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

// This is the inverse of the Fourier transform.

matrix ptoq[28][23] =
1/64,1/64,1/64,1/64,5/64,1/16,5/64,1/16,5/64,1/16,1/64,1/64,1/64,3/16,1/16,1/16,1/64,1/64,1/64,1/16,1/64,1/64,1/64,
1/32,1/32,-1/32,-1/32,5/32,-1/8,1/32,0,1/32,0,1/32,-1/32,-1/32,1/8,0,0,1/32,-1/32,-1/32,-1/8,1/32,-1/32,-1/32,
1/32,-1/32,1/32,-1/32,1/32,0,5/32,-1/8,1/32,0,-1/32,1/32,-1/32,1/8,0,-1/8,-1/32,1/32,-1/32,0,-1/32,1/32,-1/32,
1/32,-1/32,-1/32,1/32,1/32,0,1/32,0,5/32,-1/8,-1/32,-1/32,1/32,1/8,-1/8,0,-1/32,-1/32,1/32,0,-1/32,-1/32,1/32,
1/64,1/64,1/64,1/64,5/64,1/16,-3/64,-1/16,-3/64,-1/16,1/64,1/64,1/64,-1/16,-1/16,-1/16,1/64,1/64,1/64,1/16,1/64,1/64,1/64,
1/32,-1/32,-1/32,1/32,1/32,0,1/32,0,-3/32,1/8,-1/32,-1/32,1/32,-1/8,1/8,0,-1/32,-1/32,1/32,0,-1/32,-1/32,1/32,
1/32,-1/32,1/32,-1/32,1/32,0,-3/32,1/8,1/32,0,-1/32,1/32,-1/32,-1/8,0,1/8,-1/32,1/32,-1/32,0,-1/32,1/32,-1/32,
1/64,1/64,1/64,1/64,-3/64,-1/16,5/64,1/16,-3/64,-1/16,1/64,1/64,1/64,-1/16,-1/16,1/16,1/64,1/64,1/64,-1/16,1/64,1/64,1/64,
1/32,1/32,-1/32,-1/32,-3/32,1/8,1/32,0,1/32,0,1/32,-1/32,-1/32,-1/8,0,0,1/32,-1/32,-1/32,1/8,1/32,-1/32,-1/32,
1/64,1/64,1/64,1/64,-3/64,-1/16,-3/64,-1/16,5/64,1/16,1/64,1/64,1/64,-1/16,1/16,-1/16,1/64,1/64,1/64,-1/16,1/64,1/64,1/64,
1/32,1/32,1/32,1/32,5/32,1/8,-1/32,0,-1/32,0,1/32,1/32,1/32,-1/8,0,0,-1/32,-1/32,-1/32,-1/8,-1/32,-1/32,-1/32,
1/32,1/32,-1/32,-1/32,5/32,-1/8,-1/32,0,-1/32,0,1/32,-1/32,-1/32,-1/8,0,0,-1/32,1/32,1/32,1/8,-1/32,1/32,1/32,
1/16,-1/16,1/16,-1/16,1/16,0,-1/16,0,-1/16,0,-1/16,1/16,-1/16,0,0,0,1/16,-1/16,1/16,0,1/16,-1/16,1/16,
1/16,-1/16,-1/16,1/16,1/16,0,-1/16,0,-1/16,0,-1/16,-1/16,1/16,0,0,0,1/16,1/16,-1/16,0,1/16,1/16,-1/16,
1/32,1/32,1/32,1/32,-3/32,-1/8,-1/32,0,-1/32,0,1/32,1/32,1/32,1/8,0,0,-1/32,-1/32,-1/32,1/8,-1/32,-1/32,-1/32,
1/32,1/32,-1/32,-1/32,-3/32,1/8,-1/32,0,-1/32,0,1/32,-1/32,-1/32,1/8,0,0,-1/32,1/32,1/32,-1/8,-1/32,1/32,1/32,
1/32,1/32,1/32,1/32,-1/32,0,5/32,1/8,-1/32,0,-1/32,-1/32,-1/32,-1/8,0,-1/8,1/32,1/32,1/32,0,-1/32,-1/32,-1/32,
1/16,1/16,-1/16,-1/16,-1/16,0,1/16,0,-1/16,0,-1/16,1/16,1/16,0,0,0,1/16,-1/16,-1/16,0,-1/16,1/16,1/16,
1/32,-1/32,1/32,-1/32,-1/32,0,5/32,-1/8,-1/32,0,1/32,-1/32,1/32,-1/8,0,1/8,-1/32,1/32,-1/32,0,1/32,-1/32,1/32,
1/16,-1/16,-1/16,1/16,-1/16,0,1/16,0,-1/16,0,1/16,1/16,-1/16,0,0,0,-1/16,-1/16,1/16,0,1/16,1/16,-1/16,
1/32,1/32,1/32,1/32,-1/32,0,-3/32,-1/8,-1/32,0,-1/32,-1/32,-1/32,1/8,0,1/8,1/32,1/32,1/32,0,-1/32,-1/32,-1/32,
1/32,-1/32,1/32,-1/32,-1/32,0,-3/32,1/8,-1/32,0,1/32,-1/32,1/32,1/8,0,-1/8,-1/32,1/32,-1/32,0,1/32,-1/32,1/32,
1/32,1/32,1/32,1/32,-1/32,0,-1/32,0,5/32,1/8,-1/32,-1/32,-1/32,-1/8,-1/8,0,-1/32,-1/32,-1/32,0,1/32,1/32,1/32,
1/16,1/16,-1/16,-1/16,-1/16,0,-1/16,0,1/16,0,-1/16,1/16,1/16,0,0,0,-1/16,1/16,1/16,0,1/16,-1/16,-1/16,
1/16,-1/16,1/16,-1/16,-1/16,0,-1/16,0,1/16,0,1/16,-1/16,1/16,0,0,0,1/16,-1/16,1/16,0,-1/16,1/16,-1/16,
1/32,-1/32,-1/32,1/32,-1/32,0,-1/32,0,5/32,-1/8,1/32,1/32,-1/32,-1/8,1/8,0,1/32,1/32,-1/32,0,-1/32,-1/32,1/32,
1/32,1/32,1/32,1/32,-1/32,0,-1/32,0,-3/32,-1/8,-1/32,-1/32,-1/32,1/8,1/8,0,-1/32,-1/32,-1/32,0,1/32,1/32,1/32,
1/32,-1/32,-1/32,1/32,-1/32,0,-1/32,0,-3/32,1/8,1/32,1/32,-1/32,1/8,-1/8,0,1/32,1/32,-1/32,0,-1/32,-1/32,1/32;

// This is the ring of probability distributions. 

ring rP = 0,(p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,p13,p14,p15,p16,p17,p18,p19,p20,p21,p22,p23,p24,p25,p26,p27,p28),dp;

//This is the Fourier transform.

matrix qtop[23][28] =
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,-1,1,1,1,1,1,-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,-1,1,-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,-1,1,-1,-1,1,1,1,
1,1,1/5,1/5,1,1/5,1/5,-3/5,-3/5,-3/5,1,1,1/5,1/5,-3/5,-3/5,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,
1,-1,0,0,1,0,0,-1,1,-1,1,-1,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,
1,1/5,1,1/5,-3/5,1/5,-3/5,1,1/5,-3/5,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,1,1/5,1,1/5,-3/5,-3/5,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,
1,0,-1,0,-1,0,1,1,0,-1,0,0,0,0,0,0,1,0,-1,0,-1,1,0,0,0,0,0,0,
1,1/5,1/5,1,-3/5,-3/5,1/5,-3/5,1/5,1,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,-1/5,1,1/5,1/5,1,-3/5,-3/5,
1,0,0,-1,-1,1,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-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,-1,1,-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,-1,-1,1,1,-1,-1,1,1,-1,-1,1,-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,-1/3,0,0,1/3,1/3,-1/3,0,-1/3,0,1/3,1/3,-1/3,0,0,-1/3,1/3,1/3,
1,0,0,-1,-1,1,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,1,1,-1,
1,0,-1,0,-1,0,1,1,0,-1,0,0,0,0,0,0,-1,0,1,0,1,-1,0,0,0,0,0,0,
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,-1,1,1,-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,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,-1,
1,-1,0,0,1,0,0,-1,1,-1,-1,1,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,
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,-1,1,1,-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,-1,1,-1,1,1,-1,-1,1,1,-1,-1,1,1,1;


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,(c0,c1,c2,c3,e0,e1,e2,e3),dp;

ideal P = 
c0^3*e0^4+c0^3*e1^4+c0^3*e2^4+c0^3*e3^4+2*c0^2*c1*e0^3*e1+2*c0^2*c1*e0^2*e1^2+2*c0^2*c1*e0*e1^3+2*c0^2*c1*e2^3*e3+2*c0^2*c1*e2^2*e3^2+2*c0^2*c1*e2*e3^3+2*c0^2*c2*e0^3*e2+2*c0^2*c2*e0^2*e2^2+2*c0^2*c2*e0*e2^3+2*c0^2*c2*e1^3*e3+2*c0^2*c2*e1^2*e3^2+2*c0^2*c2*e1*e3^3+2*c0^2*c3*e0^3*e3+2*c0^2*c3*e0^2*e3^2+2*c0^2*c3*e0*e3^3+2*c0^2*c3*e1^3*e2+2*c0^2*c3*e1^2*e2^2+2*c0^2*c3*e1*e2^3+2*c0*c1^2*e0^3*e1+2*c0*c1^2*e0^2*e1^2+2*c0*c1^2*e0*e1^3+2*c0*c1^2*e2^3*e3+2*c0*c1^2*e2^2*e3^2+2*c0*c1^2*e2*e3^3+2*c0*c1*c2*e0^2*e1*e2+2*c0*c1*c2*e0^2*e1*e3+2*c0*c1*c2*e0^2*e2*e3+2*c0*c1*c2*e0*e1^2*e2+2*c0*c1*c2*e0*e1^2*e3+2*c0*c1*c2*e0*e1*e2^2+2*c0*c1*c2*e0*e1*e3^2+2*c0*c1*c2*e0*e2^2*e3+2*c0*c1*c2*e0*e2*e3^2+2*c0*c1*c2*e1^2*e2*e3+2*c0*c1*c2*e1*e2^2*e3+2*c0*c1*c2*e1*e2*e3^2+2*c0*c1*c3*e0^2*e1*e2+2*c0*c1*c3*e0^2*e1*e3+2*c0*c1*c3