//This is the ideal Fourier invariants.

ring rQ = 0,(q1,q2,q3,q4,q5,q6,q7,q8,q9,q10,q11,q12,q13,q14),dp;

ideal Invariants =  
q11*q13-q10*q14,
q10*q13-q9*q14,
q11*q12-q8*q14,
q10*q12-q8*q13,
q10*q12-q7*q14,
q9*q12-q7*q13,
q8*q12-q6*q14,
q7*q12-q6*q13,
q5*q11-q4*q14,
q10^2-q9*q11,
q8*q10-q5*q14,
q8*q10-q7*q11,
q5*q10-q3*q14,
q5*q10-q4*q13,
q4*q10-q3*q11,
q8*q9-q5*q13,
q8*q9-q7*q10,
q5*q9-q3*q13,
q4*q9-q3*q10,
q8^2-q6*q11,
q7*q8-q5*q12,
q7*q8-q6*q10,
q5*q8-q2*q14,
q5*q8-q4*q12,
q4*q8-q2*q11,
q7^2-q6*q9,
q5*q7-q2*q13,
q5*q7-q3*q12,
q3*q7-q2*q9,
q5*q6-q2*q12,
q4*q6-q2*q8,
q3*q6-q2*q7,
q5^2-q2*q10,
q5^2-q3*q8,
q5^2-q4*q7,
q4*q5-q1*q14,
q3*q5-q1*q13,
q2*q5-q1*q12,
q4^2-q1*q11,
q3*q4-q1*q10,
q2*q4-q1*q8,
q3^2-q1*q9,
q2*q3-q1*q7,
q2^2-q1*q6;

// This is the inverse of the Fourier transform.

matrix ptoq[14][14] =
1/256,5/128,5/128,5/128,15/64,5/256,15/128,15/128,5/256,15/128,5/256,5/64,5/64,5/64,
5/256,25/128,5/128,5/128,15/64,25/256,15/128,15/128,-15/256,-45/128,-15/256,5/64,-15/64,-15/64,
5/256,5/128,25/128,5/128,15/64,-15/256,15/128,-45/128,25/256,15/128,-15/256,-15/64,5/64,-15/64,
5/256,5/128,5/128,25/128,15/64,-15/256,-45/128,15/128,-15/256,15/128,25/256,-15/64,-15/64,5/64,
5/128,25/64,-5/64,-5/64,-15/32,25/128,-15/64,-15/64,5/128,15/64,5/128,-5/32,5/32,5/32,
5/64,5/32,5/32,-5/32,0,-15/64,15/32,-15/32,-15/64,-15/32,5/64,0,0,5/8,
5/64,5/32,-5/32,5/32,0,-15/64,-15/32,15/32,5/64,-15/32,-15/64,0,5/8,0,
5/128,-5/64,25/64,-5/64,-15/32,5/128,-15/64,15/64,25/128,-15/64,5/128,5/32,-5/32,5/32,
5/64,-5/32,5/32,5/32,0,5/64,-15/32,-15/32,-15/64,15/32,-15/64,5/8,0,0,
5/128,-5/64,-5/64,25/64,-15/32,5/128,15/64,-15/64,5/128,-15/64,25/128,5/32,5/32,-5/32,
15/128,15/64,-15/64,-15/64,-15/32,-45/128,15/64,15/64,15/128,45/64,15/128,15/32,-15/32,-15/32,
15/128,-15/64,15/64,-15/64,-15/32,15/128,15/64,45/64,-45/128,15/64,15/128,-15/32,15/32,-15/32,
15/64,-15/32,-15/32,-15/32,15/8,15/64,-15/32,-15/32,15/64,-15/32,15/64,0,0,0,
15/128,-15/64,-15/64,15/64,-15/32,15/128,45/64,15/64,15/128,15/64,-45/128,-15/32,-15/32,15/32;

// This is the ring of probability distributions. 

ring rP = 0,(p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,p13,p14),dp;

//This is the Fourier transform.

matrix qtop[14][14] =
1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1/5,1/5,1,1/5,1/5,-1/5,-1/5,-1/5,1/5,-1/5,-1/5,-1/5,
1,1/5,1,1/5,-1/5,1/5,-1/5,1,1/5,-1/5,-1/5,1/5,-1/5,-1/5,
1,1/5,1/5,1,-1/5,-1/5,1/5,-1/5,1/5,1,-1/5,-1/5,-1/5,1/5,
1,1/5,1/5,1/5,-1/5,0,0,-1/5,0,-1/5,-1/15,-1/15,2/15,-1/15,
1,1,-3/5,-3/5,1,-3/5,-3/5,1/5,1/5,1/5,-3/5,1/5,1/5,1/5,
1,1/5,1/5,-3/5,-1/5,1/5,-1/5,-1/5,-1/5,1/5,1/15,1/15,-1/15,1/5,
1,1/5,-3/5,1/5,-1/5,-1/5,1/5,1/5,-1/5,-1/5,1/15,1/5,-1/15,1/15,
1,-3/5,1,-3/5,1/5,-3/5,1/5,1,-3/5,1/5,1/5,-3/5,1/5,1/5,
1,-3/5,1/5,1/5,1/5,-1/5,-1/5,-1/5,1/5,-1/5,1/5,1/15,-1/15,1/15,
1,-3/5,-3/5,1,1/5,1/5,-3/5,1/5,-3/5,1,1/5,1/5,1/5,-3/5,
1,1/5,-3/5,-3/5,-1/5,0,0,1/5,2/5,1/5,1/5,-1/5,0,-1/5,
1,-3/5,1/5,-3/5,1/5,0,2/5,-1/5,0,1/5,-1/5,1/5,0,-1/5,
1,-3/5,-3/5,1/5,1/5,2/5,0,1/5,0,-1/5,-1/5,-1/5,0,1/5;

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,(e0,e1,e2,e3),dp;

ideal P = 
e0^5+e1^5+e2^5+e3^5,
5*e0^4*e1+5*e0*e1^4+5*e2^4*e3+5*e2*e3^4,
5*e0^4*e2+5*e0*e2^4+5*e1^4*e3+5*e1*e3^4,
5*e0^4*e3+5*e0*e3^4+5*e1^4*e2+5*e1*e2^4,
10*e0^3*e1^2+10*e0^2*e1^3+10*e2^3*e3^2+10*e2^2*e3^3,
20*e0^3*e1*e2+20*e0*e1^3*e3+20*e0*e2^3*e3+20*e1*e2*e3^3,
20*e0^3*e1*e3+20*e0*e1^3*e2+20*e0*e2*e3^3+20*e1*e2^3*e3,
10*e0^3*e2^2+10*e0^2*e2^3+10*e1^3*e3^2+10*e1^2*e3^3,
20*e0^3*e2*e3+20*e0*e1*e2^3+20*e0*e1*e3^3+20*e1^3*e2*e3,
10*e0^3*e3^2+10*e0^2*e3^3+10*e1^3*e2^2+10*e1^2*e2^3,
30*e0^2*e1^2*e2+30*e0^2*e1^2*e3+30*e0*e2^2*e3^2+30*e1*e2^2*e3^2,
30*e0^2*e1*e2^2+30*e0^2*e2^2*e3+30*e0*e1^2*e3^2+30*e1^2*e2*e3^2,
60*e0^2*e1*e2*e3+60*e0*e1^2*e2*e3+60*e0*e1*e2^2*e3+60*e0*e1*e2*e3^2,
30*e0^2*e1*e3^2+30*e0^2*e2*e3^2+30*e0*e1^2*e2^2+30*e1^2*e2^2*e3;

// 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(3,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;
}