//This is the ideal Fourier invariants.

ring rQ = 0,(q1,q2,q3,q4,q5,q6,q7,q8,q9,q10,q11,q12,q13,q14,q15,q16,q17,q18,q19,q20),dp; 

ideal Invariants = 
q13*q19-q14*q20,
q11*q19-q10*q20,
q3*q19-q2*q20,
q16*q18-q17*q19,
q13*q17-q12*q18,
q11*q17-q9*q18,
q9*q17-q8*q18,
q5*q17-q4*q18,
q13*q16-q12*q19,
q11*q16-q9*q19,
q5*q16-q4*q19,
q14*q15-q12*q16,
q11*q15-q8*q20,
q10*q15-q8*q19,
q10*q15-q9*q16,
q7*q15-q6*q16,
q3*q15-q1*q20,
q2*q15-q1*q19,
q10*q13-q11*q14,
q2*q13-q3*q14,
q11*q12-q9*q13,
q10*q12-q9*q14,
q9*q12-q8*q13,
q7*q12-q6*q14,
q5*q12-q4*q13,
q3*q10-q2*q11,
q9^2-q8*q11,
q7*q9-q6*q10,
q5*q9-q4*q11,
q5*q8-q4*q9,
q3*q8-q1*q11,
q2*q8-q1*q10,
q15*q19^2-q16^2*q20,
q6*q19^2-q7*q16*q20,
q15*q18*q19-q16*q17*q20,
q6*q18*q19-q7*q17*q20,
q5*q15*q19-q4*q16*q20,
q5*q6*q19-q4*q7*q20,
q15*q18^2-q17^2*q20,
q13*q15*q18-q12*q17*q20,
q5*q15*q18-q4*q17*q20,
q3*q13*q18-q5*q9*q20,
q7*q11*q18-q5*q14*q20,
q6*q11*q18-q5*q12*q20,
q7*q10*q18-q5*q14*q19,
q3*q17^2-q1*q18^2,
q3*q16*q17-q2*q15*q18,
q7*q11*q17-q4*q14*q20,
q6*q11*q17-q4*q12*q20,
q3*q7*q17-q2*q6*q18,
q2*q6*q17-q1*q7*q18,
q3*q4*q17-q1*q5*q18,
q3*q16^2-q2*q15*q19,
q5*q14*q16-q7*q10*q17,
q5*q11*q16-q3*q14*q18,
q5*q10*q16-q2*q14*q18,
q4*q10*q16-q2*q14*q17,
q5*q8*q16-q2*q12*q17,
q5*q8*q16-q1*q14*q18,
q4*q8*q16-q1*q14*q17,
q3*q7*q16-q2*q6*q19,
q5*q14*q15-q7*q9*q17,
q4*q14*q15-q7*q8*q17,
q13^2*q15-q12^2*q20,
q5*q13*q15-q6*q11*q17,
q5*q12*q15-q6*q9*q17,
q4*q12*q15-q6*q8*q17,
q5*q11*q15-q3*q13*q17,
q5*q10*q15-q3*q14*q17,
q5*q9*q15-q3*q12*q17,
q5*q9*q15-q1*q13*q18,
q5*q8*q15-q1*q13*q17,
q4*q8*q15-q1*q12*q17,
q5^2*q15-q4^2*q20,
q2*q5*q15-q3*q4*q16,
q2*q6*q12-q1*q7*q13,
q3*q4*q12-q1*q5*q13,
q2*q4*q12-q1*q5*q14,
q6*q11^2-q3*q13^2,
q7*q9*q11-q3*q13*q14,
q6*q9*q11-q3*q12*q13,
q7*q9*q10-q3*q14^2,
q7*q8*q10-q2*q12*q14,
q7*q9^2-q3*q12*q14,
q6*q9^2-q3*q12^2,
q6*q9^2-q1*q13^2,
q7*q8*q9-q2*q12^2,
q7*q8*q9-q1*q13*q14,
q6*q8*q9-q1*q12*q13,
q7*q8^2-q1*q12*q14,
q6*q8^2-q1*q12^2,
q3*q7*q8-q2*q6*q9,
q2*q5*q6-q3*q4*q7,
q2*q4*q6-q1*q5*q7,
q3*q4^2-q1*q5^2,
q3*q7*q18^2-q5^2*q16*q20,
q2*q7*q18^2-q5^2*q16*q19,
q3*q6*q18^2-q5^2*q15*q20,
q3*q7*q17*q18-q5^2*q15*q19,
q3*q6*q17*q18-q4*q5*q15*q20,
q5^2*q16^2-q2*q7*q17*q18,
q4*q5*q16^2-q2*q7*q17^2,
q5^2*q15*q16-q3*q7*q17^2,
q4*q5*q15*q16-q2*q6*q17^2,
q4^2*q15*q16-q1*q7*q17^2,
q7*q10^2*q16-q2*q14^2*q19,
q5^2*q15^2-q3*q6*q17^2,
q4*q5*q15^2-q1*q6*q17*q18,
q4^2*q15^2-q1*q6*q17^2,
q4*q7*q10^2-q2*q5*q14^2,
q2^2*q6^2-q1*q3*q7^2;

// This is the inverse of the Fourier transform.

matrix ptoq[24][20] =
1/64,1/32,1/64,1/16,1/16,1/32,1/32,3/32,1/16,1/16,1/32,1/8,1/16,1/16,3/64,1/32,1/16,1/16,1/32,1/64,
1/16,0,-1/16,1/8,-1/8,0,0,1/4,-1/8,0,-1/8,1/8,-1/8,0,1/16,0,1/8,-1/8,0,-1/16,
1/32,-1/16,1/32,0,0,1/16,-1/16,1/16,0,-1/8,1/16,1/8,0,-1/8,3/32,-1/16,0,0,-1/16,1/32,
1/32,1/16,1/32,0,0,-1/16,-1/16,1/16,0,1/8,1/16,-1/8,0,-1/8,-1/32,-1/16,0,0,1/16,1/32,
1/16,0,-1/16,-1/8,1/8,0,0,0,1/8,0,-1/8,-1/8,1/8,0,1/16,0,-1/8,1/8,0,-1/16,
1/32,-1/16,1/32,0,0,-1/16,1/16,1/16,0,-1/8,1/16,-1/8,0,1/8,-1/32,1/16,0,0,-1/16,1/32,
1/64,1/32,1/64,-1/16,-1/16,1/32,1/32,-1/32,-1/16,1/16,1/32,0,-1/16,1/16,3/64,1/32,-1/16,-1/16,1/32,1/64,
1/32,1/16,1/32,0,0,-1/16,-1/16,1/8,1/8,0,0,-1/8,-1/8,0,1/32,1/16,0,0,-1/16,-1/32,
1/16,0,-1/16,1/8,-1/8,0,0,1/8,-1/8,0,0,-1/8,1/8,0,-1/16,0,-1/8,1/8,0,1/16,
1/16,-1/8,1/16,0,0,-1/8,1/8,0,0,0,0,0,0,0,1/16,-1/8,0,0,1/8,-1/16,
1/16,0,-1/16,-1/8,1/8,0,0,1/8,-1/8,0,0,-1/8,1/8,0,-1/16,0,1/8,-1/8,0,1/16,
1/32,1/16,1/32,1/8,1/8,1/16,1/16,0,0,0,0,0,0,0,-3/32,-1/16,-1/8,-1/8,-1/16,-1/32,
1/16,0,-1/16,1/8,-1/8,0,0,-1/8,1/8,0,0,1/8,-1/8,0,-1/16,0,-1/8,1/8,0,1/16,
1/16,-1/8,1/16,0,0,1/8,-1/8,0,0,0,0,0,0,0,-3/16,1/8,0,0,1/8,-1/16,
1/32,1/16,1/32,0,0,-1/16,-1/16,-1/8,-1/8,0,0,1/8,1/8,0,1/32,1/16,0,0,-1/16,-1/32,
1/16,0,-1/16,-1/8,1/8,0,0,-1/8,1/8,0,0,1/8,-1/8,0,-1/16,0,1/8,-1/8,0,1/16,
1/32,1/16,1/32,-1/8,-1/8,1/16,1/16,0,0,0,0,0,0,0,-3/32,-1/16,1/8,1/8,-1/16,-1/32,
1/64,1/32,1/64,-1/16,-1/16,1/32,1/32,1/32,1/16,-1/16,-1/32,0,1/16,-1/16,3/64,1/32,-1/16,-1/16,1/32,1/64,
1/16,0,-1/16,-1/8,1/8,0,0,0,-1/8,0,1/8,1/8,-1/8,0,1/16,0,-1/8,1/8,0,-1/16,
1/32,-1/16,1/32,0,0,1/16,-1/16,-1/16,0,1/8,-1/16,-1/8,0,1/8,3/32,-1/16,0,0,-1/16,1/32,
1/32,1/16,1/32,0,0,-1/16,-1/16,-1/16,0,-1/8,-1/16,1/8,0,1/8,-1/32,-1/16,0,0,1/16,1/32,
1/16,0,-1/16,1/8,-1/8,0,0,-1/4,1/8,0,1/8,-1/8,1/8,0,1/16,0,1/8,-1/8,0,-1/16,
1/32,-1/16,1/32,0,0,-1/16,1/16,-1/16,0,1/8,-1/16,1/8,0,-1/8,-1/32,1/16,0,0,-1/16,1/32,
1/64,1/32,1/64,1/16,1/16,1/32,1/32,-3/32,-1/16,-1/16,-1/32,-1/8,-1/16,-1/16,3/64,1/32,1/16,1/16,1/32,1/64;

// This is the ring of probability distributions. 

ring rP = 0,(p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,p13,p14,p15,p16,p17,p18,p19,p20,p21,p22,p23,p24),dp;

//This is the Fourier transform.