*e0^2*e2*e3+2*c0*c1*c3*e0*e1^2*e2+2*c0*c1*c3*e0*e1^2*e3+2*c0*c1*c3*e0*e1*e2^2+2*c0*c1*c3*e0*e1*e3^2+2*c0*c1*c3*e0*e2^2*e3+2*c0*c1*c3*e0*e2*e3^2+2*c0*c1*c3*e1^2*e2*e3+2*c0*c1*c3*e1*e2^2*e3+2*c0*c1*c3*e1*e2*e3^2+2*c0*c2^2*e0^3*e2+2*c0*c2^2*e0^2*e2^2+2*c0*c2^2*e0*e2^3+2*c0*c2^2*e1^3*e3+2*c0*c2^2*e1^2*e3^2+2*c0*c2^2*e1*e3^3+2*c0*c2*c3*e0^2*e1*e2+2*c0*c2*c3*e0^2*e1*e3+2*c0*c2*c3*e0^2*e2*e3+2*c0*c2*c3*e0*e1^2*e2+2*c0*c2*c3*e0*e1^2*e3+2*c0*c2*c3*e0*e1*e2^2+2*c0*c2*c3*e0*e1*e3^2+2*c0*c2*c3*e0*e2^2*e3+2*c0*c2*c3*e0*e2*e3^2+2*c0*c2*c3*e1^2*e2*e3+2*c0*c2*c3*e1*e2^2*e3+2*c0*c2*c3*e1*e2*e3^2+2*c0*c3^2*e0^3*e3+2*c0*c3^2*e0^2*e3^2+2*c0*c3^2*e0*e3^3+2*c0*c3^2*e1^3*e2+2*c0*c3^2*e1^2*e2^2+2*c0*c3^2*e1*e2^3+c1^3*e0^4+c1^3*e1^4+c1^3*e2^4+c1^3*e3^4+2*c1^2*c2*e0^3*e3+2*c1^2*c2*e0^2*e3^2+2*c1^2*c2*e0*e3^3+2*c1^2*c2*e1^3*e2+2*c1^2*c2*e1^2*e2^2+2*c1^2*c2*e1*e2^3+2*c1^2*c3*e0^3*e2+2*c1^2*c3*e0^2*e2^2+2*c1^2*c3*e0*e2^3+2*c1^2*c3*e1^3*e3+2*c1^2*c3*e1^2*e3^2+2*c1^2*c3*e1*e3^3+2*c1*c2^2*e0^3*e3+2*c1*c2^2*e0^2*e3^2+2*c1*c2^2*e0*e3^3+2*c1*c2^2*e1^3*e2+2*c1*c2^2*e1^2*e2^2+2*c1*c2^2*e1*e2^3+2*c1*c2*c3*e0^2*e1*e2+2*c1*c2*c3*e0^2*e1*e3+2*c1*c2*c3*e0^2*e2*e3+2*c1*c2*c3*e0*e1^2*e2+2*c1*c2*c3*e0*e1^2*e3+2*c1*c2*c3*e0*e1*e2^2+2*c1*c2*c3*e0*e1*e3^2+2*c1*c2*c3*e0*e2^2*e3+2*c1*c2*c3*e0*e2*e3^2+2*c1*c2*c3*e1^2*e2*e3+2*c1*c2*c3*e1*e2^2*e3+2*c1*c2*c3*e1*e2*e3^2+2*c1*c3^2*e0^3*e2+2*c1*c3^2*e0^2*e2^2+2*c1*c3^2*e0*e2^3+2*c1*c3^2*e1^3*e3+2*c1*c3^2*e1^2*e3^2+2*c1*c3^2*e1*e3^3+c2^3*e0^4+c2^3*e1^4+c2^3*e2^4+c2^3*e3^4+2*c2^2*c3*e0^3*e1+2*c2^2*c3*e0^2*e1^2+2*c2^2*c3*e0*e1^3+2*c2^2*c3*e2^3*e3+2*c2^2*c3*e2^2*e3^2+2*c2^2*c3*e2*e3^3+2*c2*c3^2*e0^3*e1+2*c2*c3^2*e0^2*e1^2+2*c2*c3^2*e0*e1^3+2*c2*c3^2*e2^3*e3+2*c2*c3^2*e2^2*e3^2+2*c2*c3^2*e2*e3^3+c3^3*e0^4+c3^3*e1^4+c3^3*e2^4+c3^3*e3^4,
