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

// This is the inverse of the Fourier transform.

matrix ptoq[22][19] =
1/256,3/64,3/128,3/32,1/32,3/64,1/256,1/32,3/32,1/8,3/32,1/8,1/32,1/64,3/64,3/32,1/64,1/32,3/64,
1/32,3/16,0,0,0,-3/16,-1/32,3/16,3/16,1/4,-3/16,-1/4,-3/16,1/16,3/16,0,-1/16,0,-3/16,
1/64,0,3/32,0,-1/8,0,1/64,1/16,3/16,-1/4,3/16,-1/4,1/16,1/16,0,0,1/16,-1/8,0,
3/64,3/16,-3/32,-3/8,0,3/16,3/64,3/16,-3/16,0,-3/16,0,3/16,0,0,0,0,0,0,
3/32,-3/16,0,0,0,3/16,-3/32,3/16,3/16,-3/4,-3/16,3/4,-3/16,3/16,-3/16,0,-3/16,0,3/16,
3/64,0,-3/32,0,0,0,3/64,3/16,-3/16,0,-3/16,0,3/16,0,3/16,-3/8,0,0,3/16,
3/128,-3/32,9/64,-3/16,3/16,-3/32,3/128,0,0,0,0,0,0,3/32,-3/32,-3/16,3/32,3/16,-3/32,
1/32,3/16,0,0,0,-3/16,-1/32,1/16,-3/16,0,3/16,0,-1/16,-1/16,-3/16,0,1/16,0,3/16,
3/32,0,-3/16,0,0,0,3/32,0,0,0,0,0,0,0,-3/8,3/4,0,0,-3/8,
3/32,-3/16,0,0,0,3/16,-3/32,3/16,-9/16,0,9/16,0,-3/16,-3/16,3/16,0,3/16,0,-3/16,
3/32,-3/16,0,0,0,3/16,-3/32,-3/16,-3/16,3/4,3/16,-3/4,3/16,3/16,-3/16,0,-3/16,0,3/16,
3/32,-3/8,-3/16,3/4,0,-3/8,3/32,0,0,0,0,0,0,0,0,0,0,0,0,
1/64,0,3/32,0,-1/8,0,1/64,-1/16,-3/16,1/4,-3/16,1/4,-1/16,1/16,0,0,1/16,-1/8,0,
1/128,3/32,3/64,3/16,1/16,3/32,1/128,0,0,0,0,0,0,-1/32,-3/32,-3/16,-1/32,-1/16,-3/32,
1/32,3/16,0,0,0,-3/16,-1/32,-1/16,3/16,0,-3/16,0,1/16,-1/16,-3/16,0,1/16,0,3/16,
1/32,0,3/16,0,-1/4,0,1/32,0,0,0,0,0,0,-1/8,0,0,-1/8,1/4,0,
3/64,3/16,-3/32,-3/8,0,3/16,3/64,-3/16,3/16,0,3/16,0,-3/16,0,0,0,0,0,0,
3/32,-3/16,0,0,0,3/16,-3/32,-3/16,9/16,0,-9/16,0,3/16,-3/16,3/16,0,3/16,0,-3/16,
3/128,-3/32,9/64,-3/16,3/16,-3/32,3/128,0,0,0,0,0,0,-3/32,3/32,3/16,-3/32,-3/16,3/32,
1/32,3/16,0,0,0,-3/16,-1/32,-3/16,-3/16,-1/4,3/16,1/4,3/16,1/16,3/16,0,-1/16,0,-3/16,
3/64,0,-3/32,0,0,0,3/64,-3/16,3/16,0,3/16,0,-3/16,0,3/16,-3/8,0,0,3/16,
1/256,3/64,3/128,3/32,1/32,3/64,1/256,-1/32,-3/32,-1/8,-3/32,-1/8,-1/32,1/64,3/64,3/32,1/64,1/32,3/64;

// 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),dp;

//This is the Fourier transform.

