//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,q20,q21,q22,q23), dp;

ideal Invariants = 
q20*q22-q11*q23,
q18*q22-q17*q23,
q12*q22-q10*q23,
q8*q22-q5*q23,
q7*q22-q6*q23,
q19*q20-q18*q21,
q14*q20-q13*q21,
q11*q20-q12*q21,
q9*q20-q8*q21,
q3*q20-q2*q21,
q18*q19-q16*q23,
q17*q19-q16*q22,
q13*q19-q9*q23,
q12*q19-q17*q20,
q11*q19-q17*q21,
q9*q19-q11*q21,
q8*q19-q12*q21,
q7*q19-q5*q23,
q6*q19-q5*q22,
q5*q19-q10*q21,
q17*q18-q15*q23,
q14*q18-q9*q23,
q13*q18-q8*q23,
q11*q18-q17*q20,
q9*q18-q12*q21,
q8*q18-q12*q20,
q6*q18-q4*q23,
q5*q18-q10*q20,
q3*q18-q2*q19,
q17^2-q15*q22,
q16*q17-q15*q19,
q14*q17-q9*q22,
q13*q17-q5*q23,
q12*q17-q10*q18,
q11*q17-q10*q19,
q9*q17-q10*q21,
q8*q17-q10*q20,
q7*q17-q4*q23,
q6*q17-q4*q22,
q5*q17-q4*q19,
q14*q16-q11*q21,
q13*q16-q12*q21,
q12*q16-q15*q20,
q11*q16-q15*q21,
q7*q16-q10*q20,
q6*q16-q4*q19,
q14*q15-q10*q21,
q13*q15-q10*q20,
q11*q15-q10*q16,
q9*q15-q5*q16,
q7*q15-q4*q18,
q6*q15-q4*q17,
q5*q15-q4*q16,
q12*q14-q7*q21,
q10*q14-q6*q21,
q7*q14-q3*q23,
q6*q14-q3*q22,
q5*q14-q3*q19,
q4*q14-q3*q17,
q12*q13-q7*q20,
q11*q13-q7*q21,
q10*q13-q6*q20,
q9*q13-q8*q14,
q7*q13-q2*q23,
q6*q13-q2*q22,
q5*q13-q2*q19,
q4*q13-q2*q17,
q3*q13-q2*q14,
q11*q12-q10*q20,
q10*q12-q4*q18,
q9*q12-q5*q20,
q5*q12-q4*q20,
q11^2-q10*q21,
q10*q11-q4*q19,
q9*q11-q5*q21,
q8*q11-q5*q20,
q7*q11-q6*q20,
q5*q11-q4*q21,
q2*q11-q3*q12,
q10^2-q4*q17,
q9*q10-q4*q21,
q8*q10-q4*q20,
q7*q10-q6*q12,
q5*q10-q4*q11,
q3*q10-q1*q21,
q2*q10-q1*q20,
q7*q9-q2*q19,
q6*q9-q3*q17,
q5*q9-q3*q16,
q4*q9-q3*q15,
q7*q8-q2*q18,
q6*q8-q2*q17,
q5*q8-q2*q16,
q4*q8-q2*q15,
q3*q8-q2*q9,
q6*q7-q1*q23,
q5*q7-q2*q17,
q4*q7-q1*q18,
q6^2-q1*q22,
q5*q6-q1*q19,
q4*q6-q1*q17,
q3*q6-q1*q14,
q2*q6-q1*q13,
q5^2-q3*q15,
q4*q5-q1*q16,
q4^2-q1*q15,
q3*q4-q1*q9,
q2*q4-q1*q8,
q13^2*q22-q3*q23^2,
q2*q13*q22-q3*q7*q23,
q14*q19^2-q21^2*q22,
q7*q18^2-q12^2*q23,
q9^2*q14-q3*q21^2,
q8*q9*q14-q2*q21^2,
q8^2*q14-q2*q20*q21,
q8^2*q13-q2*q20^2,
q3*q7^2-q2^2*q22;

// This is the inverse of the Fourier transform.

matrix ptoq[31][23] =
1/256,1/128,1/256,9/128,3/64,9/256,3/128,3/64,3/128,21/128,3/32,3/64,3/128,3/256,1/16,1/16,3/32,3/64,3/64,3/64,3/128,1/128,1/128,
1/64,0,-1/64,3/16,-3/32,3/64,0,0,-3/32,9/32,-3/16,0,0,-3/64,1/8,-1/8,3/32,0,-3/32,0,-3/32,0,0,
1/128,-1/64,1/128,3/64,0,9/128,-3/64,-3/32,3/64,9/64,0,-3/32,-3/64,3/128,0,0,3/32,-3/32,0,-3/32,3/64,1/64,-1/64,
1/128,1/64,1/128,3/64,0,-3/128,-3/64,3/32,3/64,-3/64,0,-3/32,3/64,3/128,0,0,-3/32,-3/32,0,3/32,3/64,-1/64,-1/64,
1/64,0,-1/64,0,3/32,3/64,0,0,-3/32,-3/32,3/16,0,0,-3/64,-1/8,1/8,-3/32,0,3/32,0,-3/32,0,0,
1/128,-1/64,1/128,3/64,0,-3/128,3/64,-3/32,3/64,-3/64,0,3/32,-3/64,3/128,0,0,-3/32,3/32,0,-3/32,3/64,-1/64,1/64,
1/256,1/128,1/256,-3/128,-3/64,9/256,3/128,3/64,3/128,-3/128,-3/32,3/64,3/128,3/256,-1/16,-1/16,0,3/64,-3/64,3/64,3/128,1/128,1/128,
3/128,3/64,3/128,15/64,3/16,3/128,3/64,3/32,3/64,3/64,0,3/32,-3/64,-3/128,0,0,-9/32,-3/32,-3/16,-3/32,-3/64,-3/64,-3/64,
3/64,0,-3/64,3/8,-9/32,-3/64,0,0,-3/32,-9/32,3/16,0,0,3/64,0,0,-3/32,0,3/32,0,3/32,0,0,
3/64,-3/32,3/64,3/32,0,3/64,-3/32,-3/16,3/32,3/32,0,-3/16,3/32,-3/64,0,0,-3/16,3/16,0,3/16,-3/32,-3/32,3/32,
3/64,0,-3/64,3/16,-3/32,-3/64,0,0,-3/32,3/32,-3/16,0,0,3/64,0,0,-9/32,0,9/32,0,3/32,0,0,
3/128,3/64,3/128,9/64,3/32,-9/128,-3/64,3/32,3/64,-21/64,-3/16,-3/32,-3/64,-3/128,0,0,3/16,3/32,3/32,-3/32,-3/64,3/64,3/64,
3/64,0,-3/64,0,3/32,-3/64,0,0,-3/32,-9/32,3/16,0,0,3/64,0,0,9/32,0,-9/32,0,3/32,0,0,
3/64,-3/32,3/64,3/32,0,-9/64,3/32,-3/16,3/32,-9/32,0,3/16,3/32,-3/64,0,0,3/16,-3/16,0,3/16,-3/32,3/32,-3/32,
3/128,3/64,3/128,-9/64,-3/16,3/128,3/64,3/32,3/64,3/64,0,3/32,-3/64,-3/128,0,0,3/32,-3/32,3/16,-3/32,-3/64,-3/64,-3/64,
3/64,0,-3/64,-3/16,9/32,-3/64,0,0,-3/32,3/32,-3/16,0,0,3/64,0,0,3/32,0,-3/32,0,3/32,0,0,
3/128,3/64,3/128,-3/64,-3/32,-9/128,-3/64,3/32,3/64,3/64,3/16,-3/32,-3/64,-3/128,0,0,0,3/32,-3/32,-3/32,-3/64,3/64,3/64,
3/256,3/128,3/256,3/128,3/64,27/256,9/128,-3/64,-3/128,3/128,3/32,-3/64,9/128,9/256,-3/16,-3/16,0,-3/64,3/64,-3/64,-3/128,3/128,3/128,
3/64,0,-3/64,0,-3/32,9/64,0,0,3/32,3/32,-3/16,0,0,-9/64,-3/8,3/8,3/32,0,-3/32,0,3/32,0,0,
3/128,-3/64,3/128,-3/64,0,27/128,-9/64,3/32,-3/64,-9/64,0,3/32,-9/64,9/128,0,0,-3/32,3/32,0,3/32,-3/64,3/64,-3/64,
3/128,3/64,3/128,-3/64,0,-9/128,-9/64,-3/32,-3/64,3/64,0,3/32,9/64,9/128,0,0,3/32,3/32,0,-3/32,-3/64,-3/64,-3/64,
3/64,0,-3/64,-3/16,3/32,9/64,0,0,3/32,-9/32,3/16,0,0,-9/64,3/8,-3/8,-3/32,0,3/32,0,3/32,0,0,
3/128,-3/64,3/128,-3/64,0,-9/128,9/64,3/32,-3/64,3/64,0,-3/32,-9/64,9/128,0,0,3/32,-3/32,0,3/32,-3/64,-3/64,3/64,
3/256,3/128,3/256,-9/128,-3/64,27/256,9/128,-3/64,-3/128,-21/128,-3/32,-3/64,9/128,9/256,3/16,3/16,-3/32,-3/64,-3/64,-3/64,-3/128,3/128,3/128,
3/64,3/32,3/64,-3/32,0,3/64,3/32,-3/16,-3/32,-3/32,0,-3/16,-3/32,-3/64,0,0,3/16,3/16,0,3/16,3/32,-3/32,-3/32,
3/32,0,-3/32,0,-3/16,-3/32,0,0,3/16,-3/16,3/8,0,0,3/32,0,0,3/16,0,-3/16,0,-3/16,0,0,
3/64,-3/32,3/64,-3/32,0,3/64,-3/32,3/16,-3/32,-3/32,0,3/16,3/32,-3/64,0,0,3/16,-3/16,0,-3/16,3/32,-3/32,3/32,
3/32,0,-3/32,-3/8,3/16,-3/32,0,0,3/16,9/16,-3/8,0,0,3/32,0,0,-3/16,0,3/16,0,-3/16,0,0,
3/128,3/64,3/128,3/64,3/32,-9/128,-3/64,-3/32,-3/64,-3/64,-3/16,3/32,-3/64,-3/128,0,0,0,-3/32,3/32,3/32,3/64,3/64,3/64,
3/64,-3/32,3/64,-3/32,0,-9/64,3/32,3/16,-3/32,9/32,0,-3/16,3/32,-3/64,0,0,-3/16,3/16,0,-3/16,3/32,3/32,-3/32,
3/128,3/64,3/128,-9/64,-3/32,-9/128,-3/64,-3/32,-3/64,21/64,3/16,3/32,-3/64,-3/128,0,0,-3/16,-3/32,-3/32,3/32,3/64,3/64,3/64;

// 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,p23,p24,p25,p26,p27,p28,p29,p30,p31),dp;

//This is the Fourier transform.

matrix qtop[23][31] =
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,0,-1,1,0,-1,1,1,0,-1,0,1,0,-1,1,0,1,1,0,-1,1,0,-1,1,1,0,-1,0,1,-1,1,
1,-1,1,1,-1,1,1,1,-1,1,-1,1,-1,1,1,-1,1,1,-1,1,1,-1,1,1,1,-1,1,-1,1,1,1,
1,2/3,1/3,1/3,0,1/3,-1/3,5/9,4/9,1/9,2/9,1/3,0,1/9,-1/3,-2/9,-1/9,1/9,0,-1/9,-1/9,-2/9,-1/9,-1/3,-1/9,0,-1/9,-2/9,1/9,-1/9,-1/3,
1,-1/2,0,0,1/2,0,-1,2/3,-1/2,0,-1/6,1/3,1/6,0,-2/3,1/2,-1/3,1/3,-1/6,0,0,1/6,0,-1/3,0,-1/6,0,1/6,1/3,0,-1/3,
1,1/3,1,-1/3,1/3,-1/3,1,1/9,-1/9,1/9,-1/9,-1/3,-1/9,-1/3,1/9,-1/9,-1/3,1,1/3,1,-1/3,1/3,-1/3,1,1/9,-1/9,1/9,-1/9,-1/3,-1/3,-1/3,
1,0,-1,-1,0,1,1,1/3,0,-1/3,0,-1/3,0,1/3,1/3,0,-1/3,1,0,-1,-1,0,1,1,1/3,0,-1/3,0,-1/3,1/3,-1/3,
1,0,-1,1,0,-1,1,1/3,0,-1/3,0,1/3,0,-1/3,1/3,0,1/3,-1/3,0,1/3,-1/3,0,1/3,-1/3,-1/3,0,1/3,0,-1/3,1/3,-1/3,
1,-1,1,1,-1,1,1,1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,1/3,-1/3,1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,-1/3,1/3,-1/3,1/3,-1/3,-1/3,-1/3,
1,3/7,3/7,-1/7,-1/7,-1/7,-1/7,1/21,-1/7,1/21,1/21,-1/3,-1/7,-1/7,1/21,1/21,1/21,1/21,1/21,-1/7,1/21,-1/7,1/21,-1/3,-1/21,-1/21,-1/21,1/7,-1/21,1/7,1/3,
1,-1/2,0,0,1/2,0,-1,0,1/6,0,-1/6,-1/3,1/6,0,0,-1/6,1/3,1/3,-1/6,0,0,1/6,0,-1/3,0,1/6,0,-1/6,-1/3,0,1/3,
1,0,-1,-1,0,1,1,1/3,0,-1/3,0,-1/3,0,1/3,1/3,0,-1/3,-1/3,0,1/3,1/3,0,-1/3,-1/3,-1/3,0,1/3,0,1/3,-1/3,1/3,
1,0,-1,1,0,-1,1,-1/3,0,1/3,0,-1/3,0,1/3,-1/3,0,-1/3,1,0,-1,1,0,-1,1,-1/3,0,1/3,0,-1/3,1/3,-1/3,
1,-1,1,1,-1,1,1,-1/3,1/3,-1/3,1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,1,-1,1,1,-1,1,1,-1/3,1/3,-1/3,1/3,-1/3,-1/3,-1/3,
1,1/2,0,0,-1/2,0,-1,0,0,0,0,0,0,0,0,0,0,-1,-1/2,0,0,1/2,0,1,0,0,0,0,0,0,0,
1,-1/2,0,0,1/2,0,-1,0,0,0,0,0,0,0,0,0,0,-1,1/2,0,0,-1/2,0,1,0,0,0,0,0,0,0,
1,1/4,1/2,-1/2,-1/4,-1/2,0,-1/2,-1/12,-1/6,-1/4,1/3,1/4,1/6,1/6,1/12,0,0,1/12,-1/6,1/6,-1/12,1/6,-1/3,1/6,1/12,1/6,-1/12,0,-1/6,-1/3,
1,0,-1,-1,0,1,1,-1/3,0,1/3,0,1/3,0,-1/3,-1/3,0,1/3,-1/3,0,1/3,1/3,0,-1/3,-1/3,1/3,0,-1/3,0,-1/3,1/3,-1/3,
1,-1/2,0,0,1/2,0,-1,-2/3,1/6,0,1/2,1/3,-1/2,0,2/3,-1/6,-1/3,1/3,-1/6,0,0,1/6,0,-1/3,0,-1/6,0,1/6,1/3,0,-1/3,
1,0,-1,1,0,-1,1,-1/3,0,1/3,0,-1/3,0,1/3,-1/3,0,-1/3,-1/3,0,1/3,-1/3,0,1/3,-1/3,1/3,0,-1/3,0,1/3,-1/3,1/3,
1,-1,1,1,-1,1,1,-1/3,1/3,-1/3,1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,1/3,1/3,
