^2*d1*f1^3-4/(d0^2-d1^2)*b0*b1*d0*d1^2*f0^3-4/(d0^2-d1^2)*b0*b1*d0*d1^2*f0^2*f1-4/(d0^2-d1^2)*b0*b1*d0*d1^2*f0*f1^2-4/(d0^2-d1^2)*b0*b1*d0*d1^2*f1^3+2/(d0^2-d1^2)*b1^2*d0^2*d1*f0^3+2/(d0^2-d1^2)*b1^2*d0^2*d1*f0^2*f1+2/(d0^2-d1^2)*b1^2*d0^2*d1*f0*f1^2+2/(d0^2-d1^2)*b1^2*d0^2*d1*f1^3-2/(d0^2-d1^2)*b1^2*d0*d1^2*f0^3-2/(d0^2-d1^2)*b1^2*d0*d1^2*f0^2*f1-2/(d0^2-d1^2)*b1^2*d0*d1^2*f0*f1^2-2/(d0^2-d1^2)*b1^2*d0*d1^2*f1^3,
4/(d0^2-d1^2)*b0^2*d0^2*d1*f0^2*f1+4/(d0^2-d1^2)*b0^2*d0^2*d1*f0*f1^2-4/(d0^2-d1^2)*b0^2*d0*d1^2*f0^2*f1-4/(d0^2-d1^2)*b0^2*d0*d1^2*f0*f1^2+8/(d0^2-d1^2)*b0*b1*d0^2*d1*f0^2*f1+8/(d0^2-d1^2)*b0*b1*d0^2*d1*f0*f1^2-8/(d0^2-d1^2)*b0*b1*d0*d1^2*f0^2*f1-8/(d0^2-d1^2)*b0*b1*d0*d1^2*f0*f1^2+4/(d0^2-d1^2)*b1^2*d0^2*d1*f0^2*f1+4/(d0^2-d1^2)*b1^2*d0^2*d1*f0*f1^2-4/(d0^2-d1^2)*b1^2*d0*d1^2*f0^2*f1-4/(d0^2-d1^2)*b1^2*d0*d1^2*f0*f1^2

]:

# Substitutions based on the model

P := P0:
P := subs(d0 = 1-d1, P):
P := subs(b0 = 1-b1, 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*q4-q1*q5
]):

# 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: