restart;
with(LinearAlgebra):

# Define indeterminates

QParam := [q1,q2,q3,q4,q5,q6,q7,q8,q9,q10,q11,q12]:
PParam := Matrix([[p1],[p2],[p3],[p4],[p5],[p6],[p7],[p8],[p9],[p10],[p11],[p12],[p13],[p14],[p15]]):

# Special Fourier parameterization and inverse

F := Matrix([
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1],
[1,-1/3,-1/3,1,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,-1/3,1,-1/3],
[1,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3],
[1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1,1,1,-1/3,-1/3,-1/3,-1/3],
[1,-1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,1,-1/3,1/3,-1/3,-1/3,1/3],
[1,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,1,-1/3,-1/3,-1/3],
[1,1,-1/3,-1/3,-1/3,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3],
[1,-1/3,-1/3,-1/3,1/3,1,-1/3,-1/3,-1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3],
[1,1,1,1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3],
[1,-1/3,-1/3,5/21,1/21,-1/3,5/21,1/21,5/21,-1/3,1/21,1/21,1/21,1/21,-1/7],
[1,-1/3,1,-1/3,-1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,1/3],
[1,1,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1/3,1/3,1/3,1/3]]):

FI := Matrix([
[1/64,3/64,3/64,3/64,3/32,3/64,3/64,3/32,3/64,21/64,3/32,3/32],
[3/64,-3/64,-3/64,9/64,-3/32,-3/64,9/64,-3/32,9/64,-21/64,-3/32,9/32],
[3/64,-3/64,9/64,-3/64,-3/32,9/64,-3/64,-3/32,9/64,-21/64,9/32,-3/32],
[3/64,9/64,-3/64,-3/64,-3/32,-3/64,-3/64,-3/32,9/64,15/64,-3/32,-3/32],
[3/32,-3/32,-3/32,-3/32,3/16,-3/32,-3/32,3/16,9/32,3/32,-3/16,-3/16],
[3/64,9/64,-3/64,-3/64,-3/32,9/64,9/64,9/32,-3/64,-21/64,-3/32,-3/32],
[3/64,-3/64,9/64,-3/64,-3/32,-3/64,9/64,-3/32,-3/64,15/64,-3/32,-3/32],
[3/32,-3/32,-3/32,-3/32,3/16,-3/32,9/32,-3/16,-3/32,3/32,3/16,-3/16],
[3/64,-3/64,-3/64,9/64,-3/32,9/64,-3/64,-3/32,-3/64,15/64,-3/32,-3/32],
[3/64,9/64,9/64,9/64,9/32,-3/64,-3/64,-3/32,-3/64,-21/64,-3/32,-3/32],
[3/32,-3/32,-3/32,9/32,-3/16,-3/32,-3/32,3/16,-3/32,3/32,3/16,-3/16],
[3/32,-3/32,-3/32,-3/32,3/16,9/32,-3/32,-3/16,-3/32,3/32,-3/16,3/16],
[3/32,-3/32,9/32,-3/32,-3/16,-3/32,-3/32,3/16,-3/32,3/32,-3/16,3/16],
[3/32,9/32,-3/32,-3/32,-3/16,-3/32,-3/32,-3/16,-3/32,3/32,3/16,3/16],
[3/32,-3/32,-3/32,-3/32,3/16,-3/32,-3/32,3/16,-3/32,-9/32,3/16,3/16]]):

# List of polynomial parametrizations

P0 := [
a0*b0*c0*d0+3*a1*b1*c1*d1,
3*a0*b1*c0*d0+3*a1*b0*c1*d1+6*a1*b1*c1*d1,
3*a0*b0*c0*d1+3*a1*b1*c1*d0+6*a1*b1*c1*d1,
3*a0*b1*c0*d1+3*a1*b0*c1*d0+6*a1*b1*c1*d1,
6*a0*b1*c0*d1+6*a1*b0*c1*d1+6*a1*b1*c1*d0+6*a1*b1*c1*d1,
3*a0*b0*c1*d0+3*a1*b1*c0*d1+6*a1*b1*c1*d1,
3*a0*b1*c1*d0+3*a1*b0*c0*d1+6*a1*b1*c1*d1,
6*a0*b1*c1*d0+6*a1*b0*c1*d1+6*a1*b1*c0*d1+6*a1*b1*c1*d1,
3*a0*b0*c1*d1+3*a1*b1*c0*d0+6*a1*b1*c1*d1,
3*a0*b1*c1*d1+3*a1*b0*c0*d0+6*a1*b1*c1*d1,
6*a0*b1*c1*d1+6*a1*b0*c1*d1+6*a1*b1*c0*d0+6*a1*b1*c1*d1,
6*a0*b0*c1*d1+6*a1*b1*c0*d1+6*a1*b1*c1*d0+6*a1*b1*c1*d1,
6*a0*b1*c1*d1+6*a1*b0*c0*d1+6*a1*b1*c1*d0+6*a1*b1*c1*d1,
6*a0*b1*c1*d1+6*a1*b0*c1*d0+6*a1*b1*c0*d1+6*a1*b1*c1*d1,
6*a0*b1*c1*d1+6*a1*b0*c1*d1+6*a1*b1*c0*d1+6*a1*b1*c1*d0]:

# Substitutions based on the model

P := P0:
P := subs(a0 = 1-3*a1, P):
P := subs(b0 = 1-3*b1, P):
P := subs(c0 = 1-3*c1, P):
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([
q9*q10-q11*q12,
q7*q10-q8*q12,
q6*q10-q8*q11,
q4*q10-q5*q12,
q3*q10-q5*q11,
q8*q9-q7*q11,
q8*q9-q6*q12,
q5*q9-q3*q12,
q5*q9-q4*q11,
q5*q8-q2*q10,
q5*q7-q2*q12,
q5*q7-q4*q8,
q5*q6-q2*q11,
q5*q6-q3*q8,
q4*q6-q1*q10,
q4*q6-q2*q9,
q4*q6-q3*q7,
q4*q7*q9-q1*q12^2,
q3*q6*q9-q1*q11^2,
q2*q6*q7-q1*q8^2,
q2*q3*q4-q1*q5^2]):

# 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: