//This is the ideal Fourier invariants.

ring rQ = 0, (q1,q2,q3,q4,q5,q6,q7,q8), dp;
ideal Invariants = 
q7^2-q6*q8,
q6*q7-q5*q8,
q4*q7-q3*q8,
q3*q7-q2*q8,
q6^2-q5*q7,
q4*q6-q2*q8,
q3*q6-q2*q7,
q2*q6-q1*q8,
q4*q5-q2*q7,
q3*q5-q1*q8,
q2*q5-q1*q7,
q3^2-q2*q4,
q2^2-q1*q4;

// This is the inverse of the Fourier transform.

matrix ptoq[11][8] =
1/256,9/128,3/32,21/256,3/64,9/64,21/64,15/64,
3/64,9/32,0,-21/64,3/8,9/16,0,-15/16,
9/128,9/64,-9/16,45/128,9/32,-9/32,-9/32,9/32,
9/64,-9/32,0,9/64,9/16,0,-27/16,9/8,
3/64,9/32,0,-21/64,0,-9/16,3/8,3/16,
9/32,-9/16,0,9/32,0,-9/8,9/4,-9/8,
3/32,-9/16,3/4,-9/32,0,0,0,0,
3/256,27/128,9/32,63/256,-3/64,-9/64,-21/64,-15/64,
3/32,9/16,0,-21/32,-3/8,0,-3/8,3/4,
9/128,9/64,-9/16,45/128,-9/32,9/32,9/32,-9/32,
9/64,-9/32,0,9/64,-9/16,9/8,-9/16,0;

// This is the ring of probability distributions. 

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

//This is the Fourier transform.

matrix qtop[8][11] =
1,1,1,1,1,1,1,1,1,1,1,
1,1/3,1/9,-1/9,1/3,-1/9,-1/3,1,1/3,1/9,-1/9,
1,0,-1/3,0,0,0,1/3,1,0,-1/3,0,
1,-1/3,5/21,1/21,-1/3,1/21,-1/7,1,-1/3,5/21,1/21,
1,2/3,1/3,1/3,0,0,0,-1/3,-1/3,-1/3,-1/3,
1,1/3,-1/9,0,-1/3,-1/9,0,-1/3,0,1/9,2/9,
1,0,-1/21,-1/7,2/21,2/21,0,-1/3,-1/21,1/21,-1/21,
1,-1/3,1/15,2/15,1/15,-1/15,0,-1/3,2/15,-1/15,0;

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,(b0,b1,f0,f1),dp;

ideal P = 
b0^2*f0^5+3*b0^2*f1^5+6*b0*b1*f0^4*f1+6*b0*b1*f0*f1^4+12*b0*b1*f1^5+3*b1^2*f0^5+6*b1^2*f0^4*f1+6*b1^2*f0*f1^4+21*b1^2*f1^5,
12*b0^2*f0^4*f1+12*b0^2*f0*f1^4+24*b0^2*f1^5+72*b0*b1*f0^3*f1^2+24*b0*b1*f0^2*f1^3+96*b0*b1*f0*f1^4+96*b0*b1*f1^5+36*b1^2*f0^4*f1+72*b1^2*f0^3*f1^2+24*b1^2*f0^2*f1^3+132*b1^2*f0*f1^4+168*b1^2*f1^5,
18*b0^2*f0^3*f1^2+18*b0^2*f0^2*f1^3+36*b0^2*f1^5+36*b0*b1*f0^3*f1^2+180*b0*b1*f0^2*f1^3+72*b0*b1*f0*f1^4+144*b0*b1*f1^5+90*b1^2*f0^3*f1^2+234*b1^2*f0^2*f1^3+72*b1^2*f0*f1^4+252*b1^2*f1^5,
36*b0^2*f0^3*f1^2+72*b0^2*f0*f1^4+36*b0^2*f1^5+360*b0*b1*f0^2*f1^3+360*b0*b1*f0*f1^4+144*b0*b1*f1^5+108*b1^2*f0^3*f1^2+360*b1^2*f0^2*f1^3+576*b1^2*f0*f1^4+252*b1^2*f1^5,
12*b0^2*f0^3*f1^2+12*b0^2*f0^2*f1^3+24*b0^2*f1^5+24*b0*b1*f0^4*f1+48*b0*b1*f0^3*f1^2+120*b0*b1*f0*f1^4+96*b0*b1*f1^5+24*b1^2*f0^4*f1+84*b1^2*f0^3*f1^2+36*b1^2*f0^2*f1^3+120*b1^2*f0*f1^4+168*b1^2*f1^5,
144*b0^2*f0^2*f1^3+72*b0^2*f0*f1^4+72*b0^2*f1^5+144*b0*b1*f0^3*f1^2+432*b0*b1*f0^2*f1^3+864*b0*b1*f0*f1^4+288*b0*b1*f1^5+144*b1^2*f0^3*f1^2+864*b1^2*f0^2*f1^3+1080*b1^2*f0*f1^4+504*b1^2*f1^5,
24*b0^2*f0^2*f1^3+72*b0^2*f0*f1^4+144*b0*b1*f0^2*f1^3+432*b0*b1*f0*f1^4+216*b1^2*f0^2*f1^3+648*b1^2*f0*f1^4,
3*b0^2*f0^4*f1+3*b0^2*f0*f1^4+6*b0^2*f1^5+6*b0*b1*f0^5+12*b0*b1*f0^4*f1+12*b0*b1*f0*f1^4+42*b0*b1*f1^5+6*b1^2*f0^5+21*b1^2*f0^4*f1+21*b1^2*f0*f1^4+60*b1^2*f1^5,
24*b0^2*f0^3*f1^2+48*b0^2*f0*f1^4+24*b0^2*f1^5+48*b0*b1*f0^4*f1+96*b0*b1*f0^3*f1^2+48*b0*b1*f0^2*f1^3+144*b0*b1*f0*f1^4+240*b0*b1*f1^5+48*b1^2*f0^4*f1+168*b1^2*f0^3*f1^2+48*b1^2*f0^2*f1^3+288*b1^2*f0*f1^4+312*b1^2*f1^5,
36*b0^2*f0^2*f1^3+18*b0^2*f0*f1^4+18*b0^2*f1^5+72*b0*b1*f0^3*f1^2+144*b0*b1*f0^2*f1^3+36*b0*b1*f0*f1^4+180*b0*b1*f1^5+72*b1^2*f0^3*f1^2+252*b1^2*f0^2*f1^3+90*b1^2*f0*f1^4+234*b1^2*f1^5,
36*b0^2*f0^2*f1^3+108*b0^2*f0*f1^4+72*b0*b1*f0^3*f1^2+288*b0*b1*f0^2*f1^3+288*b0*b1*f0*f1^4+216*b0*b1*f1^5+72*b1^2*f0^3*f1^2+396*b1^2*f0^2*f1^3+612*b1^2*f0*f1^4+216*b1^2*f1^5;

// 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;
}