//This is the ideal Fourier invariants.

ring rQ = 0, (q1,q2,q3,q4,q5,q6,q7), dp;
ideal Invariants = 
q3*q6-q2*q7,
q4*q5-q2*q7,
q3*q5-q1*q7,
q2*q5-q1*q6,
q2*q3-q1*q4;

// This is the inverse of the Fourier transform.

matrix ptoq[9][7] =
1/16,1/16,5/16,1/16,1/4,1/8,1/8,
1/8,-1/8,1/8,-1/8,1/4,-1/4,0,
1/16,1/16,-3/16,1/16,0,1/8,-1/8,
1/8,1/8,-1/8,-1/8,1/4,0,-1/4,
1/4,-1/4,-1/4,1/4,0,0,0,
1/8,1/8,-1/8,-1/8,-1/4,0,1/4,
1/16,1/16,-3/16,1/16,0,-1/8,1/8,
1/8,-1/8,1/8,-1/8,-1/4,1/4,0,
1/16,1/16,5/16,1/16,-1/4,-1/8,-1/8;

// This is the ring of probability distributions. 

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

//This is the Fourier transform.

matrix qtop[7][9] =
1,1,1,1,1,1,1,1,1,
1,-1,1,1,-1,1,1,-1,1,
1,1/5,-3/5,-1/5,-1/5,-1/5,-3/5,1/5,1,
1,-1,1,-1,1,-1,1,-1,1,
1,1/2,0,1/2,0,-1/2,0,-1/2,-1,
1,-1,1,0,0,0,-1,1,-1,
1,0,-1,-1,0,1,1,0,-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,e0,e1,g0,g1),dp;

ideal P = 
b0^2*e0^4*g0^5+b0^2*e0^4*g1^5+3*b0^2*e0^3*e1*g0^4*g1+b0^2*e0^3*e1*g0^3*g1^2+b0^2*e0^3*e1*g0^2*g1^3+3*b0^2*e0^3*e1*g0*g1^4+6*b0^2*e0^2*e1^2*g0^3*g1^2+6*b0^2*e0^2*e1^2*g0^2*g1^3+3*b0^2*e0*e1^3*g0^4*g1+b0^2*e0*e1^3*g0^3*g1^2+b0^2*e0*e1^3*g0^2*g1^3+3*b0^2*e0*e1^3*g0*g1^4+b0^2*e1^4*g0^5+b0^2*e1^4*g1^5+2*b0*b1*e0^4*g0^4*g1+2*b0*b1*e0^4*g0*g1^4+2*b0*b1*e0^3*e1*g0^5+6*b0*b1*e0^3*e1*g0^3*g1^2+6*b0*b1*e0^3*e1*g0^2*g1^3+2*b0*b1*e0^3*e1*g1^5+8*b0*b1*e0^2*e1^2*g0^4*g1+4*b0*b1*e0^2*e1^2*g0^3*g1^2+4*b0*b1*e0^2*e1^2*g0^2*g1^3+8*b0*b1*e0^2*e1^2*g0*g1^4+2*b0*b1*e0*e1^3*g0^5+6*b0*b1*e0*e1^3*g0^3*g1^2+6*b0*b1*e0*e1^3*g0^2*g1^3+2*b0*b1*e0*e1^3*g1^5+2*b0*b1*e1^4*g0^4*g1+2*b0*b1*e1^4*g0*g1^4+b1^2*e0^4*g0^5+b1^2*e0^4*g1^5+3*b1^2*e0^3*e1*g0^4*g1+b1^2*e0^3*e1*g0^3*g1^2+b1^2*e0^3*e1*g0^2*g1^3+3*b1^2*e0^3*e1*g0*g1^4+6*b1^2*e0^2*e1^2*g0^3*g1^2+6*b1^2*e0^2*e1^2*g0^2*g1^3+3*b1^2*e0*e1^3*g0^4*g1+b1^2*e0*e1^3*g0^3*g1^2+b1^2*e0*e1^3*g0^2*g1^3+3*b1^2*e0*e1^3*g0*g1^4+b1^2*e1^4*g0^5+b1^2*e1^4*g1^5,
2*b0^2*e0^4*g0^4*g1+2*b0^2*e0^4*g0*g1^4+2*b0^2*e0^3*e1*g0^4*g1+6*b0^2*e0^3*e1*g0^3*g1^2+6*b0^2*e0^3*e1*g0^2*g1^3+2*b0^2*e0^3*e1*g0*g1^4+12*b0^2*e0^2*e1^2*g0^3*g1^2+12*b0^2*e0^2*e1^2*g0^2*g1^3+2*b0^2*e0*e1^3*g0^4*g1+6*b0^2*e0*e1^3*g0^3*g1^2+6*b0^2*e0*e1^3*g0^2*g1^3+2*b0^2*e0*e1^3*g0*g1^4+2*b0^2*e1^4*g0^4*g1+2*b0^2*e1^4*g0*g1^4+4*b0*b1*e0^4*g0^3*g1^2+4*b0*b1*e0^4*g0^2*g1^3+4*b0*b1*e0^3*e1*g0^4*g1+12*b0*b1*e0^3*e1*g0^3*g1^2+12*b0*b1*e0^3*e1*g0^2*g1^3+4*b0*b1*e0^3*e1*g0*g1^4+8*b0*b1*e0^2*e1^2*g0^4*g1+16*b0*b1*e0^2*e1^2*g0^3*g1^2+16*b0*b1*e0^2*e1^2*g0^2*g1^3+8*b0*b1*e0^2*e1^2*g0*g1^4+4*b0*b1*e0*e1^3*g0^4*g1+12*b0*b1*e0*e1^3*g0^3*g1^2+12*b0*b1*e0*e1^3*g0^2*g1^3+4*b0*b1*e0*e1^3*g0*g1^4+4*b0*b1*e1^4*g0^3*g1^2+4*b0*b1*e1^4*g0^2*g1^3+2*b1^2*e0^4*g0^4*g1+2*b1^2*e0^4*g0*g1^4+2*b1^2*e0^3*e1*g0^4*g1+6*b1^2*e0^3*e1*g0^3*g1^2+6*b1^2*e0^3*e1*g0^2*g1^3+2*b1^2*e0^3*e1*g0*g1^4+12*b1^2*e0^2*e1^2*g0^3*g1^2+12*b1^2*e0^2*e1^2*g0^2*g1^3+2*b1^2*e0*e1^3*g0^4*g1+6*b1^2*e0*e1^3*g0^3*g1^2+6*b1^2*e0*e1^3*g0^2*g1^3+2*b1^2*e0*e1^3*g0*g1^4+2*b1^2*e1^4*g0^4*g1+2*b1^2*e1^4*g0*g1^4,
b0^2*e0^4*g0^3*g1^2+b0^2*e0^4*g0^2*g1^3+b0^2*e0^3*e1*g0^5+3*b0^2*e0^3*e1*g0^3*g1^2+3*b0^2*e0^3*e1*g0^2*g1^3+b0^2*e0^3*e1*g1^5+6*b0^2*e0^2*e1^2*g0^4*g1+6*b0^2*e0^2*e1^2*g0*g1^4+b0^2*e0*e1^3*g0^5+3*b0^2*e0*e1^3*g0^3*g1^2+3*b0^2*e0*e1^3*g0^2*g1^3+b0^2*e0*e1^3*g1^5+b0^2*e1^4*g0^3*g1^2+b0^2*e1^4*g0^2*g1^3+2*b0*b1*e0^4*g0^3*g1^2+2*b0*b1*e0^4*g0^2*g1^3+6*b0*b1*e0^3*e1*g0^4*g1+2*b0*b1*e0^3*e1*g0^3*g1^2+2*b0*b1*e0^3*e1*g0^2*g1^3+6*b0*b1*e0^3*e1*g0*g1^4+4*b0*b1*e0^2*e1^2*g0^5+8*b0*b1*e0^2*e1^2*g0^3*g1^2+8*b0*b1*e0^2*e1^2*g0^2*g1^3+4*b0*b1*e0^2*e1^2*g1^5+6*b0*b1*e0*e1^3*g0^4*g1+2*b0*b1*e0*e1^3*g0^3*g1^2+2*b0*b1*e0*e1^3*g0^2*g1^3+6*b0*b1*e0*e1^3*g0*g1^4+2*b0*b1*e1^4*g0^3*g1^2+2*b0*b1*e1^4*g0^2*g1^3+b1^2*e0^4*g0^3*g1^2+b1^2*e0^4*g0^2*g1^3+b1^2*e0^3*e1*g0^5+3*b1^2*e0^3*e1*g0^3*g1^2+3*b1^2*e0^3*e1*g0^2*g1^3+b1^2*e0^3*e1*g1^5+6*b1^2*e0^2*e1^2*g0^4*g1+6*b1^2*e0^2*e1^2*g0*g1^4+b1^2*e0*e1^3*g0^5+3*b1^2*e0*e1^3*g0^3*g1^2+3*b1^2*e0*e1^3*g0^2*g1^3+b1^2*e0*e1^3*g1^5+b1^2*e1^4*g0^3*g1^2+b1^2*e1^4*g0^2*g1^3,
2*b0^2*e0^4*g0^4*g1+2*b0^2*e0^4*g0*g1^4+2*b0^2*e0^3*e1*g0^5+6*b0^2*e0^3*e1*g0^3*g1^2+6*b0^2*e0^3*e1*g0^2*g1^3+2*b0^2*e0^3*e1*g1^5+8*b0^2*e0^2*e1^2*g0^4*g1+4*b0^2*e0^2*e1^2*g0^3*g1^2+4*b0^2*e0^2*e1^2*g0^2*g1^3+8*b0^2*e0^2*e1^2*g0*g1^4+2*b0^2*e0*e1^3*g0^5+6*b0^2*e0*e1^3*g0^3*g1^2+6*b0^2*e0*e1^3*g0^2*g1^3+2*b0^2*e0*e1^3*g1^5+2*b0^2*e1^4*g0^4*g1+2*b0^2*e1^4*g0*g1^4+4*b0*b1*e0^4*g0^3*g1^2+4*b0*b1*e0^4*g0^2*g1^3+12*b0*b1*e0^3*e1*g0^4*g1+4*b0*b1*e0^3*e1*g0^3*g1^2+4*b0*b1*e0^3*e1*g0^2*g1^3+12*b0*b1*e0^3*e1*g0*g1^4+8*b0*b1*e0^2*e1^2*g0^5+16*b0*b1*e0^2*e1^2*g0^3*g1^2+16*b0*b1*e0^2*e1^2*g0^2*g1^3+8*b0*b1*e0^2*e1^2*g1^5+12*b0*b1*e0*e1^3*g0^4*g1+4*b0*b1*e0*e1^3*g0^3*g1^2+4*b0*b1*e0*e1^3*g0^2*g1^3+12*b0*b1*e0*e1^3*g0*g1^4+4*b0*b1*e1^4*g0^3*g1^2+4*b0*b1*e1^4*g0^2*g1^3+2*b1^2*e0^4*g0^4*g1+2*b1^2*e0^4*g0*g1^4+2*b1^2*e0^3*e1*g0^5+6*b1^2*e0^3*e1*g0^3*g1^2+6*b1^2*e0^3*e1*g0^2*g1^3+2*b1^2*e0^3*e1*g1^5+8*b1^2*e0^2*e1^2*g0^4*g1+4*b1^2*e0^2*e1^2*g0^3*g1^2+4*b1^2*e0^2*e1^2*g0^2*g1^3+8*b1^2*e0^2*e1^2*g0*g1^4+2*b1^2*e0*e1^3*g0^5+6*b1^2*e0*e1^3*g0^3*g1^2+6*b1^2*e0*e1^3*g0^2*g1^3+2*b1^2*e0*e1^3*g1^5+2*b1^2*e1^4*g0^4*g1+2*b1^2*e1^4*g0*g1^4,
4*b0^2*e0^4*g0^3*g1^2+4*b0^2*e0^4*g0^2*g1^3+4*b0^2*e0^3*e1*g0^4*g1+12*b0^2*e0^3*e1*g0^3*g1^2+12*b0^2*e0^3*e1*g0^2*g1^3+4*b0^2*e0^3*e1*g0*g1^4+8*b0^2*e0^2*e1^2*g0^4*g1+16*b0^2*e0^2*e1^2*g0^3*g1^2+16*b0^2*e0^2*e1^2*g0^2*g1^3+8*b0^2*e0^2*e1^2*g0*g1^4+4*b0^2*e0*e1^3*g0^4*g1+12*b0^2*e0*e1^3*g0^3*g1^2+12*b0^2*e0*e1^3*g0^2*g1^3+4*b0^2*e0*e1^3*g0*g1^4+4*b0^2*e1^4*g0^3*g1^2+4*b0^2*e1^4*g0^2*g1^3+8*b0*b1*e0^4*g0^3*g1^2+8*b0*b1*e0^4*g0^2*g1^3+8*b0*b1*e0^3*e1*g0^4*g1+24*b0*b1*e0^3*e1*g0^3*g1^2+24*b0*b1*e0^3*e1*g0^2*g1^3+8*b0*b1*e0^3*e1*g0*g1^4+16*b0*b1*e0^2*e1^2*g0^4*g1+32*b0*b1*e0^2*e1^2*g0^3*g1^2+32*b0*b1*e0^2*e1^2*g0^2*g1^3+16*b0*b1*e0^2*e1^2*g0*g1^4+8*b0*b1*e0*e1^3*g0^4*g1+24*b0*b1*e0*e1^3*g0^3*g1^2+24*b0*b1*e0*e1^3*g0^2*g1^3+8*b0*b1*e0*e1^3*g0*g1^4+8*b0*b1*e1^4*g0^3*g1^2+8*b0*b1*e1^4*g0^2*g1^3+4*b1^2*e0^4*g0^3*g1^2+4*b1^2*e0^4*g0^2*g1^3+4*b1^2*e0^3*e1*g0^4*g1+12*b1^2*e0^3*e1*g0^3*g1^2+12*b1^2*e0^3*e1*g0^2*g1^3+4*b1^2*e0^3*e1*g0*g1^4+8*b1^2*e0^2*e1^2*g0^4*g1+16*b1^2*e0^2*e1^2*g0^3*g1^2+16*b1^2*e0^2*e1^2*g0^2*g1^3+8*b1^2*e0^2*e1^2*g0*g1^4+4*b1^2*e0*e1^3*g0^4*g1+12*b1^2*e0*e1^3*g0^3*g1^2+12*b1^2*e0*e1^3*g0^2*g1^3+4*b1^2*e0*e1^3*g0*g1^4+4*b1^2*e1^4*g0^3*g1^2+4*b1^2*e1^4*g0^2*g1^3,
2*b0^2*e0^4*g0^3*g1^2+2*b0^2*e0^4*g0^2*g1^3+6*b0^2*e0^3*e1*g0^4*g1+2*b0^2*e0^3*e1*g0^3*g1^2+2*b0^2*e0^3*e1*g0^2*g1^3+6*b0^2*e0^3*e1*g0*g1^4+4*b0^2*e0^2*e1^2*g0^5+8*b0^2*e0^2*e1^2*g0^3*g1^2+8*b0^2*e0^2*e1^2*g0^2*g1^3+4*b0^2*e0^2*e1^2*g1^5+6*b0^2*e0*e1^3*g0^4*g1+2*b0^2*e0*e1^3*g0^3*g1^2+2*b0^2*e0*e1^3*g0^2*g1^3+6*b0^2*e0*e1^3*g0*g1^4+2*b0^2*e1^4*g0^3*g1^2+2*b0^2*e1^4*g0^2*g1^3+4*b0*b1*e0^4*g0^4*g1+4*b0*b1*e0^4*g0*g1^4+4*b0*b1*e0^3*e1*g0^5+12*b0*b1*e0^3*e1*g0^3*g1^2+12*b0*b1*e0^3*e1*g0^2*g1^3+4*b0*b1*e0^3*e1*g1^5+16*b0*b1*e0^2*e1^2*g0^4*g1+8*b0*b1*e0^2*e1^2*g0^3*g1^2+8*b0*b1*e0^2*e1^2*g0^2*g1^3+16*b0*b1*e0^2*e1^2*g0*g1^4+4*b0*b1*e0*e1^3*g0^5+12*b0*b1*e0*e1^3*g0^3*g1^2+12*b0*b1*e0*e1^3*g0^2*g1^3+4*b0*b1*e0*e1^3*g1^5+4*b0*b1*e1^4*g0^4*g1+4*b0*b1*e1^4*g0*g1^4+2*b1^2*e0^4*g0^3*g1^2+2*b1^2*e0^4*g0^2*g1^3+6*b1^2*e0^3*e1*g0^4*g1+2*b1^2*e0^3*e1*g0^3*g1^2+2*b1^2*e0^3*e1*g0^2*g1^3+6*b1^2*e0^3*e1*g0*g1^4+4*b1^2*e0^2*e1^2*g0^5+8*b1^2*e0^2*e1^2*g0^3*g1^2+8*b1^2*e0^2*e1^2*g0^2*g1^3+4*b1^2*e0^2*e1^2*g1^5+6*b1^2*e0*e1^3*g0^4*g1+2*b1^2*e0*e1^3*g0^3*g1^2+2*b1^2*e0*e1^3*g0^2*g1^3+6*b1^2*e0*e1^3*g0*g1^4+2*b1^2*e1^4*g0^3*g1^2+2*b1^2*e1^4*g0^2*g1^3,
b0^2*e0^4*g0^3*g1^2+b0^2*e0^4*g0^2*g1^3+3*b0^2*e0^3*e1*g0^4*g1+b0^2*e0^3*e1*g0^3*g1^2+b0^2*e0^3*e1*g0^2*g1^3+3*b0^2*e0^3*e1*g0*g1^4+2*b0^2*e0^2*e1^2*g0^5+4*b0^2*e0^2*e1^2*g0^3*g1^2+4*b0^2*e0^2*e1^2*g0^2*g1^3+2*b0^2*e0^2*e1^2*g1^5+3*b0^2*e0*e1^3*g0^4*g1+b0^2*e0*e1^3*g0^3*g1^2+b0^2*e0*e1^3*g0^2*g1^3+3*b0^2*e0*e1^3*g0*g1^4+b0^2*e1^4*g0^3*g1^2+b0^2*e1^4*g0^2*g1^3+2*b0*b1*e0^4*g0^3*g1^2+2*b0*b1*e0^4*g0^2*g1^3+2*b0*b1*e0^3*e1*g0^5+6*b0*b1*e0^3*e1*g0^3*g1^2+6*b0*b1*e0^3*e1*g0^2*g1^3+2*b0*b1*e0^3*e1*g1^5+12*b0*b1*e0^2*e1^2*g0^4*g1+12*b0*b1*e0^2*e1^2*g0*g1^4+2*b0*b1*e0*e1^3*g0^5+6*b0*b1*e0*e1^3*g0^3*g1^2+6*b0*b1*e0*e1^3*g0^2*g1^3+2*b0*b1*e0*e1^3*g1^5+2*b0*b1*e1^4*g0^3*g1^2+2*b0*b1*e1^4*g0^2*g1^3+b1^2*e0^4*g0^3*g1^2+b1^2*e0^4*g0^2*g1^3+3*b1^2*e0^3*e1*g0^4*g1+b1^2*e0^3*e1*g0^3*g1^2+b1^2*e0^3*e1*g0^2*g1^3+3*b1^2*e0^3*e1*g0*g1^4+2*b1^2*e0^2*e1^2*g0^5+4*b1^2*e0^2*e1^2*g0^3*g1^2+4*b1^2*e0^2*e1^2*g0^2*g1^3+2*b1^2*e0^2*e1^2*g1^5+3*b1^2*e0*e1^3*g0^4*g1+b1^2*e0*e1^3*g0^3*g1^2+b1^2*e0*e1^3*g0^2*g1^3+3*b1^2*e0*e1^3*g0*g1^4+b1^2*e1^4*g0^3*g1^2+b1^2*e1^4*g0^2*g1^3,
2*b0^2*e0^4*g0^3*g1^2+2*b0^2*e0^4*g0^2*g1^3+2*b0^2*e0^3*e1*g0^4*g1+6*b0^2*e0^3*e1*g0^3*g1^2+6*b0^2*e0^3*e1*g0^2*g1^3+2*b0^2*e0^3*e1*g0*g1^4+4*b0^2*e0^2*e1^2*g0^4*g1+8*b0^2*e0^2*e1^2*g0^3*g1^2+8*b0^2*e0^2*e1^2*g0^2*g1^3+4*b0^2*e0^2*e1^2*g0*g1^4+2*b0^2*e0*e1^3*g0^4*g1+6*b0^2*e0*e1^3*g0^3*g1^2+6*b0^2*e0*e1^3*g0^2*g1^3+2*b0^2*e0*e1^3*g0*g1^4+2*b0^2*e1^4*g0^3*g1^2+2*b0^2*e1^4*g0^2*g1^3+4*b0*b1*e0^4*g0^4*g1+4*b0*b1*e0^4*g0*g1^4+4*b0*b1*e0^3*e1*g0^4*g1+12*b0*b1*e0^3*e1*g0^3*g1^2+12*b0*b1*e0^3*e1*g0^2*g1^3+4*b0*b1*e0^3*e1*g0*g1^4+24*b0*b1*e0^2*e1^2*g0^3*g1^2+24*b0*b1*e0^2*e1^2*g0^2*g1^3+4*b0*b1*e0*e1^3*g0^4*g1+12*b0*b1*e0*e1^3*g0^3*g1^2+12*b0*b1*e0*e1^3*g0^2*g1^3+4*b0*b1*e0*e1^3*g0*g1^4+4*b0*b1*e1^4*g0^4*g1+4*b0*b1*e1^4*g0*g1^4+2*b1^2*e0^4*g0^3*g1^2+2*b1^2*e0^4*g0^2*g1^3+2*b1^2*e0^3*e1*g0^4*g1+6*b1^2*e0^3*e1*g0^3*g1^2+6*b1^2*e0^3*e1*g0^2*g1^3+2*b1^2*e0^3*e1*g0*g1^4+4*b1^2*e0^2*e1^2*g0^4*g1+8*b1^2*e0^2*e1^2*g0^3*g1^2+8*b1^2*e0^2*e1^2*g0^2*g1^3+4*b1^2*e0^2*e1^2*g0*g1^4+2*b1^2*e0*e1^3*g0^4*g1+6*b1^2*e0*e1^3*g0^3*g1^2+6*b1^2*e0*e1^3*g0^2*g1^3+2*b1^2*e0*e1^3*g0*g1^4+2*b1^2*e1^4*g0^3*g1^2+2*b1^2*e1^4*g0^2*g1^3,
b0^2*e0^4*g0^4*g1+b0^2*e0^4*g0*g1^4+b0^2*e0^3*e1*g0^5+3*b0^2*e0^3*e1*g0^3*g1^2+3*b0^2*e0^3*e1*g0^2*g1^3+b0^2*e0^3*e1*g1^5+4*b0^2*e0^2*e1^2*g0^4*g1+2*b0^2*e0^2*e1^2*g0^3*g1^2+2*b0^2*e0^2*e1^2*g0^2*g1^3+4*b0^2*e0^2*e1^2*g0*g1^4+b0^2*e0*e1^3*g0^5+3*b0^2*e0*e1^3*g0^3*g1^2+3*b0^2*e0*e1^3*g0^2*g1^3+b0^2*e0*e1^3*g1^5+b0^2*e1^4*g0^4*g1+b0^2*e1^4*g0*g1^4+2*b0*b1*e0^4*g0^5+2*b0*b1*e0^4*g1^5+6*b0*b1*e0^3*e1*g0^4*g1+2*b0*b1*e0^3*e1*g0^3*g1^2+2*b0*b1*e0^3*e1*g0^2*g1^3+6*b0*b1*e0^3*e1*g0*g1^4+12*b0*b1*e0^2*e1^2*g0^3*g1^2+12*b0*b1*e0^2*e1^2*g0^2*g1^3+6*b0*b1*e0*e1^3*g0^4*g1+2*b0*b1*e0*e1^3*g0^3*g1^2+2*b0*b1*e0*e1^3*g0^2*g1^3+6*b0*b1*e0*e1^3*g0*g1^4+2*b0*b1*e1^4*g0^5+2*b0*b1*e1^4*g1^5+b1^2*e0^4*g0^4*g1+b1^2*e0^4*g0*g1^4+b1^2*e0^3*e1*g0^5+3*b1^2*e0^3*e1*g0^3*g1^2+3*b1^2*e0^3*e1*g0^2*g1^3+b1^2*e0^3*e1*g1^5+4*b1^2*e0^2*e1^2*g0^4*g1+2*b1^2*e0^2*e1^2*g0^3*g1^2+2*b1^2*e0^2*e1^2*g0^2*g1^3+4*b1^2*e0^2*e1^2*g0*g1^4+b1^2*e0*e1^3*g0^5+3*b1^2*e0*e1^3*g0^3*g1^2+3*b1^2*e0*e1^3*g0^2*g1^3+b1^2*e0*e1^3*g1^5+b1^2*e1^4*g0^4*g1+b1^2*e1^4*g0*g1^4;

// 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(0,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;
}