//This is the ideal Fourier invariants.

ring rQ = 0, (q1,q2,q3,q4,q5), dp;
ideal Invariants = 
q3*q4-q2*q5,
q3^2-q1*q5,
q2*q3-q1*q4;

// This is the inverse of the Fourier transform.

matrix ptoq[6][5] =
1/16,1/16,9/16,3/16,1/8,
1/8,-1/8,3/8,-3/8,0,
1/16,1/16,-3/16,3/16,-1/8,
3/16,3/16,3/16,-3/16,-3/8,
3/8,-3/8,-3/8,3/8,0,
3/16,3/16,-9/16,-3/16,3/8;

// This is the ring of probability distributions. 

ring rP = 0,(p1,p2,p3,p4,p5,p6),dp;

//This is the Fourier transform.

matrix qtop[5][6] =
1,1,1,1,1,1,
1,-1,1,1,-1,1,
1,1/3,-1/3,1/9,-1/9,-1/3,
1,-1,1,-1/3,1/3,-1/3,
1,0,-1,-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,(d0,d1,f0,f1),dp;

ideal P = 
d0^4*f0^5+d0^4*f1^5+3*d0^3*d1*f0^4*f1+d0^3*d1*f0^3*f1^2+d0^3*d1*f0^2*f1^3+3*d0^3*d1*f0*f1^4+6*d0^2*d1^2*f0^3*f1^2+6*d0^2*d1^2*f0^2*f1^3+3*d0*d1^3*f0^4*f1+d0*d1^3*f0^3*f1^2+d0*d1^3*f0^2*f1^3+3*d0*d1^3*f0*f1^4+d1^4*f0^5+d1^4*f1^5,
2*d0^4*f0^4*f1+2*d0^4*f0*f1^4+2*d0^3*d1*f0^4*f1+6*d0^3*d1*f0^3*f1^2+6*d0^3*d1*f0^2*f1^3+2*d0^3*d1*f0*f1^4+12*d0^2*d1^2*f0^3*f1^2+12*d0^2*d1^2*f0^2*f1^3+2*d0*d1^3*f0^4*f1+6*d0*d1^3*f0^3*f1^2+6*d0*d1^3*f0^2*f1^3+2*d0*d1^3*f0*f1^4+2*d1^4*f0^4*f1+2*d1^4*f0*f1^4,
d0^4*f0^3*f1^2+d0^4*f0^2*f1^3+d0^3*d1*f0^5+3*d0^3*d1*f0^3*f1^2+3*d0^3*d1*f0^2*f1^3+d0^3*d1*f1^5+6*d0^2*d1^2*f0^4*f1+6*d0^2*d1^2*f0*f1^4+d0*d1^3*f0^5+3*d0*d1^3*f0^3*f1^2+3*d0*d1^3*f0^2*f1^3+d0*d1^3*f1^5+d1^4*f0^3*f1^2+d1^4*f0^2*f1^3,
3*d0^4*f0^4*f1+3*d0^4*f0*f1^4+3*d0^3*d1*f0^5+9*d0^3*d1*f0^3*f1^2+9*d0^3*d1*f0^2*f1^3+3*d0^3*d1*f1^5+12*d0^2*d1^2*f0^4*f1+6*d0^2*d1^2*f0^3*f1^2+6*d0^2*d1^2*f0^2*f1^3+12*d0^2*d1^2*f0*f1^4+3*d0*d1^3*f0^5+9*d0*d1^3*f0^3*f1^2+9*d0*d1^3*f0^2*f1^3+3*d0*d1^3*f1^5+3*d1^4*f0^4*f1+3*d1^4*f0*f1^4,
6*d0^4*f0^3*f1^2+6*d0^4*f0^2*f1^3+6*d0^3*d1*f0^4*f1+18*d0^3*d1*f0^3*f1^2+18*d0^3*d1*f0^2*f1^3+6*d0^3*d1*f0*f1^4+12*d0^2*d1^2*f0^4*f1+24*d0^2*d1^2*f0^3*f1^2+24*d0^2*d1^2*f0^2*f1^3+12*d0^2*d1^2*f0*f1^4+6*d0*d1^3*f0^4*f1+18*d0*d1^3*f0^3*f1^2+18*d0*d1^3*f0^2*f1^3+6*d0*d1^3*f0*f1^4+6*d1^4*f0^3*f1^2+6*d1^4*f0^2*f1^3,
3*d0^4*f0^3*f1^2+3*d0^4*f0^2*f1^3+9*d0^3*d1*f0^4*f1+3*d0^3*d1*f0^3*f1^2+3*d0^3*d1*f0^2*f1^3+9*d0^3*d1*f0*f1^4+6*d0^2*d1^2*f0^5+12*d0^2*d1^2*f0^3*f1^2+12*d0^2*d1^2*f0^2*f1^3+6*d0^2*d1^2*f1^5+9*d0*d1^3*f0^4*f1+3*d0*d1^3*f0^3*f1^2+3*d0*d1^3*f0^2*f1^3+9*d0*d1^3*f0*f1^4+3*d1^4*f0^3*f1^2+3*d1^4*f0^2*f1^3;

// 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;
}