//This is the ideal Fourier invariants.

ring rQ = 0,(q1,q2,q3,q4,q5,q6,q7,q8,q9,q10),dp; 

ideal Invariants = 
q3*q8*q9-q4*q7*q10,
q2*q7*q9-q1*q8*q10,
q2*q6*q9-q4*q5*q10,
q3*q5*q9-q1*q6*q10,
q2*q6*q7-q3*q5*q8,
q4*q5*q7-q1*q6*q8,
q2*q3*q9^2-q1*q4*q10^2,
q2*q4*q7^2-q1*q3*q8^2,
q3*q4*q5^2-q1*q2*q6^2;

// This is the inverse of the Fourier transform.

matrix ptoq[10][10] =
1/16,1/16,1/16,1/16,1/8,1/8,1/8,1/8,1/8,1/8,
1/8,1/8,-1/8,-1/8,1/4,-1/4,0,0,0,0,
1/8,-1/8,1/8,-1/8,0,0,1/4,-1/4,0,0,
1/8,-1/8,-1/8,1/8,0,0,0,0,1/4,-1/4,
1/16,1/16,1/16,1/16,1/8,1/8,-1/8,-1/8,-1/8,-1/8,
1/8,-1/8,-1/8,1/8,0,0,0,0,-1/4,1/4,
1/8,-1/8,1/8,-1/8,0,0,-1/4,1/4,0,0,
1/16,1/16,1/16,1/16,-1/8,-1/8,1/8,1/8,-1/8,-1/8,
1/8,1/8,-1/8,-1/8,-1/4,1/4,0,0,0,0,
1/16,1/16,1/16,1/16,-1/8,-1/8,-1/8,-1/8,1/8,1/8;

// This is the ring of probability distributions. 

ring rP = 0,(p1,p2,p3,p4,p5,p6,p7,p8,p9,p10),dp;

//This is the Fourier transform.

matrix qtop[10][10] =
1,1,1,1,1,1,1,1,1,1,
1,1,-1,-1,1,-1,-1,1,1,1,
1,-1,1,-1,1,-1,1,1,-1,1,
1,-1,-1,1,1,1,-1,1,-1,1,
1,1,0,0,1,0,0,-1,-1,-1,
1,-1,0,0,1,0,0,-1,1,-1,
1,0,1,0,-1,0,-1,1,0,-1,
1,0,-1,0,-1,0,1,1,0,-1,
1,0,0,1,-1,-1,0,-1,0,1,
1,0,0,-1,-1,1,0,-1,0,1;

ideal Fourier = qtop*transpose(maxideal(1));

// This is the list of polynomial invariants.

map F = rQ, Fourier;
ideal PInvariants =  F(Invariants); 

// This is the polynomial parametrization.

ring r = 0,(b0,b1,b2,b3,d0,d1,d2,d3),dp;

