//This is the ideal of Fourier invariants.

ring rQ = 0,(q1,q2,q3,q4),dp; 

ideal Invariants = 
0;

//This is the inverse of the Fourier transform.

matrix ptoq[4][4] = 
1/4,1/4,1/4,1/4,
1/4,-1/4,-1/4,1/4,
1/4,-1/4,1/4,-1/4,
1/4,1/4,-1/4,-1/4;


// This is the ring of probability distributions. 

ring rP = 0,(p1,p2,p3,p4),dp;

//This is the Fourier transform. 

matrix qtop[4][4] =
1,1,1,1,
1,-1,-1,1,
1,-1,1,-1,
1,1,-1,-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,(a0,a1,b0,b1,c0,c1),dp;

ideal P = 
a0*b0*c0+a1*b1*c1,
a0*b1*c1+a1*b0*c0,
a0*b0*c1+a1*b1*c0,
a0*b1*c0+a1*b0*c1;

// 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(0,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;
}