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],
[1,7/15,1/5,1/15,-1/15,-1/5],
[1,1/5,-1/5,0,-1/15,2/15],
[1,-1/15,1/105,-11/105,3/35,-1/35],
[1,-1/3,1/15,2/15,-1/15,0]]):

FI := Matrix([
[1/256,15/128,15/64,105/256,15/64],
[15/256,105/128,45/64,-105/256,-75/64],
[15/128,45/64,-45/32,15/128,15/32],
[15/64,15/32,0,-165/64,15/8],
[45/128,-45/64,-45/32,405/128,-45/32],
[15/64,-45/32,15/8,-45/64,0]]):

# List of polynomial parametrizations

P0 := [
e0^5+3*e1^5,
15*e0^4*e1+15*e0*e1^4+30*e1^5,
30*e0^3*e1^2+30*e0^2*e1^3+60*e1^5,
60*e0^3*e1^2+120*e0*e1^4+60*e1^5,
180*e0^2*e1^3+90*e0*e1^4+90*e1^5,
60*e0^2*e1^3+180*e0*e1^4]:

# Substitutions based on the model

P := P0:
P := subs(e0 = 1-3*e1, 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([
q4^2-q3*q5,
q3*q4-q2*q5,
q3^2-q2*q4,
q2*q3-q1*q5,
q2^2-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: