//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), dp;
ideal Invariants = 
q13*q15-q14*q16,
q9*q15-q10*q16,
q7*q15-q9*q16,
q6*q15-q8*q16,
q13*q14-q12*q15,
q11*q14-q10*q15,
q9*q14-q8*q15,
q7*q14-q8*q16,
q6*q14-q5*q15,
q4*q14-q3*q15,
q13^2-q12*q16,
q11*q13-q10*q16,
q10*q13-q8*q15,
q9*q13-q8*q16,
q8*q13-q5*q15,
q7*q13-q6*q16,
q6*q13-q5*q16,
q4*q13-q3*q16,
q11*q12-q8*q15,
q10*q12-q8*q14,
q9*q12-q5*q15,
q8*q12-q5*q14,
q7*q12-q5*q16,
q6*q12-q5*q13,
q4*q12-q3*q13,
q2*q12-q1*q16,
q9*q10-q8*q11,
q7*q10-q6*q11,
q6*q10-q5*q11,
q4*q10-q3*q11,
q9^2-q6*q11,
q8*q9-q5*q11,
q8^2-q5*q10,
q7*q8-q6*q9,
q6*q8-q5*q9,
q4*q8-q3*q9,
q4*q7-q2*q11,
q3*q7-q2*q10,
q6^2-q5*q7,
q4*q6-q2*q10,
q3*q6-q1*q11,
q4*q5-q1*q11,
q3*q5-q1*q10,
q2*q5-q1*q7,
q2*q15^2-q3*q16^2,
q1*q15^2-q3*q12*q16,
q2*q14*q15-q3*q13*q16,
q2*q14^2-q3*q12*q16,
q2*q3*q14-q1*q4*q15,
q3*q12*q13-q1*q14*q15,
q3*q12^2-q1*q14^2,
q6*q11^2-q4*q15^2,
q5*q11^2-q3*q15^2,
q5*q10*q11-q3*q14*q15,
q7*q9*q11-q4*q15*q16,
q6*q9*q11-q3*q15*q16,
q5*q9*q11-q3*q14*q16,
q5*q8*q11-q3*q12*q15,
q7^2*q11-q4*q16^2,
q6*q7*q11-q3*q16^2,
q5*q7*q11-q3*q13*q16,
q5*q6*q11-q3*q12*q16,
q5^2*q11-q1*q14*q15,
q5*q10^2-q3*q14^2,
q5*q8*q10-q3*q12*q14,
q5^2*q10-q1*q14^2,
q7^2*q9-q2*q15*q16,
q6*q7*q9-q2*q14*q16,
q5*q7*q9-q1*q15*q16,
q5*q6*q9-q1*q14*q16,
q5^2*q9-q1*q12*q15,
q5^2*q8-q1*q12*q14,
q2*q3*q8-q1*q4*q9,
q7^3-q2*q16^2,
q6*q7^2-q2*q13*q16,
q5*q7^2-q1*q16^2,
q5*q6*q7-q1*q13*q16,
q5^2*q7-q1*q12*q16,
q5^2*q6-q1*q12*q13,
q5^3-q1*q12^2,
q2*q3^2-q1*q4^2;

// This is the inverse of the Fourier transform.

matrix ptoq[25][16] =
1/256,3/256,3/128,3/128,21/256,3/64,27/256,3/32,3/32,9/128,9/128,9/128,3/64,3/32,3/32,9/128,
3/128,-3/128,3/64,-3/64,39/128,-3/32,-27/128,3/16,-3/16,9/64,-9/64,15/64,-3/32,3/16,-3/16,-9/64,
3/256,9/256,-3/128,-3/128,15/256,-3/64,81/256,-3/32,-3/32,-9/128,-9/128,3/128,-3/64,-3/32,-3/32,27/128,
3/128,-3/128,-3/64,3/64,15/128,3/32,-27/128,-3/16,3/16,-9/64,9/64,3/64,3/32,-3/16,3/16,-9/64,
3/256,9/256,-3/128,-3/128,39/256,9/64,9/256,-3/32,-3/32,-9/128,-9/128,15/128,9/64,-3/32,-3/32,-9/128,
3/128,-3/128,3/64,-3/64,15/128,-3/32,-3/128,-3/16,3/16,9/64,-9/64,3/64,-3/32,-3/16,3/16,3/64,
3/64,-3/64,-3/32,3/32,15/64,-3/16,-3/64,0,0,-9/32,9/32,3/32,-3/16,0,0,3/32,
3/256,9/256,9/128,9/128,-9/256,-3/64,9/256,-3/32,-3/32,27/128,27/128,-9/128,-3/64,-3/32,-3/32,-9/128,
3/64,-3/64,3/32,-3/32,-9/64,3/16,-3/64,0,0,9/32,-9/32,-9/32,3/16,0,0,3/32,
3/128,9/128,-3/64,-3/64,-9/128,-3/32,9/128,3/16,3/16,-9/64,-9/64,-9/64,-3/32,3/16,3/16,-9/64,
3/128,-3/128,-3/64,3/64,-9/128,3/32,-3/128,3/16,-3/16,-9/64,9/64,-9/64,3/32,3/16,-3/16,3/64,
3/128,9/128,9/64,9/64,15/128,3/32,9/128,3/16,3/16,-9/64,-9/64,-9/64,-3/32,-3/16,-3/16,-9/64,
3/64,-3/64,3/32,-3/32,15/64,-3/16,-3/64,0,0,-3/32,3/32,-9/32,3/16,0,0,3/32,
3/32,-3/32,3/16,-3/16,3/32,0,-3/32,3/8,-3/8,-3/16,3/16,-3/16,0,-3/8,3/8,3/16,
3/128,9/128,-3/64,-3/64,15/128,3/32,9/128,-3/16,-3/16,3/64,3/64,-9/64,-3/32,3/16,3/16,-9/64,
3/32,-3/32,-3/16,3/16,3/32,0,-3/32,-3/8,3/8,3/16,-3/16,-3/16,0,3/8,-3/8,3/16,
3/64,9/64,-3/32,-3/32,-9/64,-3/16,9/64,0,0,3/32,3/32,3/32,3/16,0,0,-9/32,
3/64,-3/64,-3/32,3/32,-9/64,3/16,-3/64,0,0,3/32,-3/32,3/32,-3/16,0,0,3/32,
3/64,9/64,-3/32,-3/32,3/64,3/16,-27/64,0,0,3/32,3/32,-3/32,-3/16,0,0,9/32,
3/64,-3/64,-3/32,3/32,3/64,-3/16,9/64,0,0,3/32,-3/32,-3/32,3/16,0,0,-3/32,
3/32,-3/32,3/16,-3/16,-9/32,0,9/32,-3/8,3/8,-3/16,3/16,3/16,0,3/8,-3/8,-3/16,
3/32,-3/32,-3/16,3/16,-9/32,0,9/32,3/8,-3/8,3/16,-3/16,3/16,0,-3/8,3/8,-3/16,
3/128,9/128,9/64,9/64,-21/128,-3/32,-27/128,-3/16,-3/16,-9/64,-9/64,9/64,3/32,3/16,3/16,9/64,
3/64,-3/64,3/32,-3/32,-21/64,3/16,9/64,0,0,-3/32,3/32,9/32,-3/16,0,0,-3/32,
3/128,9/128,-3/64,-3/64,-21/128,-3/32,-27/128,3/16,3/16,3/64,3/64,9/64,3/32,-3/16,-3/16,9/64;

// 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,p25),dp;

//This is the Fourier transform.

matrix qtop[16][25] =
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,-1/3,1,-1/3,1,-1/3,-1/3,1,-1/3,1,-1/3,1,-1/3,-1/3,1,-1/3,1,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,1,
1,1/3,-1/3,-1/3,-1/3,1/3,-1/3,1,1/3,-1/3,-1/3,1,1/3,1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,1/3,-1/3,1,1/3,-1/3,
1,-1/3,-1/3,1/3,-1/3,-1/3,1/3,1,-1/3,-1/3,1/3,1,-1/3,-1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,1,-1/3,-1/3,
1,13/21,5/21,5/21,13/21,5/21,5/21,-1/7,-1/7,-1/7,-1/7,5/21,5/21,1/21,5/21,1/21,-1/7,-1/7,1/21,1/21,-1/7,-1/7,-1/3,-1/3,-1/3,
1,-1/3,-1/3,1/3,1,-1/3,-1/3,-1/3,1/3,-1/3,1/3,1/3,-1/3,0,1/3,0,-1/3,1/3,1/3,-1/3,0,0,-1/3,1/3,-1/3,
1,-1/3,1,-1/3,1/9,-1/27,-1/27,1/9,-1/27,1/9,-1/27,1/9,-1/27,-1/27,1/9,-1/27,1/9,-1/27,-1/3,1/9,1/9,1/9,-1/3,1/9,-1/3,
1,1/3,-1/3,-1/3,-1/3,-1/3,0,-1/3,0,1/3,1/3,1/3,0,1/6,-1/3,-1/6,0,0,0,0,-1/6,1/6,-1/3,0,1/3,
1,-1/3,-1/3,1/3,-1/3,1/3,0,-1/3,0,1/3,-1/3,1/3,0,-1/6,-1/3,1/6,0,0,0,0,1/6,-1/6,-1/3,0,1/3,
1,1/3,-1/3,-1/3,-1/3,1/3,-1/3,1,1/3,-1/3,-1/3,-1/3,-1/9,-1/9,1/9,1/9,1/9,1/9,1/9,1/9,-1/9,1/9,-1/3,-1/9,1/9,
1,-1/3,-1/3,1/3,-1/3,-1/3,1/3,1,-1/3,-1/3,1/3,-1/3,1/9,1/9,1/9,-1/9,1/9,-1/9,1/9,-1/9,1/9,-1/9,-1/3,1/9,1/9,
1,5/9,1/9,1/9,5/9,1/9,1/9,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/9,-1/3,-1/9,1/9,1/9,-1/9,-1/9,1/9,1/9,1/3,1/3,1/3,
1,-1/3,-1/3,1/3,1,-1/3,-1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,0,-1/3,0,1/3,-1/3,-1/3,1/3,0,0,1/3,-1/3,1/3,
1,1/3,-1/3,-1/3,-1/3,-1/3,0,-1/3,0,1/3,1/3,-1/3,0,-1/6,1/3,1/6,0,0,0,0,1/6,-1/6,1/3,0,-1/3,
1,-1/3,-1/3,1/3,-1/3,1/3,0,-1/3,0,1/3,-1/3,-1/3,0,1/6,1/3,-1/6,0,0,0,0,-1/6,1/6,1/3,0,-1/3,
1,-1/3,1,-1/3,-1/3,1/9,1/9,-1/3,1/9,-1/3,1/9,-1/3,1/9,1/9,-1/3,1/9,-1/3,1/9,1/3,-1/9,-1/9,-1/9,1/3,-1/9,1/3;

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+3*c0^3*e0^4*g1^5+9*c0^3*e0^3*e1*g0^4*g1+3*c0^3*e0^3*e1*g0^3*g1^2+3*c0^3*e0^3*e1*g0^2*g1^3+9*c0^3*e0^3*e1*g0*g1^4+24*c0^3*e0^3*e1*g1^5+36*c0^3*e0^2*e1^2*g0^3*g1^2+54*c0^3*e0^2*e1^2*g0^2*g1^3+54*c0^3*e0^2*e1^2*g0*g1^4+72*c0^3*e0^2*e1^2*g1^5+9*c0^3*e0*e1^3*g0^4*g1+39*c0^3*e0*e1^3*g0^3*g1^2+99*c0^3*e0*e1^3*g0^2*g1^3+189*c0^3*e0*e1^3*g0*g1^4+96*c0^3*e0*e1^3*g1^5+3*c0^3*e1^4*g0^5+18*c0^3*e1^4*g0^4*g1+42*c0^3*e1^4*g0^3*g1^2+60*c0^3*e1^4*g0^2*g1^3+72*c0^3*e1^4*g0*g1^4+129*c0^3*e1^4*g1^5+6*c0^2*c1*e0^4*g0^4*g1+3*c0^2*c1*e0^4*g0^3*g1^2+3*c0^2*c1*e0^4*g0^2*g1^3+6*c0^2*c1*e0^4*g0*g1^4+18*c0^2*c1*e0^4*g1^5+6*c0^2*c1*e0^3*e1*g0^5+21*c0^2*c1*e0^3*e1*g0^4*g1+57*c0^2*c1*e0^3*e1*g0^3*g1^2+69*c0^2*c1*e0^3*e1*g0^2*g1^3+117*c0^2*c1*e0^3*e1*g0*g1^4+162*c0^2*c1*e0^3*e1*g1^5+6*c0^2*c1*e0^2*e1^2*g0^5+66*c0^2*c1*e0^2*e1^2*g0^4*g1+228*c0^2*c1*e0^2*e1^2*g0^3*g1^2+402*c0^2*c1*e0^2*e1^2*g0^2*g1^3+648*c0^2*c1*e0^2*e1^2*g0*g1^4+594*c0^2*c1*e0^2*e1^2*g1^5+6*c0^2*c1*e0*e1^3*g0^5+105*c0^2*c1*e0*e1^3*g0^4*g1+441*c0^2*c1*e0*e1^3*g0^3*g1^2+921*c0^2*c1*e0*e1^3*g0^2*g1^3+1245*c0^2*c1*e0*e1^3*g0*g1^4+1170*c0^2*c1*e0*e1^3*g1^5+18*c0^2*c1*e1^4*g0^5+126*c0^2*c1*e1^4*g0^4*g1+351*c0^2*c1*e1^4*g0^3*g1^2+549*c0^2*c1*e1^4*g0^2*g1^3+900*c0^2*c1*e1^4*g0*g1^4+972*c0^2*c1*e1^4*g1^5+6*c0*c1^2*e0^4*g0^4*g1+21*c0*c1^2*e0^4*g0^3*g1^2+3*c0*c1^2*e0^4*g0^2*g1^3+42*c0*c1^2*e0^4*g0*g1^4+36*c0*c1^2*e0^4*g1^5+6*c0*c1^2*e0^3*e1*g0^5+57*c0*c1^2*e0^3*e1*g0^4*g1+165*c0*c1^2*e0^3*e1*g0^3*g1^2+213*c0*c1^2*e0^3*e1*g0^2*g1^3+441*c0*c1^2*e0^3*e1*g0*g1^4+414*c0*c1^2*e0^3*e1*g1^5+24*c0*c1^2*e0^2*e1^2*g0^5+210*c0*c1^2*e0^2*e1^2*g0^4*g1+642*c0*c1^2*e0^2*e1^2*g0^3*g1^2+1284*c0*c1^2*e0^2*e1^2*g0^2*g1^3+1836*c0*c1^2*e0^2*e1^2*g0*g1^4+1836*c0*c1^2*e0^2*e1^2*g1^5+42*c0*c1^2*e0*e1^3*g0^5+357*c0*c1^2*e0*e1^3*g0^4*g1+1377*c0*c1^2*e0*e1^3*g0^3*g1^2+2613*c0*c1^2*e0*e1^3*g0^2*g1^3+3585*c0*c1^2*e0*e1^3*g0*g1^4+3690*c0*c1^2*e0*e1^3*g1^5+36*c0*c1^2*e1^4*g0^5+342*c0*c1^2*e1^4*g0^4*g1+1035*c0*c1^2*e1^4*g0^3*g1^2+1719*c0*c1^2*e1^4*g0^2*g1^3+2844*c0*c1^2*e1^4*g0*g1^4+2772*c0*c1^2*e1^4*g1^5+3*c1^3*e0^4*g0^5+12*c1^3*e0^4*g0^4*g1+12*c1^3*e0^4*g0^3*g1^2+6*c1^3*e0^4*g0^2*g1^3+24*c1^3*e0^4*g0*g1^4+51*c1^3*e0^4*g1^5+12*c1^3*e0^3*e1*g0^5+81*c1^3*e0^3*e1*g0^4*g1+159*c1^3*e0^3*e1*g0^3*g1^2+195*c1^3*e0^3*e1*g0^2*g1^3+369*c1^3*e0^3*e1*g0*g1^4+480*c1^3*e0^3*e1*g1^5+18*c1^3*e0^2*e1^2*g0^5+180*c1^3*e0^2*e1^2*g0^4*g1+702*c1^3*e0^2*e1^2*g0^3*g1^2+1260*c1^3*e0^2*e1^2*g0^2*g1^3+1854*c1^3*e0^2*e1^2*g0*g1^4+1818*c1^3*e0^2*e1^2*g1^5+24*c1^3*e0*e1^3*g0^5+321*c1^3*e0*e1^3*g0^4*g1+1311*c1^3*e0*e1^3*g0^3*g1^2+2703*c1^3*e0*e1^3*g0^2*g1^3+3837*c1^3*e0*e1^3*g0*g1^4+3468*c1^3*e0*e1^3*g1^5+51*c1^3*e1^4*g0^5+378*c1^3*e1^4*g0^4*g1+1056*c1^3*e1^4*g0^3*g1^2+1668*c1^3*e1^4*g0^2*g1^3+2664*c1^3*e1^4*g0*g1^4+2931*c1^3*e1^4*g1^5,
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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(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;
}