matrix qtop[19][22] =
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1/2,0,1/3,-1/6,0,-1/3,1/2,0,-1/6,-1/6,-1/3,0,1,1/2,0,1/3,-1/6,-1/3,1/2,0,1,
1,0,1,-1/3,0,-1/3,1,0,-1/3,0,0,-1/3,1,1,0,1,-1/3,0,1,0,-1/3,1,
1,0,0,-1/3,0,0,-1/3,0,0,0,0,1/3,0,1,0,0,-1/3,0,-1/3,0,0,1,
1,0,-1,0,0,0,1,0,0,0,0,0,-1,1,0,-1,0,0,1,0,0,1,
1,-1/2,0,1/3,1/6,0,-1/3,-1/2,0,1/6,1/6,-1/3,0,1,-1/2,0,1/3,1/6,-1/3,-1/2,0,1,
1,-1,1,1,-1,1,1,-1,1,-1,-1,1,1,1,-1,1,1,-1,1,-1,1,1,
1,3/4,1/2,1/2,1/4,1/2,0,1/4,0,1/4,-1/4,0,-1/2,0,-1/4,0,-1/2,-1/4,0,-3/4,-1/2,-1,
1,1/4,1/2,-1/6,1/12,-1/6,0,-1/4,0,-1/4,-1/12,0,-1/2,0,1/4,0,1/6,1/4,0,-1/4,1/6,-1,
1,1/4,-1/2,0,-1/4,0,0,0,0,0,1/4,0,1/2,0,0,0,0,0,0,-1/4,0,-1,
1,-1/4,1/2,-1/6,-1/12,-1/6,0,1/4,0,1/4,1/12,0,-1/2,0,-1/4,0,1/6,-1/4,0,1/4,1/6,-1,
1,-1/4,-1/2,0,1/4,0,0,0,0,0,-1/4,0,1/2,0,0,0,0,0,0,1/4,0,-1,
1,-3/4,1/2,1/2,-1/4,1/2,0,-1/4,0,-1/4,1/4,0,-1/2,0,1/4,0,-1/2,1/4,0,3/4,-1/2,-1,
1,1/2,1,0,1/2,0,1,-1/2,0,-1/2,1/2,0,1,-1,-1/2,-1,0,-1/2,-1,1/2,0,1,
1,1/2,0,0,-1/6,1/3,-1/3,-1/2,-1/3,1/6,-1/6,0,0,-1,-1/2,0,0,1/6,1/3,1/2,1/3,1,
1,0,0,0,0,-1/3,-1/3,0,1/3,0,0,0,0,-1,0,0,0,0,1/3,0,-1/3,1,
1,-1/2,1,0,-1/2,0,1,1/2,0,1/2,-1/2,0,1,-1,1/2,-1,0,1/2,-1,-1/2,0,1,
1,0,-1,0,0,0,1,0,0,0,0,0,-1,-1,0,1,0,0,-1,0,0,1,
1,-1/2,0,0,1/6,1/3,-1/3,1/2,-1/3,-1/6,1/6,0,0,-1,1/2,0,0,-1/6,1/3,-1/2,1/3,1;

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,f0,f1,f2),dp;

ideal P = 
b0^2*f0^5+2*b0^2*f1^5+b0^2*f2^5+4*b0*b1*f0^4*f1+4*b0*b1*f0*f1^4+4*b0*b1*f1^4*f2+4*b0*b1*f1*f2^4+2*b0*b2*f0^4*f2+2*b0*b2*f0*f2^4+4*b0*b2*f1^5+2*b1^2*f0^5+2*b1^2*f0^4*f2+2*b1^2*f0*f2^4+8*b1^2*f1^5+2*b1^2*f2^5+4*b1*b2*f0^4*f1+4*b1*b2*f0*f1^4+4*b1*b2*f1^4*f2+4*b1*b2*f1*f2^4+b2^2*f0^5+2*b2^2*f1^5+b2^2*f2^5,
8*b0^2*f0^4*f1+8*b0^2*f0*f1^4+8*b0^2*f1^4*f2+8*b0^2*f1*f2^4+32*b0*b1*f0^3*f1^2+16*b0*b1*f0^2*f1^3+32*b0*b1*f0*f1^3*f2+16*b0*b1*f1^3*f2^2+32*b0*b1*f1^2*f2^3+16*b0*b2*f0^3*f1*f2+16*b0*b2*f0*f1^4+16*b0*b2*f0*f1*f2^3+16*b0*b2*f1^4*f2+16*b1^2*f0^4*f1+16*b1^2*f0^3*f1*f2+32*b1^2*f0*f1^4+16*b1^2*f0*f1*f2^3+32*b1^2*f1^4*f2+16*b1^2*f1*f2^4+32*b1*b2*f0^3*f1^2+16*b1*b2*f0^2*f1^3+32*b1*b2*f0*f1^3*f2+16*b1*b2*f1^3*f2^2+32*b1*b2*f1^2*f2^3+8*b2^2*f0^4*f1+8*b2^2*f0*f1^4+8*b2^2*f1^4*f2+8*b2^2*f1*f2^4,
4*b0^2*f0^4*f2+4*b0^2*f0*f2^4+8*b0^2*f1^5+16*b0*b1*f0^3*f1*f2+16*b0*b1*f0*f1^4+16*b0*b1*f0*f1*f2^3+16*b0*b1*f1^4*f2+8*b0*b2*f0^3*f2^2+8*b0*b2*f0^2*f2^3+16*b0*b2*f1^5+8*b1^2*f0^4*f2+8*b1^2*f0^3*f2^2+8*b1^2*f0^2*f2^3+8*b1^2*f0*f2^4+32*b1^2*f1^5+16*b1*b2*f0^3*f1*f2+16*b1*b2*f0*f1^4+16*b1*b2*f0*f1*f2^3+16*b1*b2*f1^4*f2+4*b2^2*f0^4*f2+4*b2^2*f0*f2^4+8*b2^2*f1^5,
12*b0^2*f0^3*f1^2+12*b0^2*f0^2*f1^3+12*b0^2*f1^3*f2^2+12*b0^2*f1^2*f2^3+24*b0*b1*f0^3*f1^2+48*b0*b1*f0^2*f1^3+24*b0*b1*f0^2*f1^2*f2+24*b0*b1*f0*f1^2*f2^2+48*b0*b1*f1^3*f2^2+24*b0*b1*f1^2*f2^3+24*b0*b2*f0^2*f1^3+24*b0*b2*f0^2*f1^2*f2+24*b0*b2*f0*f1^2*f2^2+24*b0*b2*f1^3*f2^2+24*b1^2*f0^3*f1^2+48*b1^2*f0^2*f1^3+24*b1^2*f0^2*f1^2*f2+24*b1^2*f0*f1^2*f2^2+48*b1^2*f1^3*f2^2+24*b1^2*f1^2*f2^3+24*b1*b2*f0^3*f1^2+48*b1*b2*f0^2*f1^3+24*b1*b2*f0^2*f1^2*f2+24*b1*b2*f0*f1^2*f2^2+48*b1*b2*f1^3*f2^2+24*b1*b2*f1^2*f2^3+12*b2^2*f0^3*f1^2+12*b2^2*f0^2*f1^3+12*b2^2*f1^3*f2^2+12*b2^2*f1^2*f2^3,
24*b0^2*f0^3*f1*f2+24*b0^2*f0*f1^4+24*b0^2*f0*f1*f2^3+24*b0^2*f1^4*f2+48*b0*b1*f0^2*f1^3+96*b0*b1*f0^2*f1^2*f2+96*b0*b1*f0*f1^3*f2+96*b0*b1*f0*f1^2*f2^2+48*b0*b1*f1^3*f2^2+96*b0*b2*f0^2*f1*f2^2+48*b0*b2*f0*f1^4+48*b0*b2*f1^4*f2+48*b1^2*f0^3*f1*f2+96*b1^2*f0^2*f1*f2^2+96*b1^2*f0*f1^4+48*b1^2*f0*f1*f2^3+96*b1^2*f1^4*f2+48*b1*b2*f0^2*f1^3+96*b1*b2*f0^2*f1^2*f2+96*b1*b2*f0*f1^3*f2+96*b1*b2*f0*f1^2*f2^2+48*b1*b2*f1^3*f2^2+24*b2^2*f0^3*f1*f2+24*b2^2*f0*f1^4+24*b2^2*f0*f1*f2^3+24*b2^2*f1^4*f2,
12*b0^2*f0^3*f1^2+24*b0^2*f0*f1^3*f2+12*b0^2*f1^2*f2^3+48*b0*b1*f0^2*f1^3+48*b0*b1*f0^2*f1^2*f2+48*b0*b1*f0*f1^2*f2^2+48*b0*b1*f1^3*f2^2+24*b0*b2*f0^2*f1^2*f2+48*b0*b2*f0*f1^3*f2+24*b0*b2*f0*f1^2*f2^2+24*b1^2*f0^3*f1^2+24*b1^2*f0^2*f1^2*f2+96*b1^2*f0*f1^3*f2+24*b1^2*f0*f1^2*f2^2+24*b1^2*f1^2*f2^3+48*b1*b2*f0^2*f1^3+48*b1*b2*f0^2*f1^2*f2+48*b1*b2*f0*f1^2*f2^2+48*b1*b2*f1^3*f2^2+12*b2^2*f0^3*f1^2+24*b2^2*f0*f1^3*f2+12*b2^2*f1^2*f2^3,
6*b0^2*f0^3*f2^2+6*b0^2*f0^2*f2^3+12*b0^2*f1^5+48*b0*b1*f0^2*f1*f2^2+24*b0*b1*f0*f1^4+24*b0*b1*f1^4*f2+12*b0*b2*f0^3*f2^2+12*b0*b2*f0^2*f2^3+24*b0*b2*f1^5+24*b1^2*f0^3*f2^2+24*b1^2*f0^2*f2^3+48*b1^2*f1^5+48*b1*b2*f0^2*f1*f2^2+24*b1*b2*f0*f1^4+24*b1*b2*f1^4*f2+6*b2^2*f0^3*f2^2+6*b2^2*f0^2*f2^3+12*b2^2*f1^5,
8*b0^2*f0^3*f1^2+8*b0^2*f0^2*f1^3+8*b0^2*f1^3*f2^2+8*b0^2*f1^2*f2^3+16*b0*b1*f0^4*f1+16*b0*b1*f0^3*f1*f2+32*b0*b1*f0*f1^4+16*b0*b1*f0*f1*f2^3+32*b0*b1*f1^4*f2+16*b0*b1*f1*f2^4+16*b0*b2*f0^3*f1^2+32*b0*b2*f0*f1^3*f2+16*b0*b2*f1^2*f2^3+32*b1^2*f0^3*f1^2+16*b1^2*f0^2*f1^3+32*b1^2*f0*f1^3*f2+16*b1^2*f1^3*f2^2+32*b1^2*f1^2*f2^3+16*b1*b2*f0^4*f1+16*b1*b2*f0^3*f1*f2+32*b1*b2*f0*f1^4+16*b1*b2*f0*f1*f2^3+32*b1*b2*f1^4*f2+16*b1*b2*f1*f2^4+8*b2^2*f0^3*f1^2+8*b2^2*f0^2*f1^3+8*b2^2*f1^3*f2^2+8*b2^2*f1^2*f2^3,
24*b0^2*f0^2*f1^3+24*b0^2*f0^2*f1^2*f2+24*b0^2*f0*f1^2*f2^2+24*b0^2*f1^3*f2^2+48*b0*b1*f0^3*f1^2+48*b0*b1*f0^2*f1^2*f2+192*b0*b1*f0*f1^3*f2+48*b0*b1*f0*f1^2*f2^2+48*b0*b1*f1^2*f2^3+48*b0*b2*f0^2*f1^3+48*b0*b2*f0^2*f1^2*f2+48*b0*b2*f0*f1^2*f2^2+48*b0*b2*f1^3*f2^2+96*b1^2*f0^2*f1^3+96*b1^2*f0^2*f1^2*f2+96*b1^2*f0*f1^2*f2^2+96*b1^2*f1^3*f2^2+48*b1*b2*f0^3*f1^2+48*b1*b2*f0^2*f1^2*f2+192*b1*b2*f0*f1^3*f2+48*b1*b2*f0*f1^2*f2^2+48*b1*b2*f1^2*f2^3+24*b2^2*f0^2*f1^3+24*b2^2*f0^2*f1^2*f2+24*b2^2*f0*f1^2*f2^2+24*b2^2*f1^3*f2^2,
24*b0^2*f0^2*f1^3+24*b0^2*f0^2*f1^2*f2+24*b0^2*f0*f1^2*f2^2+24*b0^2*f1^3*f2^2+48*b0*b1*f0^3*f1*f2+96*b0*b1*f0^2*f1*f2^2+96*b0*b1*f0*f1^4+48*b0*b1*f0*f1*f2^3+96*b0*b1*f1^4*f2+48*b0*b2*f0^2*f1^2*f2+96*b0*b2*f0*f1^3*f2+48*b0*b2*f0*f1^2*f2^2+48*b1^2*f0^2*f1^3+96*b1^2*f0^2*f1^2*f2+96*b1^2*f0*f1^3*f2+96*b1^2*f0*f1^2*f2^2+48*b1^2*f1^3*f2^2+48*b1*b2*f0^3*f1*f2+96*b1*b2*f0^2*f1*f2^2+96*b1*b2*f0*f1^4+48*b1*b2*f0*f1*f2^3+96*b1*b2*f1^4*f2+24*b2^2*f0^2*f1^3+24*b2^2*f0^2*f1^2*f2+24*b2^2*f0*f1^2*f2^2+24*b2^2*f1^3*f2^2,
48*b0^2*f0^2*f1*f2^2+24*b0^2*f0*f1^4+24*b0^2*f1^4*f2+48*b0*b1*f0^2*f1^3+96*b0*b1*f0^2*f1^2*f2+96*b0*b1*f0*f1^3*f2+96*b0*b1*f0*f1^2*f2^2+48*b0*b1*f1^3*f2^2+48*b0*b2*f0^3*f1*f2+48*b0*b2*f0*f1^4+48*b0*b2*f0*f1*f2^3+48*b0*b2*f1^4*f2+48*b1^2*f0^3*f1*f2+96*b1^2*f0^2*f1*f2^2+96*b1^2*f0*f1^4+48*b1^2*f0*f1*f2^3+96*b1^2*f1^4*f2+48*b1*b2*f0^2*f1^3+96*b1*b2*f0^2*f1^2*f2+96*b1*b2*f0*f1^3*f2+96*b1*b2*f0*f1^2*f2^2+48*b1*b2*f1^3*f2^2+48*b2^2*f0^2*f1*f2^2+24*b2^2*f0*f1^4+24*b2^2*f1^4*f2,
24*b0^2*f0^2*f1^2*f2+48*b0^2*f0*f1^3*f2+24*b0^2*f0*f1^2*f2^2+96*b0*b1*f0^2*f1^2*f2+192*b0*b1*f0*f1^3*f2+96*b0*b1*f0*f1^2*f2^2+48*b0*b2*f0^2*f1^2*f2+96*b0*b2*f0*f1^3*f2+48*b0*b2*f0*f1^2*f2^2+96*b1^2*f0^2*f1^2*f2+192*b1^2*f0*f1^3*f2+96*b1^2*f0*f1^2*f2^2+96*b1*b2*f0^2*f1^2*f2+192*b1*b2*f0*f1^3*f2+96*b1*b2*f0*f1^2*f2^2+24*b2^2*f0^2*f1^2*f2+48*b2^2*f0*f1^3*f2+24*b2^2*f0*f1^2*f2^2,
4*b0^2*f0^3*f2^2+4*b0^2*f0^2*f2^3+8*b0^2*f1^5+16*b0*b1*f0^3*f1*f2+16*b0*b1*f0*f1^4+16*b0*b1*f0*f1*f2^3+16*b0*b1*f1^4*f2+8*b0*b2*f0^4*f2+8*b0*b2*f0*f2^4+16*b0*b2*f1^5+8*b1^2*f0^4*f2+8*b1^2*f0^3*f2^2+8*b1^2*f0^2*f2^3+8*b1^2*f0*f2^4+32*b1^2*f1^5+16*b1*b2*f0^3*f1*f2+16*b1*b2*f0*f1^4+16*b1*b2*f0*f1*f2^3+16*b1*b2*f1^4*f2+4*b2^2*f0^3*f2^2+4*b2^2*f0^2*f2^3+8*b2^2*f1^5,
2*b0^2*f0^4*f1+2*b0^2*f0*f1^4+2*b0^2*f1^4*f2+2*b0^2*f1*f2^4+4*b0*b1*f0^5+4*b0*b1*f0^4*f2+4*b0*b1*f0*f2^4+16*b0*b1*f1^5+4*b0*b1*f2^5+4*b0*b2*f0^4*f1+4*b0*b2*f0*f1^4+4*b0*b2*f1^4*f2+4*b0*b2*f1*f2^4+8*b1^2*f0^4*f1+8*b1^2*f0*f1^4+8*b1^2*f1^4*f2+8*b1^2*f1*f2^4+4*b1*b2*f0^5+4*b1*b2*f0^4*f2+4*b1*b2*f0*f2^4+16*b1*b2*f1^5+4*b1*b2*f2^5+2*b2^2*f0^4*f1+2*b2^2*f0*f1^4+2*b2^2*f1^4*f2+2*b2^2*f1*f2^4,
8*b0^2*f0^3*f1^2+16*b0^2*f0*f1^3*f2+8*b0^2*f1^2*f2^3+16*b0*b1*f0^4*f1+16*b0*b1*f0^3*f1*f2+32*b0*b1*f0*f1^4+16*b0*b1*f0*f1*f2^3+32*b0*b1*f1^4*f2+16*b0*b1*f1*f2^4+16*b0*b2*f0^3*f1^2+16*b0*b2*f0^2*f1^3+16*b0*b2*f1^3*f2^2+16*b0*b2*f1^2*f2^3+32*b1^2*f0^3*f1^2+16*b1^2*f0^2*f1^3+32*b1^2*f0*f1^3*f2+16*b1^2*f1^3*f2^2+32*b1^2*f1^2*f2^3+16*b1*b2*f0^4*f1+16*b1*b2*f0^3*f1*f2+32*b1*b2*f0*f1^4+16*b1*b2*f0*f1*f2^3+32*b1*b2*f1^4*f2+16*b1*b2*f1*f2^4+8*b2^2*f0^3*f1^2+16*b2^2*f0*f1^3*f2+8*b2^2*f1^2*f2^3,
8*b0^2*f0^3*f1*f2+8*b0^2*f0*f1^4+8*b0^2*f0*f1*f2^3+8*b0^2*f1^4*f2+16*b0*b1*f0^4*f2+16*b0*b1*f0^3*f2^2+16*b0*b1*f0^2*f2^3+16*b0*b1*f0*f2^4+64*b0*b1*f1^5+16*b0*b2*f0^3*f1*f2+16*b0*b2*f0*f1^4+16*b0*b2*f0*f1*f2^3+16*b0*b2*f1^4*f2+32*b1^2*f0^3*f1*f2+32*b1^2*f0*f1^4+32*b1^2*f0*f1*f2^3+32*b1^2*f1^4*f2+16*b1*b2*f0^4*f2+16*b1*b2*f0^3*f2^2+16*b1*b2*f0^2*f2^3+16*b1*b2*f0*f2^4+64*b1*b2*f1^5+8*b2^2*f0^3*f1*f2+8*b2^2*f0*f1^4+8*b2^2*f0*f1*f2^3+8*b2^2*f1^4*f2,
12*b0^2*f0^2*f1^3+12*b0^2*f0^2*f1^2*f2+12*b0^2*f0*f1^2*f2^2+12*b0^2*f1^3*f2^2+24*b0*b1*f0^3*f1^2+48*b0*b1*f0^2*f1^3+24*b0*b1*f0^2*f1^2*f2+24*b0*b1*f0*f1^2*f2^2+48*b0*b1*f1^3*f2^2+24*b0*b1*f1^2*f2^3+24*b0*b2*f0^3*f1^2+24*b0*b2*f0^2*f1^3+24*b0*b2*f1^3*f2^2+24*b0*b2*f1^2*f2^3+24*b1^2*f0^3*f1^2+48*b1^2*f0^2*f1^3+24*b1^2*f0^2*f1^2*f2+24*b1^2*f0*f1^2*f2^2+48*b1^2*f1^3*f2^2+24*b1^2*f1^2*f2^3+24*b1*b2*f0^3*f1^2+48*b1*b2*f0^2*f1^3+24*b1*b2*f0^2*f1^2*f2+24*b1*b2*f0*f1^2*f2^2+48*b1*b2*f1^3*f2^2+24*b1*b2*f1^2*f2^3+12*b2^2*f0^2*f1^3+12*b2^2*f0^2*f1^2*f2+12*b2^2*f0*f1^2*f2^2+12*b2^2*f1^3*f2^2,
24*b0^2*f0^2*f1^2*f2+48*b0^2*f0*f1^3*f2+24*b0^2*f0*f1^2*f2^2+48*b0*b1*f0^3*f1*f2+96*b0*b1*f0^2*f1*f2^2+96*b0*b1*f0*f1^4+48*b0*b1*f0*f1*f2^3+96*b0*b1*f1^4*f2+48*b0*b2*f0^2*f1^3+48*b0*b2*f0^2*f1^2*f2+48*b0*b2*f0*f1^2*f2^2+48*b0*b2*f1^3*f2^2+48*b1^2*f0^2*f1^3+96*b1^2*f0^2*f1^2*f2+96*b1^2*f0*f1^3*f2+96*b1^2*f0*f1^2*f2^2+48*b1^2*f1^3*f2^2+48*b1*b2*f0^3*f1*f2+96*b1*b2*f0^2*f1*f2^2+96*b1*b2*f0*f1^4+48*b1*b2*f0*f1*f2^3+96*b1*b2*f1^4*f2+24*b2^2*f0^2*f1^2*f2+48*b2^2*f0*f1^3*f2+24*b2^2*f0*f1^2*f2^2,
