restart;
with(LinearAlgebra):

# Define indeterminates

QParam := [q1,q2,q3,q4,q5,q6]:
PParam := Matrix([[p1],[p2],[p3],[p4],[p5],[p6],[p7],[p8]]):

# Special Fourier parameterization and inverse

F := Matrix([[1,1,1,1,1,1,1,1],
[1,-1,1,1,-1,1,1,1],
[1,0,-1,-1,0,1,1,-1],
[1,3/7,-1/7,3/7,-1/7,-5/7,-1/7,-1/7],
[1,-1,1,-1/3,1/3,-1/3,-1/3,-1/3],
[1,0,-1,-1,0,1,-1,1]]):

FI := Matrix([
[1/16,1/16,1/8,7/16,3/16,1/8],
[1/8,-1/8,0,3/8,-3/8,0],
[1/16,1/16,-1/8,-1/16,3/16,-1/8],
[1/16,1/16,-1/8,3/16,-1/16,-1/8],
[3/8,-3/8,0,-3/8,3/8,0],
[1/16,1/16,1/8,-5/16,-1/16,1/8],
[1/8,1/8,1/4,-1/8,-1/8,-1/4],
[1/8,1/8,-1/4,-1/8,-1/8,1/4]
]):

# List of polynomial parametrizations

P0 := [
c0^3*e0^4*g0^5+c0^3*e0^4*g1^5+3*c0^3*e0^3*e1*g0^4*g1+c0^3*e0^3*e1*g0^3*g1^2+c0^3*e0^3*e1*g0^2*g1^3+3*c0^3*e0^3*e1*g0*g1^4+6*c0^3*e0^2*e1^2*g0^3*g1^2+6*c0^3*e0^2*e1^2*g0^2*g1^3+3*c0^3*e0*e1^3*g0^4*g1+c0^3*e0*e1^3*g0^3*g1^2+c0^3*e0*e1^3*g0^2*g1^3+3*c0^3*e0*e1^3*g0*g1^4+c0^3*e1^4*g0^5+c0^3*e1^4*g1^5+2*c0^2*c1*e0^4*g0^4*g1+c0^2*c1*e0^4*g0^3*g1^2+c0^2*c1*e0^4*g0^2*g1^3+2*c0^2*c1*e0^4*g0*g1^4+2*c0^2*c1*e0^3*e1*g0^5+3*c0^2*c1*e0^3*e1*g0^4*g1+7*c0^2*c1*e0^3*e1*g0^3*g1^2+7*c0^2*c1*e0^3*e1*g0^2*g1^3+3*c0^2*c1*e0^3*e1*g0*g1^4+2*c0^2*c1*e0^3*e1*g1^5+2*c0^2*c1*e0^2*e1^2*g0^5+8*c0^2*c1*e0^2*e1^2*g0^4*g1+8*c0^2*c1*e0^2*e1^2*g0^3*g1^2+8*c0^2*c1*e0^2*e1^2*g0^2*g1^3+8*c0^2*c1*e0^2*e1^2*g0*g1^4+2*c0^2*c1*e0^2*e1^2*g1^5+2*c0^2*c1*e0*e1^3*g0^5+3*c0^2*c1*e0*e1^3*g0^4*g1+7*c0^2*c1*e0*e1^3*g0^3*g1^2+7*c0^2*c1*e0*e1^3*g0^2*g1^3+3*c0^2*c1*e0*e1^3*g0*g1^4+2*c0^2*c1*e0*e1^3*g1^5+2*c0^2*c1*e1^4*g0^4*g1+c0^2*c1*e1^4*g0^3*g1^2+c0^2*c1*e1^4*g0^2*g1^3+2*c0^2*c1*e1^4*g0*g1^4+2*c0*c1^2*e0^4*g0^4*g1+c0*c1^2*e0^4*g0^3*g1^2+c0*c1^2*e0^4*g0^2*g1^3+2*c0*c1^2*e0^4*g0*g1^4+2*c0*c1^2*e0^3*e1*g0^5+3*c0*c1^2*e0^3*e1*g0^4*g1+7*c0*c1^2*e0^3*e1*g0^3*g1^2+7*c0*c1^2*e0^3*e1*g0^2*g1^3+3*c0*c1^2*e0^3*e1*g0*g1^4+2*c0*c1^2*e0^3*e1*g1^5+2*c0*c1^2*e0^2*e1^2*g0^5+8*c0*c1^2*e0^2*e1^2*g0^4*g1+8*c0*c1^2*e0^2*e1^2*g0^3*g1^2+8*c0*c1^2*e0^2*e1^2*g0^2*g1^3+8*c0*c1^2*e0^2*e1^2*g0*g1^4+2*c0*c1^2*e0^2*e1^2*g1^5+2*c0*c1^2*e0*e1^3*g0^5+3*c0*c1^2*e0*e1^3*g0^4*g1+7*c0*c1^2*e0*e1^3*g0^3*g1^2+7*c0*c1^2*e0*e1^3*g0^2*g1^3+3*c0*c1^2*e0*e1^3*g0*g1^4+2*c0*c1^2*e0*e1^3*g1^5+2*c0*c1^2*e1^4*g0^4*g1+c0*c1^2*e1^4*g0^3*g1^2+c0*c1^2*e1^4*g0^2*g1^3+2*c0*c1^2*e1^4*g0*g1^4+c1^3*e0^4*g0^5+c1^3*e0^4*g1^5+3*c1^3*e0^3*e1*g0^4*g1+c1^3*e0^3*e1*g0^3*g1^2+c1^3*e0^3*e1*g0^2*g1^3+3*c1^3*e0^3*e1*g0*g1^4+6*c1^3*e0^2*e1^2*g0^3*g1^2+6*c1^3*e0^2*e1^2*g0^2*g1^3+3*c1^3*e0*e1^3*g0^4*g1+c1^3*e0*e1^3*g0^3*g1^2+c1^3*e0*e1^3*g0^2*g1^3+3*c1^3*e0*e1^3*g0*g1^4+c1^3*e1^4*g0^5+c1^3*e1^4*g1^5,
2*c0^3*e0^4*g0^4*g1+2*c0^3*e0^4*g0*g1^4+2*c0^3*e0^3*e1*g0^4*g1+6*c0^3*e0^3*e1*g0^3*g1^2+6*c0^3*e0^3*e1*g0^2*g1^3+2*c0^3*e0^3*e1*g0*g1^4+12*c0^3*e0^2*e1^2*g0^3*g1^2+12*c0^3*e0^2*e1^2*g0^2*g1^3+2*c0^3*e0*e1^3*g0^4*g1+6*c0^3*e0*e1^3*g0^3*g1^2+6*c0^3*e0*e1^3*g0^2*g1^3+2*c0^3*e0*e1^3*g0*g1^4+2*c0^3*e1^4*g0^4*g1+2*c0^3*e1^4*g0*g1^4+6*c0^2*c1*e0^4*g0^3*g1^2+6*c0^2*c1*e0^4*g0^2*g1^3+6*c0^2*c1*e0^3*e1*g0^4*g1+18*c0^2*c1*e0^3*e1*g0^3*g1^2+18*c0^2*c1*e0^3*e1*g0^2*g1^3+6*c0^2*c1*e0^3*e1*g0*g1^4+12*c0^2*c1*e0^2*e1^2*g0^4*g1+24*c0^2*c1*e0^2*e1^2*g0^3*g1^2+24*c0^2*c1*e0^2*e1^2*g0^2*g1^3+12*c0^2*c1*e0^2*e1^2*g0*g1^4+6*c0^2*c1*e0*e1^3*g0^4*g1+18*c0^2*c1*e0*e1^3*g0^3*g1^2+18*c0^2*c1*e0*e1^3*g0^2*g1^3+6*c0^2*c1*e0*e1^3*g0*g1^4+6*c0^2*c1*e1^4*g0^3*g1^2+6*c0^2*c1*e1^4*g0^2*g1^3+6*c0*c1^2*e0^4*g0^3*g1^2+6*c0*c1^2*e0^4*g0^2*g1^3+6*c0*c1^2*e0^3*e1*g0^4*g1+18*c0*c1^2*e0^3*e1*g0^3*g1^2+18*c0*c1^2*e0^3*e1*g0^2*g1^3+6*c0*c1^2*e0^3*e1*g0*g1^4+12*c0*c1^2*e0^2*e1^2*g0^4*g1+24*c0*c1^2*e0^2*e1^2*g0^3*g1^2+24*c0*c1^2*e0^2*e1^2*g0^2*g1^3+12*c0*c1^2*e0^2*e1^2*g0*g1^4+6*c0*c1^2*e0*e1^3*g0^4*g1+18*c0*c1^2*e0*e1^3*g0^3*g1^2+18*c0*c1^2*e0*e1^3*g0^2*g1^3+6*c0*c1^2*e0*e1^3*g0*g1^4+6*c0*c1^2*e1^4*g0^3*g1^2+6*c0*c1^2*e1^4*g0^2*g1^3+2*c1^3*e0^4*g0^4*g1+2*c1^3*e0^4*g0*g1^4+2*c1^3*e0^3*e1*g0^4*g1+6*c1^3*e0^3*e1*g0^3*g1^2+6*c1^3*e0^3*e1*g0^2*g1^3+2*c1^3*e0^3*e1*g0*g1^4+12*c1^3*e0^2*e1^2*g0^3*g1^2+12*c1^3*e0^2*e1^2*g0^2*g1^3+2*c1^3*e0*e1^3*g0^4*g1+6*c1^3*e0*e1^3*g0^3*g1^2+6*c1^3*e0*e1^3*g0^2*g1^3+2*c1^3*e0*e1^3*g0*g1^4+2*c1^3*e1^4*g0^4*g1+2*c1^3*e1^4*g0*g1^4,
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2*c0^3*e0^4*g0^3*g1^2+2*c0^3*e0^4*g0^2*g1^3+6*c0^3*e0^3*e1*g0^4*g1+2*c0^3*e0^3*e1*g0^3*g1^2+2*c0^3*e0^3*e1*g0^2*g1^3+6*c0^3*e0^3*e1*g0*g1^4+4*c0^3*e0^2*e1^2*g0^5+8*c0^3*e0^2*e1^2*g0^3*g1^2+8*c0^3*e0^2*e1^2*g0^2*g1^3+4*c0^3*e0^2*e1^2*g1^5+6*c0^3*e0*e1^3*g0^4*g1+2*c0^3*e0*e1^3*g0^3*g1^2+2*c0^3*e0*e1^3*g0^2*g1^3+6*c0^3*e0*e1^3*g0*g1^4+2*c0^3*e1^4*g0^3*g1^2+2*c0^3*e1^4*g0^2*g1^3+2*c0^2*c1*e0^4*g0^4*g1+4*c0^2*c1*e0^4*g0^3*g1^2+4*c0^2*c1*e0^4*g0^2*g1^3+2*c0^2*c1*e0^4*g0*g1^4+4*c0^2*c1*e0^3*e1*g0^5+6*c0^2*c1*e0^3*e1*g0^4*g1+14*c0^2*c1*e0^3*e1*g0^3*g1^2+14*c0^2*c1*e0^3*e1*g0^2*g1^3+6*c0^2*c1*e0^3*e1*g0*g1^4+4*c0^2*c1*e0^3*e1*g1^5+4*c0^2*c1*e0^2*e1^2*g0^5+20*c0^2*c1*e0^2*e1^2*g0^4*g1+12*c0^2*c1*e0^2*e1^2*g0^3*g1^2+12*c0^2*c1*e0^2*e1^2*g0^2*g1^3+20*c0^2*c1*e0^2*e1^2*g0*g1^4+4*c0^2*c1*e0^2*e1^2*g1^5+4*c0^2*c1*e0*e1^3*g0^5+6*c0^2*c1*e0*e1^3*g0^4*g1+14*c0^2*c1*e0*e1^3*g0^3*g1^2+14*c0^2*c1*e0*e1^3*g0^2*g1^3+6*c0^2*c1*e0*e1^3*g0*g1^4+4*c0^2*c1*e0*e1^3*g1^5+2*c0^2*c1*e1^4*g0^4*g1+4*c0^2*c1*e1^4*g0^3*g1^2+4*c0^2*c1*e1^4*g0^2*g1^3+2*c0^2*c1*e1^4*g0*g1^4+2*c0*c1^2*e0^4*g0^4*g1+4*c0*c1^2*e0^4*g0^3*g1^2+4*c0*c1^2*e0^4*g0^2*g1^3+2*c0*c1^2*e0^4*g0*g1^4+4*c0*c1^2*e0^3*e1*g0^5+6*c0*c1^2*e0^3*e1*g0^4*g1+14*c0*c1^2*e0^3*e1*g0^3*g1^2+14*c0*c1^2*e0^3*e1*g0^2*g1^3+6*c0*c1^2*e0^3*e1*g0*g1^4+4*c0*c1^2*e0^3*e1*g1^5+4*c0*c1^2*e0^2*e1^2*g0^5+20*c0*c1^2*e0^2*e1^2*g0^4*g1+12*c0*c1^2*e0^2*e1^2*g0^3*g1^2+12*c0*c1^2*e0^2*e1^2*g0^2*g1^3+20*c0*c1^2*e0^2*e1^2*g0*g1^4+4*c0*c1^2*e0^2*e1^2*g1^5+4*c0*c1^2*e0*e1^3*g0^5+6*c0*c1^2*e0*e1^3*g0^4*g1+14*c0*c1^2*e0*e1^3*g0^3*g1^2+14*c0*c1^2*e0*e1^3*g0^2*g1^3+6*c0*c1^2*e0*e1^3*g0*g1^4+4*c0*c1^2*e0*e1^3*g1^5+2*c0*c1^2*e1^4*g0^4*g1+4*c0*c1^2*e1^4*g0^3*g1^2+4*c0*c1^2*e1^4*g0^2*g1^3+2*c0*c1^2*e1^4*g0*g1^4+2*c1^3*e0^4*g0^3*g1^2+2*c1^3*e0^4*g0^2*g1^3+6*c1^3*e0^3*e1*g0^4*g1+2*c1^3*e0^3*e1*g0^3*g1^2+2*c1^3*e0^3*e1*g0^2*g1^3+6*c1^3*e0^3*e1*g0*g1^4+4*c1^3*e0^2*e1^2*g0^5+8*c1^3*e0^2*e1^2*g0^3*g1^2+8*c1^3*e0^2*e1^2*g0^2*g1^3+4*c1^3*e0^2*e1^2*g1^5+6*c1^3*e0*e1^3*g0^4*g1+2*c1^3*e0*e1^3*g0^3*g1^2+2*c1^3*e0*e1^3*g0^2*g1^3+6*c1^3*e0*e1^3*g0*g1^4+2*c1^3*e1^4*g0^3*g1^2+2*c1^3*e1^4*g0^2*g1^3]:

# Substitutions based on the model

P := P0:
P := subs(c0 = 1-c1, P):
P := subs(e0 = 1-e1, P):
P := subs(g0 = 1-g1, 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*q5-q2*q6,
q3*q4-q1*q6,
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: