//This is the ideal Fourier invariants.

ring rQ = 0,(q1,q2,q3,q4,q5,q6,q7,q8,q9,q10,q11,q12),dp; 

ideal Invariants = 
q8*q11-q7*q12,
q3*q11-q2*q12,
q8*q10-q6*q11,
q4*q10-q2*q11,
q8*q9-q5*q12,
q7*q9-q6*q10,
q7*q9-q5*q11,
q4*q9-q3*q10,
q3*q9-q1*q12,
q2*q9-q1*q11,
q6^2-q5*q8,
q4*q6-q2*q8,
q4*q6-q3*q7,
q4*q5-q2*q6,
q3*q5-q1*q8,
q2*q5-q1*q7,
q9*q11^2-q10^2*q12,
q2^2*q3-q1*q4^2;

// This is the inverse of the Fourier transform.

matrix ptoq[13][12] =
1/64,3/32,3/64,3/32,3/32,3/16,1/8,3/32,3/64,3/32,3/32,1/64,
3/32,3/16,-3/32,-3/16,3/8,0,0,-3/8,3/32,3/16,-3/16,-3/32,
3/64,-3/32,9/64,-3/32,3/32,3/16,-3/8,3/32,9/64,-3/32,-3/32,3/64,
3/32,3/16,-3/32,-3/16,3/16,-3/8,0,3/16,-3/32,-3/16,3/16,3/32,
3/16,-3/8,-3/16,3/8,0,0,0,0,3/16,-3/8,3/8,-3/16,
3/32,-3/16,-3/32,3/16,3/16,-3/8,0,3/16,-3/32,3/16,-3/16,3/32,
3/64,-3/32,9/64,-3/32,-3/32,-3/16,3/8,-3/32,9/64,-3/32,-3/32,3/64,
1/32,3/16,3/32,3/16,0,0,0,0,-3/32,-3/16,-3/16,-1/32,
3/32,3/16,-3/32,-3/16,-3/16,3/8,0,-3/16,-3/32,-3/16,3/16,3/32,
3/32,-3/16,9/32,-3/16,0,0,0,0,-9/32,3/16,3/16,-3/32,
3/32,3/16,-3/32,-3/16,-3/8,0,0,3/8,3/32,3/16,-3/16,-3/32,
3/32,-3/16,-3/32,3/16,-3/16,3/8,0,-3/16,-3/32,3/16,-3/16,3/32,
1/64,3/32,3/64,3/32,-3/32,-3/16,-1/8,-3/32,3/64,3/32,3/32,1/64;

// This is the ring of probability distributions. 

ring rP = 0,(p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,p13),dp;

//This is the Fourier transform.

matrix qtop[12][13] =
1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1/3,-1/3,1/3,-1/3,-1/3,-1/3,1,1/3,-1/3,1/3,-1/3,1,
1,-1/3,1,-1/3,-1/3,-1/3,1,1,-1/3,1,-1/3,-1/3,1,
1,-1/3,-1/3,-1/3,1/3,1/3,-1/3,1,-1/3,-1/3,-1/3,1/3,1,
1,2/3,1/3,1/3,0,1/3,-1/3,0,-1/3,0,-2/3,-1/3,-1,
1,0,1/3,-1/3,0,-1/3,-1/3,0,1/3,0,0,1/3,-1,
1,0,-1,0,0,0,1,0,0,0,0,0,-1,
1,-2/3,1/3,1/3,0,1/3,-1/3,0,-1/3,0,2/3,-1/3,-1,
1,1/3,1,-1/3,1/3,-1/3,1,-1,-1/3,-1,1/3,-1/3,1,
1,1/3,-1/3,-1/3,-1/3,1/3,-1/3,-1,-1/3,1/3,1/3,1/3,1,
1,-1/3,-1/3,1/3,1/3,-1/3,-1/3,-1,1/3,1/3,-1/3,-1/3,1,
1,-1,1,1,-1,1,1,-1,1,-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,(b0,b1,b2,e0,e1,e2),dp;

ideal P = 
b0^2*e0^4+2*b0^2*e1^4+b0^2*e2^4+4*b0*b1*e0^3*e1+4*b0*b1*e0*e1^3+4*b0*b1*e1^3*e2+4*b0*b1*e1*e2^3+2*b0*b2*e0^3*e2+2*b0*b2*e0*e2^3+4*b0*b2*e1^4+2*b1^2*e0^4+2*b1^2*e0^3*e2+2*b1^2*e0*e2^3+8*b1^2*e1^4+2*b1^2*e2^4+4*b1*b2*e0^3*e1+4*b1*b2*e0*e1^3+4*b1*b2*e1^3*e2+4*b1*b2*e1*e2^3+b2^2*e0^4+2*b2^2*e1^4+b2^2*e2^4,
6*b0^2*e0^3*e1+6*b0^2*e0*e1^3+6*b0^2*e1^3*e2+6*b0^2*e1*e2^3+36*b0*b1*e0^2*e1^2+24*b0*b1*e0*e1^2*e2+36*b0*b1*e1^2*e2^2+12*b0*b2*e0^2*e1*e2+12*b0*b2*e0*e1^3+12*b0*b2*e0*e1*e2^2+12*b0*b2*e1^3*e2+12*b1^2*e0^3*e1+12*b1^2*e0^2*e1*e2+24*b1^2*e0*e1^3+12*b1^2*e0*e1*e2^2+24*b1^2*e1^3*e2+12*b1^2*e1*e2^3+36*b1*b2*e0^2*e1^2+24*b1*b2*e0*e1^2*e2+36*b1*b2*e1^2*e2^2+6*b2^2*e0^3*e1+6*b2^2*e0*e1^3+6*b2^2*e1^3*e2+6*b2^2*e1*e2^3,
3*b0^2*e0^3*e2+3*b0^2*e0*e2^3+6*b0^2*e1^4+12*b0*b1*e0^2*e1*e2+12*b0*b1*e0*e1^3+12*b0*b1*e0*e1*e2^2+12*b0*b1*e1^3*e2+12*b0*b2*e0^2*e2^2+12*b0*b2*e1^4+6*b1^2*e0^3*e2+12*b1^2*e0^2*e2^2+6*b1^2*e0*e2^3+24*b1^2*e1^4+12*b1*b2*e0^2*e1*e2+12*b1*b2*e0*e1^3+12*b1*b2*e0*e1*e2^2+12*b1*b2*e1^3*e2+3*b2^2*e0^3*e2+3*b2^2*e0*e2^3+6*b2^2*e1^4,
12*b0^2*e0^2*e1^2+12*b0^2*e1^2*e2^2+12*b0*b1*e0^3*e1+12*b0*b1*e0^2*e1*e2+24*b0*b1*e0*e1^3+12*b0*b1*e0*e1*e2^2+24*b0*b1*e1^3*e2+12*b0*b1*e1*e2^3+12*b0*b2*e0^2*e1^2+24*b0*b2*e0*e1^2*e2+12*b0*b2*e1^2*e2^2+36*b1^2*e0^2*e1^2+24*b1^2*e0*e1^2*e2+36*b1^2*e1^2*e2^2+12*b1*b2*e0^3*e1+12*b1*b2*e0^2*e1*e2+24*b1*b2*e0*e1^3+12*b1*b2*e0*e1*e2^2+24*b1*b2*e1^3*e2+12*b1*b2*e1*e2^3+12*b2^2*e0^2*e1^2+12*b2^2*e1^2*e2^2,
12*b0^2*e0^2*e1*e2+12*b0^2*e0*e1^3+12*b0^2*e0*e1*e2^2+12*b0^2*e1^3*e2+24*b0*b1*e0^2*e1^2+144*b0*b1*e0*e1^2*e2+24*b0*b1*e1^2*e2^2+24*b0*b2*e0^2*e1*e2+24*b0*b2*e0*e1^3+24*b0*b2*e0*e1*e2^2+24*b0*b2*e1^3*e2+48*b1^2*e0^2*e1*e2+48*b1^2*e0*e1^3+48*b1^2*e0*e1*e2^2+48*b1^2*e1^3*e2+24*b1*b2*e0^2*e1^2+144*b1*b2*e0*e1^2*e2+24*b1*b2*e1^2*e2^2+12*b2^2*e0^2*e1*e2+12*b2^2*e0*e1^3+12*b2^2*e0*e1*e2^2+12*b2^2*e1^3*e2,
6*b0^2*e0^2*e1^2+12*b0^2*e0*e1^2*e2+6*b0^2*e1^2*e2^2+24*b0*b1*e0^2*e1*e2+24*b0*b1*e0*e1^3+24*b0*b1*e0*e1*e2^2+24*b0*b1*e1^3*e2+48*b0*b2*e0*e1^2*e2+12*b1^2*e0^2*e1^2+72*b1^2*e0*e1^2*e2+12*b1^2*e1^2*e2^2+24*b1*b2*e0^2*e1*e2+24*b1*b2*e0*e1^3+24*b1*b2*e0*e1*e2^2+24*b1*b2*e1^3*e2+6*b2^2*e0^2*e1^2+12*b2^2*e0*e1^2*e2+6*b2^2*e1^2*e2^2,
6*b0^2*e0^2*e2^2+6*b0^2*e1^4+12*b0*b1*e0^2*e1*e2+12*b0*b1*e0*e1^3+12*b0*b1*e0*e1*e2^2+12*b0*b1*e1^3*e2+6*b0*b2*e0^3*e2+6*b0*b2*e0*e2^3+12*b0*b2*e1^4+6*b1^2*e0^3*e2+12*b1^2*e0^2*e2^2+6*b1^2*e0*e2^3+24*b1^2*e1^4+12*b1*b2*e0^2*e1*e2+12*b1*b2*e0*e1^3+12*b1*b2*e0*e1*e2^2+12*b1*b2*e1^3*e2+6*b2^2*e0^2*e2^2+6*b2^2*e1^4,
2*b0^2*e0^3*e1+2*b0^2*e0*e1^3+2*b0^2*e1^3*e2+2*b0^2*e1*e2^3+4*b0*b1*e0^4+4*b0*b1*e0^3*e2+4*b0*b1*e0*e2^3+16*b0*b1*e1^4+4*b0*b1*e2^4+4*b0*b2*e0^3*e1+4*b0*b2*e0*e1^3+4*b0*b2*e1^3*e2+4*b0*b2*e1*e2^3+8*b1^2*e0^3*e1+8*b1^2*e0*e1^3+8*b1^2*e1^3*e2+8*b1^2*e1*e2^3+4*b1*b2*e0^4+4*b1*b2*e0^3*e2+4*b1*b2*e0*e2^3+16*b1*b2*e1^4+4*b1*b2*e2^4+2*b2^2*e0^3*e1+2*b2^2*e0*e1^3+2*b2^2*e1^3*e2+2*b2^2*e1*e2^3,
6*b0^2*e0^2*e1^2+12*b0^2*e0*e1^2*e2+6*b0^2*e1^2*e2^2+12*b0*b1*e0^3*e1+12*b0*b1*e0^2*e1*e2+24*b0*b1*e0*e1^3+12*b0*b1*e0*e1*e2^2+24*b0*b1*e1^3*e2+12*b0*b1*e1*e2^3+24*b0*b2*e0^2*e1^2+24*b0*b2*e1^2*e2^2+36*b1^2*e0^2*e1^2+24*b1^2*e0*e1^2*e2+36*b1^2*e1^2*e2^2+12*b1*b2*e0^3*e1+12*b1*b2*e0^2*e1*e2+24*b1*b2*e0*e1^3+12*b1*b2*e0*e1*e2^2+24*b1*b2*e1^3*e2+12*b1*b2*e1*e2^3+6*b2^2*e0^2*e1^2+12*b2^2*e0*e1^2*e2+6*b2^2*e1^2*e2^2,
6*b0^2*e0^2*e1*e2+6*b0^2*e0*e1^3+6*b0^2*e0*e1*e2^2+6*b0^2*e1^3*e2+12*b0*b1*e0^3*e2+24*b0*b1*e0^2*e2^2+12*b0*b1*e0*e2^3+48*b0*b1*e1^4+12*b0*b2*e0^2*e1*e2+12*b0*b2*e0*e1^3+12*b0*b2*e0*e1*e2^2+12*b0*b2*e1^3*e2+24*b1^2*e0^2*e1*e2+24*b1^2*e0*e1^3+24*b1^2*e0*e1*e2^2+24*b1^2*e1^3*e2+12*b1*b2*e0^3*e2+24*b1*b2*e0^2*e2^2+12*b1*b2*e0*e2^3+48*b1*b2*e1^4+6*b2^2*e0^2*e1*e2+6*b2^2*e0*e1^3+6*b2^2*e0*e1*e2^2+6*b2^2*e1^3*e2,
6*b0^2*e0^2*e1*e2+6*b0^2*e0*e1^3+6*b0^2*e0*e1*e2^2+6*b0^2*e1^3*e2+36*b0*b1*e0^2*e1^2+24*b0*b1*e0*e1^2*e2+36*b0*b1*e1^2*e2^2+12*b0*b2*e0^3*e1+12*b0*b2*e0*e1^3+12*b0*b2*e1^3*e2+12*b0*b2*e1*e2^3+12*b1^2*e0^3*e1+12*b1^2*e0^2*e1*e2+24*b1^2*e0*e1^3+12*b1^2*e0*e1*e2^2+24*b1^2*e1^3*e2+12*b1^2*e1*e2^3+36*b1*b2*e0^2*e1^2+24*b1*b2*e0*e1^2*e2+36*b1*b2*e1^2*e2^2+6*b2^2*e0^2*e1*e2+6*b2^2*e0*e1^3+6*b2^2*e0*e1*e2^2+6*b2^2*e1^3*e2,
24*b0^2*e0*e1^2*e2+24*b0*b1*e0^2*e1*e2+24*b0*b1*e0*e1^3+24*b0*b1*e0*e1*e2^2+24*b0*b1*e1^3*e2+12*b0*b2*e0^2*e1^2+24*b0*b2*e0*e1^2*e2+12*b0*b2*e1^2*e2^2+12*b1^2*e0^2*e1^2+72*b1^2*e0*e1^2*e2+12*b1^2*e1^2*e2^2+24*b1*b2*e0^2*e1*e2+24*b1*b2*e0*e1^3+24*b1*b2*e0*e1*e2^2+24*b1*b2*e1^3*e2+24*b2^2*e0*e1^2*e2,
b0^2*e0^3*e2+b0^2*e0*e2^3+2*b0^2*e1^4+4*b0*b1*e0^3*e1+4*b0*b1*e0*e1^3+4*b0*b1*e1^3*e2+4*b0*b1*e1*e2^3+2*b0*b2*e0^4+4*b0*b2*e1^4+2*b0*b2*e2^4+2*b1^2*e0^4+2*b1^2*e0^3*e2+2*b1^2*e0*e2^3+8*b1^2*e1^4+2*b1^2*e2^4+4*b1*b2*e0^3*e1+4*b1*b2*e0*e1^3+4*b1*b2*e1^3*e2+4*b1*b2*e1*e2^3+b2^2*e0^3*e2+b2^2*e0*e2^3+2*b2^2*e1^4;

// 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(2,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;
}