1,0,1,-1,0,-1,1,-1,0,-1,0,1,0,1,-1,0,1,1,0,1,-1,0,-1,1,-1,0,-1,0,1,1,1,
1,0,-1,-1,0,1,1,-1,0,1,0,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,-1,0,1,0,1,-1,1;

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,(d0,d1,d2,f0,f1,f2),dp;

ideal P = 
d0^4*f0^5+2*d0^4*f1^5+d0^4*f2^5+6*d0^3*d1*f0^4*f1+2*d0^3*d1*f0^3*f1^2+2*d0^3*d1*f0^2*f1^3+6*d0^3*d1*f0*f1^4+6*d0^3*d1*f1^4*f2+2*d0^3*d1*f1^3*f2^2+2*d0^3*d1*f1^2*f2^3+6*d0^3*d1*f1*f2^4+3*d0^3*d2*f0^4*f2+d0^3*d2*f0^3*f2^2+d0^3*d2*f0^2*f2^3+3*d0^3*d2*f0*f2^4+8*d0^3*d2*f1^5+18*d0^2*d1^2*f0^3*f1^2+18*d0^2*d1^2*f0^2*f1^3+6*d0^2*d1^2*f0^2*f1^2*f2+12*d0^2*d1^2*f0*f1^3*f2+6*d0^2*d1^2*f0*f1^2*f2^2+18*d0^2*d1^2*f1^3*f2^2+18*d0^2*d1^2*f1^2*f2^3+12*d0^2*d1*d2*f0^3*f1*f2+6*d0^2*d1*d2*f0^2*f1^3+6*d0^2*d1*d2*f0^2*f1^2*f2+12*d0^2*d1*d2*f0^2*f1*f2^2+18*d0^2*d1*d2*f0*f1^4+6*d0^2*d1*d2*f0*f1^2*f2^2+12*d0^2*d1*d2*f0*f1*f2^3+18*d0^2*d1*d2*f1^4*f2+6*d0^2*d1*d2*f1^3*f2^2+6*d0^2*d2^2*f0^3*f2^2+6*d0^2*d2^2*f0^2*f2^3+12*d0^2*d2^2*f1^5+6*d0*d1^3*f0^4*f1+2*d0*d1^3*f0^3*f1^2+12*d0*d1^3*f0^3*f1*f2+8*d0*d1^3*f0^2*f1^3+6*d0*d1^3*f0^2*f1^2*f2+12*d0*d1^3*f0^2*f1*f2^2+24*d0*d1^3*f0*f1^4+6*d0*d1^3*f0*f1^2*f2^2+12*d0*d1^3*f0*f1*f2^3+24*d0*d1^3*f1^4*f2+8*d0*d1^3*f1^3*f2^2+2*d0*d1^3*f1^2*f2^3+6*d0*d1^3*f1*f2^4+12*d0*d1^2*d2*f0^3*f1^2+12*d0*d1^2*d2*f0^2*f1^3+36*d0*d1^2*d2*f0^2*f1^2*f2+72*d0*d1^2*d2*f0*f1^3*f2+36*d0*d1^2*d2*f0*f1^2*f2^2+12*d0*d1^2*d2*f1^3*f2^2+12*d0*d1^2*d2*f1^2*f2^3+12*d0*d1*d2^2*f0^3*f1*f2+6*d0*d1*d2^2*f0^2*f1^3+6*d0*d1*d2^2*f0^2*f1^2*f2+12*d0*d1*d2^2*f0^2*f1*f2^2+18*d0*d1*d2^2*f0*f1^4+6*d0*d1*d2^2*f0*f1^2*f2^2+12*d0*d1*d2^2*f0*f1*f2^3+18*d0*d1*d2^2*f1^4*f2+6*d0*d1*d2^2*f1^3*f2^2+3*d0*d2^3*f0^4*f2+d0*d2^3*f0^3*f2^2+d0*d2^3*f0^2*f2^3+3*d0*d2^3*f0*f2^4+8*d0*d2^3*f1^5+2*d1^4*f0^5+6*d1^4*f0^4*f2+8*d1^4*f0^3*f2^2+8*d1^4*f0^2*f2^3+6*d1^4*f0*f2^4+32*d1^4*f1^5+2*d1^4*f2^5+6*d1^3*d2*f0^4*f1+2*d1^3*d2*f0^3*f1^2+12*d1^3*d2*f0^3*f1*f2+8*d1^3*d2