ideal P = 
b0^2*d0^3+b0^2*d1^3+b0^2*d2^3+b0^2*d3^3+2*b0*b1*d0^2*d1+2*b0*b1*d0*d1^2+2*b0*b1*d2^2*d3+2*b0*b1*d2*d3^2+2*b0*b2*d0^2*d2+2*b0*b2*d0*d2^2+2*b0*b2*d1^2*d3+2*b0*b2*d1*d3^2+2*b0*b3*d0^2*d3+2*b0*b3*d0*d3^2+2*b0*b3*d1^2*d2+2*b0*b3*d1*d2^2+b1^2*d0^3+b1^2*d1^3+b1^2*d2^3+b1^2*d3^3+2*b1*b2*d0^2*d3+2*b1*b2*d0*d3^2+2*b1*b2*d1^2*d2+2*b1*b2*d1*d2^2+2*b1*b3*d0^2*d2+2*b1*b3*d0*d2^2+2*b1*b3*d1^2*d3+2*b1*b3*d1*d3^2+b2^2*d0^3+b2^2*d1^3+b2^2*d2^3+b2^2*d3^3+2*b2*b3*d0^2*d1+2*b2*b3*d0*d1^2+2*b2*b3*d2^2*d3+2*b2*b3*d2*d3^2+b3^2*d0^3+b3^2*d1^3+b3^2*d2^3+b3^2*d3^3,
2*b0^2*d0^2*d1+2*b0^2*d0*d1^2+2*b0^2*d2^2*d3+2*b0^2*d2*d3^2+4*b0*b1*d0^2*d1+4*b0*b1*d0*d1^2+4*b0*b1*d2^2*d3+4*b0*b1*d2*d3^2+4*b0*b2*d0*d1*d2+4*b0*b2*d0*d1*d3+4*b0*b2*d0*d2*d3+4*b0*b2*d1*d2*d3+4*b0*b3*d0*d1*d2+4*b0*b3*d0*d1*d3+4*b0*b3*d0*d2*d3+4*b0*b3*d1*d2*d3+2*b1^2*d0^2*d1+2*b1^2*d0*d1^2+2*b1^2*d2^2*d3+2*b1^2*d2*d3^2+4*b1*b2*d0*d1*d2+4*b1*b2*d0*d1*d3+4*b1*b2*d0*d2*d3+4*b1*b2*d1*d2*d3+4*b1*b3*d0*d1*d2+4*b1*b3*d0*d1*d3+4*b1*b3*d0*d2*d3+4*b1*b3*d1*d2*d3+2*b2^2*d0^2*d1+2*b2^2*d0*d1^2+2*b2^2*d2^2*d3+2*b2^2*d2*d3^2+4*b2*b3*d0^2*d1+4*b2*b3*d0*d1^2+4*b2*b3*d2^2*d3+4*b2*b3*d2*d3^2+2*b3^2*d0^2*d1+2*b3^2*d0*d1^2+2*b3^2*d2^2*d3+2*b3^2*d2*d3^2,
2*b0^2*d0^2*d2+2*b0^2*d0*d2^2+2*b0^2*d1^2*d3+2*b0^2*d1*d3^2+4*b0*b1*d0*d1*d2+4*b0*b1*d0*d1*d3+4*b0*b1*d0*d2*d3+4*b0*b1*d1*d2*d3+4*b0*b2*d0^2*d2+4*b0*b2*d0*d2^2+4*b0*b2*d1^2*d3+4*b0*b2*d1*d3^2+4*b0*b3*d0*d1*d2+4*b0*b3*d0*d1*d3+4*b0*b3*d0*d2*d3+4*b0*b3*d1*d2*d3+2*b1^2*d0^2*d2+2*b1^2*d0*d2^2+2*b1^2*d1^2*d3+2*b1^2*d1*d3^2+4*b1*b2*d0*d1*d2+4*b1*b2*d0*d1*d3+4*b1*b2*d0*d2*d3+4*b1*b2*d1*d2*d3+4*b1*b3*d0^2*d2+4*b1*b3*d0*d2^2+4*b1*b3*d1^2*d3+4*b1*b3*d1*d3^2+2*b2^2*d0^2*d2+2*b2^2*d0*d2^2+2*b2^2*d1^2*d3+2*b2^2*d1*d3^2+4*b2*b3*d0*d1*d2+4*b2*b3*d0*d1*d3+4*b2*b3*d0*d2*d3+4*b2*b3*d1*d2*d3+2*b3^2*d0^2*d2+2*b3^2*d0*d2^2+2*b3^2*d1^2*d3+2*b3^2*d1*d3^2,
2*b0^2*d0^2*d3+2*b0^2*d0*d3^2+2*b0^2*d1^2*d2+2*b0^2*d1*d2^2+4*b0*b1*d0*d1*d2+4*b0*b1*d0*d1*d3+4*b0*b1*d0*d2*d3+4*b0*b1*d1*d2*d3+4*b0*b2*d0*d1*d2+4*b0*b2*d0*d1*d3+4*b0*b2*d0*d2*d3+4*b0*b2*d1*d2*d3+4*b0*b3*d0^2*d3+4*b0*b3*d0*d3^2+4*b0*b3*d1^2*d2+4*b0*b3*d1*d2^2+2*b1^2*d0^2*d3+2*b1^2*d0*d3^2+2*b1^2*d1^2*d2+2*b1^2*d1*d2^2+4*b1*b2*d0^2*d3+4*b1*b2*d0*d3^2+4*b1*b2*d1^2*d2+4*b1*b2*d1*d2^2+4*b1*b3*d0*d1*d2+4*b1*b3*d0*d1*d3+4*b1*b3*d0*d2*d3+4*b1*b3*d1*d2*d3+2*b2^2*d0^2*d3+2*b2^2*d0*d3^2+2*b2^2*d1^2*d2+2*b2^2*d1*d2^2+4*b2*b3*d0*d1*d2+4*b2*b3*d0*d1*d3+4*b2*b3*d0*d2*d3+4*b2*b3*d1*d2*d3+2*b3^2*d0^2*d3+2*b3^2*d0*d3^2+2*b3^2*d1^2*d2+2*b3^2*d1*d2^2,
b0^2*d0^2*d1+b0^2*d0*d1^2+b0^2*d2^2*d3+b0^2*d2*d3^2+2*b0*b1*d0^3+2*b0*b1*d1^3+2*b0*b1*d2^3+2*b0*b1*d3^3+2*b0*b2*d0^2*d3+2*b0*b2*d0*d3^2+2*b0*b2*d1^2*d2+2*b0*b2*d1*d2^2+2*b0*b3*d0^2*d2+2*b0*b3*d0*d2^2+2*b0*b3*d1^2*d3+2*b0*b3*d1*d3^2+b1^2*d0^2*d1+b1^2*d0*d1^2+b1^2*d2^2*d3+b1^2*d2*d3^2+2*b1*b2*d0^2*d2+2*b1*b2*d0*d2^2+2*b1*b2*d1^2*d3+2*b1*b2*d1*d3^2+2*b1*b3*d0^2*d3+2*b1*b3*d0*d3^2+2*b1*b3*d1^2*d2+2*b1*b3*d1*d2^2+b2^2*d0^2*d1+b2^2*d0*d1^2+b2^2*d2^2*d3+b2^2*d2*d3^2+2*b2*b3*d0^3+2*b2*b3*d1^3+2*b2*b3*d2^3+2*b2*b3*d3^3+b3^2*d0^2*d1+b3^2*d0*d1^2+b3^2*d2^2*d3+b3^2*d2*d3^2,
2*b0^2*d0*d1*d2+2*b0^2*d0*d1*d3+2*b0^2*d0*d2*d3+2*b0^2*d1*d2*d3+4*b0*b1*d0^2*d3+4*b0*b1*d0*d3^2+4*b0*b1*d1^2*d2+4*b0*b1*d1*d2^2+4*b0*b2*d0^2*d3+4*b0*b2*d0*d3^2+4*b0*b2*d1^2*d2+4*b0*b2*d1*d2^2+4*b0*b3*d0*d1*d2+4*b0*b3*d0*d1*d3+4*b0*b3*d0*d2*d3+4*b0*b3*d1*d2*d3+2*b1^2*d0*d1*d2+2*b1^2*d0*d1*d3+2*b1^2*d0*d2*d3+2*b1^2*d1*d2*d3+4*b1*b2*d0*d1*d2+4*b1*b2*d0*d1*d3+4*b1*b2*d0*d2*d3+4*b1*b2*d1*d2*d3+4*b1*b3*d0^2*d3+4*b1*b3*d0*d3^2+4*b1*b3*d1^2*d2+4*b1*b3*d1*d2^2+2*b2^2*d0*d1*d2+2*b2^2*d0*d1*d3+2*b2^2*d0*d2*d3+2*b2^2*d1*d2*d3+4*b2*b3*d0^2*d3+4*b2*b3*d0*d3^2+4*b2*b3*d1^2*d2+4*b2*b3*d1*d2^2+2*b3^2*d0*d1*d2+2*b3^2*d0*d1*d3+2*b3^2*d0*d2*d3+2*b3^2*d1*d2*d3,
2*b0^2*d0*d1*d2+2*b0^2*d0*d1*d3+2*b0^2*d0*d2*d3+2*b0^2*d1*d2*d3+4*b0*b1*d0^2*d2+4*b0*b1*d0*d2^2+4*b0*b1*d1^2*d3+4*b0*b1*d1*d3^2+4*b0*b2*d0*d1*d2+4*b0*b2*d0*d1*d3+4*b0*b2*d0*d2*d3+4*b0*b2*d1*d2*d3+4*b0*b3*d0^2*d2+4*b0*b3*d0*d2^2+4*b0*b3*d1^2*d3+4*b0*b3*d1*d3^2+2*b1^2*d0*d1*d2+2*b1^2*d0*d1*d3+2*b1^2*d0*d2*d3+2*b1^2*d1*d2*d3+4*b1*b2*d0^2*d2+4*b1*b2*d0*d2^2+4*b1*b2*d1^2*d3+4*b1*b2*d1*d3^2+4*b1*b3*d0*d1*d2+4*b1*b3*d0*d1*d3+4*b1*b3*d0*d2*d3+4*b1*b3*d1*d2*d3+2*b2^2*d0*d1*d2+2*b2^2*d0*d1*d3+2*b2^2*d0*d2*d3+2*b2^2*d1*d2*d3+4*b2*b3*d0^2*d2+4*b2*b3*d0*d2^2+4*b2*b3*d1^2*d3+4*b2*b3*d1*d3^2+2*b3^2*d0*d1*d2+2*b3^2*d0*d1*d3+2*b3^2*d0*d2*d3+2*b3^2*d1*d2*d3,
b0^2*d0^2*d2+b0^2*d0*d2^2+b0^2*d1^2*d3+b0^2*d1*d3^2+2*b0*b1*d0^2*d3+2*b0*b1*d0*d3^2+2*b0*b1*d1^2*d2+2*b0*b1*d1*d2^2+2*b0*b2*d0^3+2*b0*b2*d1^3+2*b0*b2*d2^3+2*b0*b2*d3^3+2*b0*b3*d0^2*d1+2*b0*b3*d0*d1^2+2*b0*b3*d2^2*d3+2*b0*b3*d2*d3^2+b1^2*d0^2*d2+b1^2*d0*d2^2+b1^2*d1^2*d3+b1^2*d1*d3^2+2*b1*b2*d0^2*d1+2*b1*b2*d0*d1^2+2*b1*b2*d2^2*d3+2*b1*b2*d2*d3^2+2*b1*b3*d0^3+2*b1*b3*d1^3+2*b1*b3*d2^3+2*b1*b3*d3^3+b2^2*d0^2*d2+b2^2*d0*d2^2+b2^2*d1^2*d3+b2^2*d1*d3^2+2*b2*b3*d0^2*d3+2*b2*b3*d0*d3^2+2*b2*b3*d1^2*d2+2*b2*b3*d1*d2^2+b3^2*d0^2*d2+b3^2*d0*d2^2+b3^2*d1^2*d3+b3^2*d1*d3^2,
2*b0^2*d0*d1*d2+2*b0^2*d0*d1*d3+2*b0^2*d0*d2*d3+2*b0^2*d1*d2*d3+4*b0*b1*d0*d1*d2+4*b0*b1*d0*d1*d3+4*b0*b1*d0*d2*d3+4*b0*b1*d1*d2*d3+4*b0*b2*d0^2*d1+4*b0*b2*d0*d1^2+4*b0*b2*d2^2*d3+4*b0*b2*d2*d3^2+4*b0*b3*d0^2*d1+4*b0*b3*d0*d1^2+4*b0*b3*d2^2*d3+4*b0*b3*d2*d3^2+2*b1^2*d0*d1*d2+2*b1^2*d0*d1*d3+2*b1^2*d0*d2*d3+2*b1^2*d1*d2*d3+4*b1*b2*d0^2*d1+4*b1*b2*d0*d1^2+4*b1*b2*d2^2*d3+4*b1*b2*d2*d3^2+4*b1*b3*d0^2*d1+4*b1*b3*d0*d1^2+4*b1*b3*d2^2*d3+4*b1*b3*d2*d3^2+2*b2^2*d0*d1*d2+2*b2^2*d0*d1*d3+2*b2^2*d0*d2*d3+2*b2^2*d1*d2*d3+4*b2*b3*d0*d1*d2+4*b2*b3*d0*d1*d3+4*b2*b3*d0*d2*d3+4*b2*b3*d1*d2*d3+2*b3^2*d0*d1*d2+2*b3^2*d0*d1*d3+2*b3^2*d0*d2*d3+2*b3^2*d1*d2*d3,
b0^2*d0^2*d3+b0^2*d0*d3^2+b0^2*d1^2*d2+b0^2*d1*d2^2+2*b0*b1*d0^2*d2+2*b0*b1*d0*d2^2+2*b0*b1*d1^2*d3+2*b0*b1*d1*d3^2+2*b0*b2*d0^2*d1+2*b0*b2*d0*d1^2+2*b0*b2*d2^2*d3+2*b0*b2*d2*d3^2+2*b0*b3*d0^3+2*b0*b3*d1^3+2*b0*b3*d2^3+2*b0*b3*d3^3+b1^2*d0^2*d3+b1^2*d0*d3^2+b1^2*d1^2*d2+b1^2*d1*d2^2+2*b1*b2*d0^3+2*b1*b2*d1^3+2*b1*b2*d2^3+2*b1*b2*d3^3+2*b1*b3*d0^2*d1+2*b1*b3*d0*d1^2+2*b1*b3*d2^2*d3+2*b1*b3*d2*d3^2+b2^2*d0^2*d3+b2^2*d0*d3^2+b2^2*d1^2*d2+b2^2*d1*d2^2+2*b2*b3*d0^2*d2+2*b2*b3*d0*d2^2+2*b2*b3*d1^2*d3+2*b2*b3*d1*d3^2+b3^2*d0^2*d3+b3^2*d0*d3^2+b3^2*d1^2*d2+b3^2*d1*d2^2;

// This checks that the polynomial parametrization
// lies on the probability simplex.
// It requires suma.sing. Most likely, you should 
// change the directory where you saved this file. 

// If you do have this file, you should uncomment
// the following two lines.

// < "/home/lgp/singular/suma.sing";

// Suma(Substitute(3,P));

// This checks that the PInvariants vanish at
// the polynomial parametrization.

map Evaluate = rP, P;

// The following command takes a lot of space and time to
// finish for larger models.

// ideal Z = Evaluate(PInvariants);

setring rP;
ideal Z;
int i;
for (i=1; i<= size(PInvariants); i++)
{
  i;
  Z = PInvariants[i];
  setring r;
  Evaluate(Z);
  setring rP;
}