12*b0^2*f0^2*f1*f2^2+6*b0^2*f0*f1^4+6*b0^2*f1^4*f2+24*b0*b1*f0^3*f2^2+24*b0*b1*f0^2*f2^3+48*b0*b1*f1^5+24*b0*b2*f0^2*f1*f2^2+12*b0*b2*f0*f1^4+12*b0*b2*f1^4*f2+48*b1^2*f0^2*f1*f2^2+24*b1^2*f0*f1^4+24*b1^2*f1^4*f2+24*b1*b2*f0^3*f2^2+24*b1*b2*f0^2*f2^3+48*b1*b2*f1^5+12*b2^2*f0^2*f1*f2^2+6*b2^2*f0*f1^4+6*b2^2*f1^4*f2,
8*b0^2*f0^3*f1*f2+8*b0^2*f0*f1^4+8*b0^2*f0*f1*f2^3+8*b0^2*f1^4*f2+32*b0*b1*f0^3*f1^2+16*b0*b1*f0^2*f1^3+32*b0*b1*f0*f1^3*f2+16*b0*b1*f1^3*f2^2+32*b0*b1*f1^2*f2^3+16*b0*b2*f0^4*f1+16*b0*b2*f0*f1^4+16*b0*b2*f1^4*f2+16*b0*b2*f1*f2^4+16*b1^2*f0^4*f1+16*b1^2*f0^3*f1*f2+32*b1^2*f0*f1^4+16*b1^2*f0*f1*f2^3+32*b1^2*f1^4*f2+16*b1^2*f1*f2^4+32*b1*b2*f0^3*f1^2+16*b1*b2*f0^2*f1^3+32*b1*b2*f0*f1^3*f2+16*b1*b2*f1^3*f2^2+32*b1*b2*f1^2*f2^3+8*b2^2*f0^3*f1*f2+8*b2^2*f0*f1^4+8*b2^2*f0*f1*f2^3+8*b2^2*f1^4*f2,
12*b0^2*f0^2*f1^2*f2+24*b0^2*f0*f1^3*f2+12*b0^2*f0*f1^2*f2^2+48*b0*b1*f0^2*f1^3+48*b0*b1*f0^2*f1^2*f2+48*b0*b1*f0*f1^2*f2^2+48*b0*b1*f1^3*f2^2+24*b0*b2*f0^3*f1^2+48*b0*b2*f0*f1^3*f2+24*b0*b2*f1^2*f2^3+24*b1^2*f0^3*f1^2+24*b1^2*f0^2*f1^2*f2+96*b1^2*f0*f1^3*f2+24*b1^2*f0*f1^2*f2^2+24*b1^2*f1^2*f2^3+48*b1*b2*f0^2*f1^3+48*b1*b2*f0^2*f1^2*f2+48*b1*b2*f0*f1^2*f2^2+48*b1*b2*f1^3*f2^2+12*b2^2*f0^2*f1^2*f2+24*b2^2*f0*f1^3*f2+12*b2^2*f0*f1^2*f2^2,
b0^2*f0^4*f2+b0^2*f0*f2^4+2*b0^2*f1^5+4*b0*b1*f0^4*f1+4*b0*b1*f0*f1^4+4*b0*b1*f1^4*f2+4*b0*b1*f1*f2^4+2*b0*b2*f0^5+4*b0*b2*f1^5+2*b0*b2*f2^5+2*b1^2*f0^5+2*b1^2*f0^4*f2+2*b1^2*f0*f2^4+8*b1^2*f1^5+2*b1^2*f2^5+4*b1*b2*f0^4*f1+4*b1*b2*f0*f1^4+4*b1*b2*f1^4*f2+4*b1*b2*f1*f2^4+b2^2*f0^4*f2+b2^2*f0*f2^4+2*b2^2*f1^5;

// 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;
}