//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] =
1/16,3/16,7/16,5/16,
3/16,-3/16,9/16,-9/16,
9/16,-9/16,-9/16,9/16,
1/16,3/16,-5/16,1/16,
1/8,3/8,-1/8,-3/8;

// 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,-1/3,-1/3,1,1,
1,3/7,-1/7,-5/7,-1/7,
1,-3/5,1/5,1/5,-3/5;

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,f0,f1),dp;

ideal P = 
c0^3*f0^5+c0^3*f1^5+2*c0^2*c1*f0^4*f1+c0^2*c1*f0^3*f1^2+c0^2*c1*f0^2*f1^3+2*c0^2*c1*f0*f1^4+2*c0*c1^2*f0^4*f1+c0*c1^2*f0^3*f1^2+c0*c1^2*f0^2*f1^3+2*c0*c1^2*f0*f1^4+c1^3*f0^5+c1^3*f1^5,
3*c0^3*f0^4*f1+3*c0^3*f0*f1^4+9*c0^2*c1*f0^3*f1^2+9*c0^2*c1*f0^2*f1^3+9*c0*c1^2*f0^3*f1^2+9*c0*c1^2*f0^2*f1^3+3*c1^3*f0^4*f1+3*c1^3*f0*f1^4,
9*c0^3*f0^3*f1^2+9*c0^3*f0^2*f1^3+9*c0^2*c1*f0^4*f1+18*c0^2*c1*f0^3*f1^2+18*c0^2*c1*f0^2*f1^3+9*c0^2*c1*f0*f1^4+9*c0*c1^2*f0^4*f1+18*c0*c1^2*f0^3*f1^2+18*c0*c1^2*f0^2*f1^3+9*c0*c1^2*f0*f1^4+9*c1^3*f0^3*f1^2+9*c1^3*f0^2*f1^3,
c0^3*f0^3*f1^2+c0^3*f0^2*f1^3+c0^2*c1*f0^5+2*c0^2*c1*f0^4*f1+2*c0^2*c1*f0*f1^4+c0^2*c1*f1^5+c0*c1^2*f0^5+2*c0*c1^2*f0^4*f1+2*c0*c1^2*f0*f1^4+c0*c1^2*f1^5+c1^3*f0^3*f1^2+c1^3*f0^2*f1^3,
2*c0^3*f0^4*f1+2*c0^3*f0*f1^4+2*c0^2*c1*f0^5+2*c0^2*c1*f0^4*f1+2*c0^2*c1*f0^3*f1^2+2*c0^2*c1*f0^2*f1^3+2*c0^2*c1*f0*f1^4+2*c0^2*c1*f1^5+2*c0*c1^2*f0^5+2*c0*c1^2*f0^4*f1+2*c0*c1^2*f0^3*f1^2+2*c0*c1^2*f0^2*f1^3+2*c0*c1^2*f0*f1^4+2*c0*c1^2*f1^5+2*c1^3*f0^4*f1+2*c1^3*f0*f1^4;

// 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;
}