*f0^2*f1^3+6*d1^3*d2*f0^2*f1^2*f2+12*d1^3*d2*f0^2*f1*f2^2+24*d1^3*d2*f0*f1^4+6*d1^3*d2*f0*f1^2*f2^2+12*d1^3*d2*f0*f1*f2^3+24*d1^3*d2*f1^4*f2+8*d1^3*d2*f1^3*f2^2+2*d1^3*d2*f1^2*f2^3+6*d1^3*d2*f1*f2^4+18*d1^2*d2^2*f0^3*f1^2+18*d1^2*d2^2*f0^2*f1^3+6*d1^2*d2^2*f0^2*f1^2*f2+12*d1^2*d2^2*f0*f1^3*f2+6*d1^2*d2^2*f0*f1^2*f2^2+18*d1^2*d2^2*f1^3*f2^2+18*d1^2*d2^2*f1^2*f2^3+6*d1*d2^3*f0^4*f1+2*d1*d2^3*f0^3*f1^2+2*d1*d2^3*f0^2*f1^3+6*d1*d2^3*f0*f1^4+6*d1*d2^3*f1^4*f2+2*d1*d2^3*f1^3*f2^2+2*d1*d2^3*f1^2*f2^3+6*d1*d2^3*f1*f2^4+d2^4*f0^5+2*d2^4*f1^5+d2^4*f2^5,
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6*d0^4*f0^2*f1^2*f2+12*d0^4*f0*f1^3*f2+6*d0^4*f0*f1^2*f2^2+24*d0^3*d1*f0^3*f1*f2+12*d0^3*d1*f0^2*f1^3+12*d0^3*d1*f0^2*f1^2*f2+24*d0^3*d1*f0^2*f1*f2^2+36*d0^3*d1*f0*f1^4+12*d0^3*d1*f0*f1^2*f2^2+24*d0^3*d1*f0*f1*f2^3+36*d0^3*d1*f1^4*f2+12*d0^3*d1*f1^3*f2^2+12*d0^3*d2*f0^3*f1^2+12*d0^3*d2*f0^2*f1^3+12*d0^3*d2*f0^2*f1^2*f2+24*d0^3*d2*f0*f1^3*f2+12*d0^3*d2*f0*f1^2*f2^2+12*d0^3*d2*f1^3*f2^2+12*d0^3*d2*f1^2*f2^3+18*d0^2*d1^2*f0^4*f2+48*d0^2*d1^2*f0^3*f1^2+30*d0^2*d1^2*f0^3*f2^2+48*d0^2*d1^2*f0^2*f1^3+48*d0^2*d1^2*f0^2*f1^2*f2+30*d0^2*d1^2*f0^2*f2^3+96*d0^2*d1^2*f0*f1^3*f2+48*d0^2*d1^2*f0*f1^2*f2^2+18*d0^2*d1^2*f0*f2^4+96*d0^2*d1^2*f1^5+48*d0^2*d1^2*f1^3*f2^2+48*d0^2*d1^2*f1^2*f2^3+36*d0^2*d1*d2*f0^4*f1+12*d0^2*d1*d2*f0^3*f1^2+48*d0^2*d1*d2*f0^3*f1*f2+36*d0^2*d1*d2*f0^2*f1^3+24*d0^2*d1*d2*f0^2*f1^2*f2+48*d0^2*d1*d2*f0^2*f1*f2^2+108*d0^2*d1*d2*f0*f1^4+24*d0^2*d1*d2*f0*f1^2*f2^2+48*d0^2*d1*d2*f0*f1*f2^3+108*d0^2*d1*d2*f1^4*f2+36*d0^2*d1*d2*f1^3*f2^2+12*d0^2*d1*d2*f1^2*f2^3+36*d0^2*d1*d2*f1*f2^4+24*d0^2*d2^2*f0^3*f1^2+24*d0^2*d2^2*f0^2*f1^3+12*d0^2*d2^2*f0^2*f1^2*f2+24*d0^2*d2^2*f0*f1^3*f2+12*d0^2*d2^2*f0*f1^2*f2^2+24*d0^2*d2^2*f1^3*f2^2+24*d0^2*d2^2*f1^2*f2^3+36*d0*d1^3*f0^4*f1+12*d0*d1^3*f0^3*f1^2+72*d0*d1^3*f0^3*f1*f2+48*d0*d1^3*f0^2*f1^3+36*d0*d1^3*f0^2*f1^2*f2+72*d0*d1^3*f0^2*f1*f2^2+144*d0*d1^3*f0*f1^4+36