//This is the ideal Fourier invariants.

ring rQ = 0,(q1,q2,q3,q4,q5,q6,q7,q8,q9,q10,q11,q12),dp; 

ideal Invariants = 
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;

// This is the inverse of the Fourier transform.

matrix ptoq[15][12] =
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;

// This is the ring of probability distributions. 

ring rP = 0,(p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,p13,p14,p15),dp;

//This is the Fourier transform.

matrix qtop[12][15] =
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;

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,d0,d1),dp;

ideal P = 
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;

// 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(1,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;
}