//This is the ideal Fourier invariants.

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

ideal Invariants = 
q2*q3-q1*q4;

// This is the inverse of the Fourier transform.

matrix ptoq[5][4] =
1/8,1/8,5/8,1/8,
1/4,-1/4,1/4,-1/4,
1/8,1/8,-3/8,1/8,
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,p5),dp;

//This is the Fourier transform.

matrix qtop[4][5] =
1,1,1,1,1,
1,-1,1,1,-1,
1,1/5,-3/5,-1/5,-1/5,
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,(c0,c1,e0,e1),dp;

ideal P = 
c0^3*e0^4+c0^3*e1^4+2*c0^2*c1*e0^3*e1+2*c0^2*c1*e0^2*e1^2+2*c0^2*c1*e0*e1^3+2*c0*c1^2*e0^3*e1+2*c0*c1^2*e0^2*e1^2+2*c0*c1^2*e0*e1^3+c1^3*e0^4+c1^3*e1^4,
2*c0^3*e0^3*e1+2*c0^3*e0*e1^3+2*c0^2*c1*e0^3*e1+8*c0^2*c1*e0^2*e1^2+2*c0^2*c1*e0*e1^3+2*c0*c1^2*e0^3*e1+8*c0*c1^2*e0^2*e1^2+2*c0*c1^2*e0*e1^3+2*c1^3*e0^3*e1+2*c1^3*e0*e1^3,
2*c0^3*e0^2*e1^2+c0^2*c1*e0^4+2*c0^2*c1*e0^3*e1+2*c0^2*c1*e0*e1^3+c0^2*c1*e1^4+c0*c1^2*e0^4+2*c0*c1^2*e0^3*e1+2*c0*c1^2*e0*e1^3+c0*c1^2*e1^4+2*c1^3*e0^2*e1^2,
2*c0^3*e0^3*e1+2*c0^3*e0*e1^3+2*c0^2*c1*e0^4+2*c0^2*c1*e0^3*e1+4*c0^2*c1*e0^2*e1^2+2*c0^2*c1*e0*e1^3+2*c0^2*c1*e1^4+2*c0*c1^2*e0^4+2*c0*c1^2*e0^3*e1+4*c0*c1^2*e0^2*e1^2+2*c0*c1^2*e0*e1^3+2*c0*c1^2*e1^4+2*c1^3*e0^3*e1+2*c1^3*e0*e1^3,
4*c0^3*e0^2*e1^2+4*c0^2*c1*e0^3*e1+4*c0^2*c1*e0^2*e1^2+4*c0^2*c1*e0*e1^3+4*c0*c1^2*e0^3*e1+4*c0*c1^2*e0^2*e1^2+4*c0*c1^2*e0*e1^3+4*c1^3*e0^2*e1^2;

// 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(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;
}