*d0*d1^3*f0*f1^2*f2^2+72*d0*d1^3*f0*f1*f2^3+144*d0*d1^3*f1^4*f2+48*d0*d1^3*f1^3*f2^2+12*d0*d1^3*f1^2*f2^3+36*d0*d1^3*f1*f2^4+24*d0*d1^2*d2*f0^5+36*d0*d1^2*d2*f0^4*f2+96*d0*d1^2*d2*f0^3*f1^2+36*d0*d1^2*d2*f0^3*f2^2+96*d0*d1^2*d2*f0^2*f1^3+96*d0*d1^2*d2*f0^2*f1^2*f2+36*d0*d1^2*d2*f0^2*f2^3+192*d0*d1^2*d2*f0*f1^3*f2+96*d0*d1^2*d2*f0*f1^2*f2^2+36*d0*d1^2*d2*f0*f2^4+192*d0*d1^2*d2*f1^5+96*d0*d1^2*d2*f1^3*f2^2+96*d0*d1^2*d2*f1^2*f2^3+24*d0*d1^2*d2*f2^5+36*d0*d1*d2^2*f0^4*f1+12*d0*d1*d2^2*f0^3*f1^2+48*d0*d1*d2^2*f0^3*f1*f2+36*d0*d1*d2^2*f0^2*f1^3+24*d0*d1*d2^2*f0^2*f1^2*f2+48*d0*d1*d2^2*f0^2*f1*f2^2+108*d0*d1*d2^2*f0*f1^4+24*d0*d1*d2^2*f0*f1^2*f2^2+48*d0*d1*d2^2*f0*f1*f2^3+108*d0*d1*d2^2*f1^4*f2+36*d0*d1*d2^2*f1^3*f2^2+12*d0*d1*d2^2*f1^2*f2^3+36*d0*d1*d2^2*f1*f2^4+12*d0*d2^3*f0^3*f1^2+12*d0*d2^3*f0^2*f1^3+12*d0*d2^3*f0^2*f1^2*f2+24*d0*d2^3*f0*f1^3*f2+12*d0*d2^3*f0*f1^2*f2^2+12*d0*d2^3*f1^3*f2^2+12*d0*d2^3*f1^2*f2^3+48*d1^4*f0^3*f1^2+48*d1^4*f0^2*f1^3+48*d1^4*f0^2*f1^2*f2+96*d1^4*f0*f1^3*f2+48*d1^4*f0*f1^2*f2^2+48*d1^4*f1^3*f2^2+48*d1^4*f1^2*f2^3+36*d1^3*d2*f0^4*f1+12*d1^3*d2*f0^3*f1^2+72*d1^3*d2*f0^3*f1*f2+48*d1^3*d2*f0^2*f1^3+36*d1^3*d2*f0^2*f1^2*f2+72*d1^3*d2*f0^2*f1*f2^2+144*d1^3*d2*f0*f1^4+36*d1^3*d2*f0*f1^2*f2^2+72*d1^3*d2*f0*f1*f2^3+144*d1^3*d2*f1^4*f2+48*d1^3*d2*f1^3*f2^2+12*d1^3*d2*f1^2*f2^3+36*d1^3*d2*f1*f2^4+18*d1^2*d2^2*f0^4*f2+48*d1^2*d2^2*f0^3*f1^2+30*d1^2*d2^2*f0^3*f2^2+48*d1^2*d2^2*f0^2*f1^3+48*d1^2*d2^2*f0^2*f1^2*f2+30*d1^2*d2^2*f0^2*f2^3+96*d1^2*d2^2*f0*f1^3*f2+48*d1^2*d2^2*f0*f1^2*f2^2+18*d1^2*d2^2*f0*f2^4+96*d1^2*d2^2*f1^5+48*d1^2*d2^2*f1^3*f2^2+48*d1^2*d2^2*f1^2*f2^3+24*d1*d2^3*f0^3*f1*f2+12*d1*d2^3*f0^2*f1^3+12*d1*d2^3*f0^2*f1^2*f2+24*d1*d2^3*f0^2*f1*f2^2+36*d1*d2^3*f0*f1^4+12*d1*d2^3*f0*f1^2*f2^2+24*d1*d2^3*f0*f1*f2^3+36*d1*d2^3*f1^4*f2+12*d1*d2^3*f1^3*f2^2+6*d2^4*f0^2*f1^2*f2+12*d2^4*f0*f1^3*f2+6*d2^4*f0*f1^2*f2^2;

// 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;
}