//This is the ideal Fourier invariants.

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

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

// This is the inverse of the Fourier transform.

matrix ptoq[10][9] =
1/256,5/64,5/128,15/64,5/32,15/64,5/256,5/32,5/64,
5/128,15/32,5/64,15/32,5/16,-15/32,-15/128,-5/16,-15/32,
5/256,5/64,25/128,15/64,-15/32,15/64,25/256,-15/32,5/64,
5/64,5/8,-5/32,-15/16,0,0,5/64,0,5/16,
5/32,0,5/16,0,-5/4,0,-15/32,5/4,0,
5/64,5/16,-5/32,0,0,-15/16,5/64,0,5/8,
5/128,-5/32,25/64,-15/32,5/16,-15/32,25/128,5/16,-5/32,
15/64,0,-15/32,-15/16,0,15/8,15/64,0,-15/16,
15/128,-15/32,15/64,-15/32,15/16,15/32,-45/128,-15/16,15/32,
15/64,-15/16,-15/32,15/8,0,-15/16,15/64,0,0;

// This is the ring of probability distributions. 

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

//This is the Fourier transform.

matrix qtop[9][10] =
1,1,1,1,1,1,1,1,1,1,
1,3/5,1/5,2/5,0,1/5,-1/5,0,-1/5,-1/5,
1,1/5,1,-1/5,1/5,-1/5,1,-1/5,1/5,-1/5,
1,1/5,1/5,-1/5,0,0,-1/5,-1/15,-1/15,2/15,
1,1/5,-3/5,0,-1/5,0,1/5,0,1/5,0,
1,-1/5,1/5,0,0,-1/5,-1/5,2/15,1/15,-1/15,
1,-3/5,1,1/5,-3/5,1/5,1,1/5,-3/5,1/5,
1,-1/5,-3/5,0,1/5,0,1/5,0,-1/5,0,
1,-3/5,1/5,1/5,0,2/5,-1/5,-1/5,1/5,0;

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,(e0,e1,e2),dp;

ideal P = 
e0^5+2*e1^5+e2^5,
10*e0^4*e1+10*e0*e1^4+10*e1^4*e2+10*e1*e2^4,
5*e0^4*e2+5*e0*e2^4+10*e1^5,
20*e0^3*e1^2+20*e0^2*e1^3+20*e1^3*e2^2+20*e1^2*e2^3,
40*e0^3*e1*e2+40*e0*e1^4+40*e0*e1*e2^3+40*e1^4*e2,
20*e0^3*e1^2+40*e0*e1^3*e2+20*e1^2*e2^3,
10*e0^3*e2^2+10*e0^2*e2^3+20*e1^5,
60*e0^2*e1^3+60*e0^2*e1^2*e2+60*e0*e1^2*e2^2+60*e1^3*e2^2,
60*e0^2*e1*e2^2+30*e0*e1^4+30*e1^4*e2,
60*e0^2*e1^2*e2+120*e0*e1^3*e2+60*e0*e1^2*e2^2;

// 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;
}