matrix qtop[20][24] =
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,0,-1,1,0,-1,1,1,0,-1,0,1,0,-1,1,0,1,1,0,-1,1,0,-1,1,
1,-1,1,1,-1,1,1,1,-1,1,-1,1,-1,1,1,-1,1,1,-1,1,1,-1,1,1,
1,1/2,0,0,-1/2,0,-1,0,1/2,0,-1/2,1,1/2,0,0,-1/2,-1,-1,-1/2,0,0,1/2,0,1,
1,-1/2,0,0,1/2,0,-1,0,-1/2,0,1/2,1,-1/2,0,0,1/2,-1,-1,1/2,0,0,-1/2,0,1,
1,0,1,-1,0,-1,1,-1,0,-1,0,1,0,1,-1,0,1,1,0,1,-1,0,-1,1,
1,0,-1,-1,0,1,1,-1,0,1,0,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,
1,2/3,1/3,1/3,0,1/3,-1/3,2/3,1/3,0,1/3,0,-1/3,0,-2/3,-1/3,0,1/3,0,-1/3,-1/3,-2/3,-1/3,-1,
1,-1/2,0,0,1/2,0,-1,1,-1/2,0,-1/2,0,1/2,0,-1,1/2,0,1,-1/2,0,0,1/2,0,-1,
1,0,-1,1,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,-1,0,1,-1,
1,-1,1,1,-1,1,1,0,0,0,0,0,0,0,0,0,0,-1,1,-1,-1,1,-1,-1,
1,1/4,1/2,-1/2,-1/4,-1/2,0,-1/2,-1/4,0,-1/4,0,1/4,0,1/2,1/4,0,0,1/4,-1/2,1/2,-1/4,1/2,-1,
1,-1/2,0,0,1/2,0,-1,-1,1/2,0,1/2,0,-1/2,0,1,-1/2,0,1,-1/2,0,0,1/2,0,-1,
1,0,-1,-1,0,1,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,1,0,-1,-1,
1,1/3,1,-1/3,1/3,-1/3,1,1/3,-1/3,1/3,-1/3,-1,-1/3,-1,1/3,-1/3,-1,1,1/3,1,-1/3,1/3,-1/3,1,
1,0,-1,-1,0,1,1,1,0,-1,0,-1,0,1,1,0,-1,1,0,-1,-1,0,1,1,
1,1/2,0,0,-1/2,0,-1,0,-1/2,0,1/2,-1,-1/2,0,0,1/2,1,-1,-1/2,0,0,1/2,0,1,
1,-1/2,0,0,1/2,0,-1,0,1/2,0,-1/2,-1,1/2,0,0,-1/2,1,-1,1/2,0,0,-1/2,0,1,
1,0,-1,1,0,-1,1,-1,0,1,0,-1,0,1,-1,0,-1,1,0,-1,1,0,-1,1,
1,-1,1,1,-1,1,1,-1,1,-1,1,-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,d0,d1,d2,f0,f1,f2),dp;

ideal P = 
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d0*d1^2*f1*f2^3+4*b2^2*d0*d1*d2*f0^3*f1+8*b2^2*d0*d1*d2*f0^2*f1^2+4*b2^2*d0*d1*d2*f0^2*f1*f2+8*b2^2*d0*d1*d2*f0*f1^3+4*b2^2*d0*d1*d2*f0*f1*f2^2+8*b2^2*d0*d1*d2*f1^3*f2+8*b2^2*d0*d1*d2*f1^2*f2^2+4*b2^2*d0*d1*d2*f1*f2^3+b2^2*d0*d2^2*f0^4+b2^2*d0*d2^2*f0^3*f2+2*b2^2*d0*d2^2*f0^2*f2^2+b2^2*d0*d2^2*f0*f2^3+6*b2^2*d0*d2^2*f1^4+b2^2*d0*d2^2*f2^4+2*b2^2*d1^3*f0^4+4*b2^2*d1^3*f0^3*f2+4*b2^2*d1^3*f0^2*f2^2+4*b2^2*d1^3*f0*f2^3+16*b2^2*d1^3*f1^4+2*b2^2*d1^3*f2^4+4*b2^2*d1^2*d2*f0^3*f1+6*b2^2*d1^2*d2*f0^2*f1^2+4*b2^2*d1^2*d2*f0^2*f1*f2+8*b2^2*d1^2*d2*f0*f1^3+4*b2^2*d1^2*d2*f0*f1^2*f2+4*b2^2*d1^2*d2*f0*f1*f2^2+8*b2^2*d1^2*d2*f1^3*f2+6*b2^2*d1^2*d2*f1^2*f2^2+4*b2^2*d1^2*d2*f1*f2^3+2*b2^2*d1*d2^2*f0^3*f1+2*b2^2*d1*d2^2*f0^2*f1^2+2*b2^2*d1*d2^2*f0^2*f1*f2+4*b2^2*d1*d2^2*f0*f1^3+4*b2^2*d1*d2^2*f0*f1^2*f2+2*b2^2*d1*d2^2*f0*f1*f2^2+4*b2^2*d1*d2^2*f1^3*f2+2*b2^2*d1*d2^2*f1^2*f2^2+2*b2^2*d1*d2^2*f1*f2^3+b2^2*d2^3*f0^3*f2+b2^2*d2^3*f0*f2^3+2*b2^2*d2^3*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(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;
}