//This is the ideal Fourier invariants.

ring rQ = 0, (q1,q2,q3,q4,q5,q6), dp;;
ideal Invariants = 
q3*q5-q2*q6,
q3*q4-q1*q6,
q2*q4-q1*q5;

// This is the inverse of the Fourier transform.

matrix ptoq[8][6] =
1/16,1/16,1/8,7/16,3/16,1/8,
1/8,-1/8,0,3/8,-3/8,0,
1/16,1/16,-1/8,-1/16,3/16,-1/8,
1/16,1/16,-1/8,3/16,-1/16,-1/8,
3/8,-3/8,0,-3/8,3/8,0,
1/16,1/16,1/8,-5/16,-1/16,1/8,
1/8,1/8,1/4,-1/8,-1/8,-1/4,
1/8,1/8,-1/4,-1/8,-1/8,1/4;

// This is the ring of probability distributions. 

ring rP = 0,(p1,p2,p3,p4,p5,p6,p7,p8),dp;

//This is the Fourier transform.

matrix qtop[6][8] =
1,1,1,1,1,1,1,1,
1,-1,1,1,-1,1,1,1,
1,0,-1,-1,0,1,1,-1,
1,3/7,-1/7,3/7,-1/7,-5/7,-1/7,-1/7,
1,-1,1,-1/3,1/3,-1/3,-1/3,-1/3,
1,0,-1,-1,0,1,-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,g0,g1),dp;

ideal P = 
c0^3*e0^4*g0^5+c0^3*e0^4*g1^5+3*c0^3*e0^3*e1*g0^4*g1+c0^3*e0^3*e1*g0^3*g1^2+c0^3*e0^3*e1*g0^2*g1^3+3*c0^3*e0^3*e1*g0*g1^4+6*c0^3*e0^2*e1^2*g0^3*g1^2+6*c0^3*e0^2*e1^2*g0^2*g1^3+3*c0^3*e0*e1^3*g0^4*g1+c0^3*e0*e1^3*g0^3*g1^2+c0^3*e0*e1^3*g0^2*g1^3+3*c0^3*e0*e1^3*g0*g1^4+c0^3*e1^4*g0^5+c0^3*e1^4*g1^5+2*c0^2*c1*e0^4*g0^4*g1+c0^2*c1*e0^4*g0^3*g1^2+c0^2*c1*e0^4*g0^2*g1^3+2*c0^2*c1*e0^4*g0*g1^4+2*c0^2*c1*e0^3*e1*g0^5+3*c0^2*c1*e0^3*e1*g0^4*g1+7*c0^2*c1*e0^3*e1*g0^3*g1^2+7*c0^2*c1*e0^3*e1*g0^2*g1^3+3*c0^2*c1*e0^3*e1*g0*g1^4+2*c0^2*c1*e0^3*e1*g1^5+2*c0^2*c1*e0^2*e1^2*g0^5+8*c0^2*c1*e0^2*e1^2*g0^4*g1+8*c0^2*c1*e0^2*e1^2*g0^3*g1^2+8*c0^2*c1*e0^2*e1^2*g0^2*g1^3+8*c0^2*c1*e0^2*e1^2*g0*g1^4+2*c0^2*c1*e0^2*e1^2*g1^5+2*c0^2*c1*e0*e1^3*g0^5+3*c0^2*c1*e0*e1^3*g0^4*g1+7*c0^2*c1*e0*e1^3*g0^3*g1^2+7*c0^2*c1*e0*e1^3*g0^2*g1^3+3*c0^2*c1*e0*e1^3*g0*g1^4+2*c0^2*c1*e0*e1^3*g1^5+2*c0^2*c1*e1^4*g0^4*g1+c0^2*c1*e1^4*g0^3*g1^2+c0^2*c1*e1^4*g0^2*g1^3+2*c0^2*c1*e1^4*g0*g1^4+2*c0*c1^2*e0^4*g0^4*g1+c0*c1^2*e0^4*g0^3*g1^2+c0*c1^2*e0^4*g0^2*g1^3+2*c0*c1^2*e0^4*g0*g1^4+2*c0*c1^2*e0^3*e1*g0^5+3*c0*c1^2*e0^3*e1*g0^4*g1+7*c0*c1^2*e0^3*e1*g0^3*g1^2+7*c0*c1^2*e0^3*e1*g0^2*g1^3+3*c0*c1^2*e0^3*e1*g0*g1^4+2*c0*c1^2*e0^3*e1*g1^5+2*c0*c1^2*e0^2*e1^2*g0^5+8*c0*c1^2*e0^2*e1^2*g0^4*g1+8*c0*c1^2*e0^2*e1^2*g0^3*g1^2+8*c0*c1^2*e0^2*e1^2*g0^2*g1^3+8*c0*c1^2*e0^2*e1^2*g0*g1^4+2*c0*c1^2*e0^2*e1^2*g1^5+2*c0*c1^2*e0*e1^3*g0^5+3*c0*c1^2*e0*e1^3*g0^4*g1+7*c0*c1^2*e0*e1^3*g0^3*g1^2+7*c0*c1^2*e0*e1^3*g0^2*g1^3+3*c0*c1^2*e0*e1^3*g0*g1^4+2*c0*c1^2*e0*e1^3*g1^5+2*c0*c1^2*e1^4*g0^4*g1+c0*c1^2*e1^4*g0^3*g1^2+c0*c1^2*e1^4*g0^2*g1^3+2*c0*c1^2*e1^4*g0*g1^4+c1^3*e0^4*g0^5+c1^3*e0^4*g1^5+3*c1^3*e0^3*e1*g0^4*g1+c1^3*e0^3*e1*g0^3*g1^2+c1^3*e0^3*e1*g0^2*g1^3+3*c1^3*e0^3*e1*g0*g1^4+6*c1^3*e0^2*e1^2*g0^3*g1^2+6*c1^3*e0^2*e1^2*g0^2*g1^3+3*c1^3*e0*e1^3*g0^4*g1+c1^3*e0*e1^3*g0^3*g1^2+c1^3*e0*e1^3*g0^2*g1^3+3*c1^3*e0*e1^3*g0*g1^4+c1^3*e1^4*g0^5+c1^3*e1^4*g1^5,
2*c0^3*e0^4*g0^4*g1+2*c0^3*e0^4*g0*g1^4+2*c0^3*e0^3*e1*g0^4*g1+6*c0^3*e0^3*e1*g0^3*g1^2+6*c0^3*e0^3*e1*g0^2*g1^3+2*c0^3*e0^3*e1*g0*g1^4+12*c0^3*e0^2*e1^2*g0^3*g1^2+12*c0^3*e0^2*e1^2*g0^2*g1^3+2*c0^3*e0*e1^3*g0^4*g1+6*c0^3*e0*e1^3*g0^3*g1^2+6*c0^3*e0*e1^3*g0^2*g1^3+2*c0^3*e0*e1^3*g0*g1^4+2*c0^3*e1^4*g0^4*g1+2*c0^3*e1^4*g0*g1^4+6*c0^2*c1*e0^4*g0^3*g1^2+6*c0^2*c1*e0^4*g0^2*g1^3+6*c0^2*c1*e0^3*e1*g0^4*g1+18*c0^2*c1*e0^3*e1*g0^3*g1^2+18*c0^2*c1*e0^3*e1*g0^2*g1^3+6*c0^2*c1*e0^3*e1*g0*g1^4+12*c0^2*c1*e0^2*e1^2*g0^4*g1+24*c0^2*c1*e0^2*e1^2*g0^3*g1^2+24*c0^2*c1*e0^2*e1^2*g0^2*g1^3+12*c0^2*c1*e0^2*e1^2*g0*g1^4+6*c0^2*c1*e0*e1^3*g0^4*g1+18*c0^2*c1*e0*e1^3*g0^3*g1^2+18*c0^2*c1*e0*e1^3*g0^2*g1^3+6*c0^2*c1*e0*e1^3*g0*g1^4+6*c0^2*c1*e1^4*g0^3*g1^2+6*c0^2*c1*e1^4*g0^2*g1^3+6*c0*c1^2*e0^4*g0^3*g1^2+6*c0*c1^2*e0^4*g0^2*g1^3+6*c0*c1^2*e0^3*e1*g0^4*g1+18*c0*c1^2*e0^3*e1*g0^3*g1^2+18*c0*c1^2*e0^3*e1*g0^2*g1^3+6*c0*c1^2*e0^3*e1*g0*g1^4+12*c0*c1^2*e0^2*e1^2*g0^4*g1+24*c0*c1^2*e0^2*e1^2*g0^3*g1^2+24*c0*c1^2*e0^2*e1^2*g0^2*g1^3+12*c0*c1^2*e0^2*e1^2*g0*g1^4+6*c0*c1^2*e0*e1^3*g0^4*g1+18*c0*c1^2*e0*e1^3*g0^3*g1^2+18*c0*c1^2*e0*e1^3*g0^2*g1^3+6*c0*c1^2*e0*e1^3*g0*g1^4+6*c0*c1^2*e1^4*g0^3*g1^2+6*c0*c1^2*e1^4*g0^2*g1^3+2*c1^3*e0^4*g0^4*g1+2*c1^3*e0^4*g0*g1^4+2*c1^3*e0^3*e1*g0^4*g1+6*c1^3*e0^3*e1*g0^3*g1^2+6*c1^3*e0^3*e1*g0^2*g1^3+2*c1^3*e0^3*e1*g0*g1^4+12*c1^3*e0^2*e1^2*g0^3*g1^2+12*c1^3*e0^2*e1^2*g0^2*g1^3+2*c1^3*e0*e1^3*g0^4*g1+6*c1^3*e0*e1^3*g0^3*g1^2+6*c1^3*e0*e1^3*g0^2*g1^3+2*c1^3*e0*e1^3*g0*g1^4+2*c1^3*e1^4*g0^4*g1+2*c1^3*e1^4*g0*g1^4,
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// 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;
}