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8*c0^3*e0*e1*e2*e3+4*c0^2*c1*e0^2*e2*e3+4*c0^2*c1*e0^2*e3^2+4*c0^2*c1*e0*e1*e2^2+4*c0^2*c1*e0*e1*e3^2+4*c0^2*c1*e1^2*e2^2+4*c0^2*c1*e1^2*e2*e3+4*c0^2*c2*e0^2*e1*e3+4*c0^2*c2*e0^2*e3^2+4*c0^2*c2*e0*e1^2*e2+4*c0^2*c2*e0*e2*e3^2+4*c0^2*c2*e1^2*e2^2+4*c0^2*c2*e1*e2^2*e3+4*c0^2*c3*e0^2*e1*e2+4*c0^2*c3*e0*e1^2*e3+8*c0^2*c3*e0*e1*e2*e3+4*c0^2*c3*e0*e2^2*e3+4*c0^2*c3*e1*e2*e3^2+4*c0*c1^2*e0^2*e2*e3+4*c0*c1^2*e0^2*e3^2+4*c0*c1^2*e0*e1*e2^2+4*c0*c1^2*e0*e1*e3^2+4*c0*c1^2*e1^2*e2^2+4*c0*c1^2*e1^2*e2*e3+4*c0*c1*c2*e0^3*e3+4*c0*c1*c2*e0^2*e1*e3+4*c0*c1*c2*e0^2*e2*e3+4*c0*c1*c2*e0*e1^2*e2+4*c0*c1*c2*e0*e1*e2^2+4*c0*c1*c2*e0*e1*e3^2+4*c0*c1*c2*e0*e2*e3^2+4*c0*c1*c2*e0*e3^3+4*c0*c1*c2*e1^3*e2+4*c0*c1*c2*e1^2*e2*e3+4*c0*c1*c2*e1*e2^3+4*c0*c1*c2*e1*e2^2*e3+4*c0*c1*c3*e0^3*e3+4*c0*c1*c3*e0^2*e1*e3+4*c0*c1*c3*e0^2*e2*e3+4*c0*c1*c3*e0*e1^2*e2+4*c0*c1*c3*e0*e1*e2^2+4*c0*c1*c3*e0*e1*e3^2+4*c0*c1*c3*e0*e2*e3^2+4*c0*c1*c3*e0*e3^3+4*c0*c1*c3*e1^3*e2+4*c0*c1*c3*e1^2*e2*e3+4*c0*c1*c3*e1*e2^3+4*c0*c1*c3*e1*e2^2*e3+4*c0*c2^2*e0^2*e1*e3+4*c0*c2^2*e0^2*e3^2+4*c0*c2^2*e0*e1^2*e2+4*c0*c2^2*e0*e2*e3^2+4*c0*c2^2*e1^2*e2^2+4*c0*c2^2*e1*e2^2*e3+4*c0*c2*c3*e0^3*e3+4*c0*c2*c3*e0^2*e1*e3+4*c0*c2*c3*e0^2*e2*e3+4*c0*c2*c3*e0*e1^2*e2+4*c0*c2*c3*e0*e1*e2^2+4*c0*c2*c3*e0*e1*e3^2+4*c0*c2*c3*e0*e2*e3^2+4*c0*c2*c3*e0*e3^3+4*c0*c2*c3*e1^3*e2+4*c0*c2*c3*e1^2*e2*e3+4*c0*c2*c3*e1*e2^3+4*c0*c2*c3*e1*e2^2*e3+4*c0*c3^2*e0^2*e1*e2+4*c0*c3^2*e0*e1^2*e3+8*c0*c3^2*e0*e1*e2*e3+4*c0*c3^2*e0*e2^2*e3+4*c0*c3^2*e1*e2*e3^2+8*c1^3*e0*e1*e2*e3+4*c1^2*c2*e0^2*e1*e2+4*c1^2*c2*e0*e1^2*e3+8*c1^2*c2*e0*e1*e2*e3+4*c1^2*c2*e0*e2^2*e3+4*c1^2*c2*e1*e2*e3^2+4*c1^2*c3*e0^2*e1*e3+4*c1^2*c3*e0^2*e3^2+4*c1^2*c3*e0*e1^2*e2+4*c1^2*c3*e0*e2*e3^2+4*c1^2*c3*e1^2*e2^2+4*c1^2*c3*e1*e2^2*e3+4*c1*c2^2*e0^2*e1*e2+4*c1*c2^2*e0*e1^2*e3+8*c1*c2^2*e0*e1*e2*e3+4*c1*c2^2*e0*e2^2*e3+4*c1*c2^2*e1*e2*e3^2+4*c1*c2*c3*e0^3*e3+4*c1*c2*c3*e0^2*e1*e3+4*c1*c2*c3*e0^2*e2*e3+4*c1*c2*c3*e0*e1^2*e2+4*c1*c2*c3*e0*e1*e2^2+4*c1*c2*c3*e0*e1*e3^2+4*c1*c2*c3*e0*e2*e3^2+4*c1*c2*c3*e0*e3^3+4*c1*c2*c3*e1^3*e2+4*c1*c2*c3*e1^2*e2*e3+4*c1*c2*c3*e1*e2^3+4*c1*c2*c3*e1*e2^2*e3+4*c1*c3^2*e0^2*e1*e3+4*c1*c3^2*e0^2*e3^2+4*c1*c3^2*e0*e1^2*e2+4*c1*c3^2*e0*e2*e3^2+4*c1*c3^2*e1^2*e2^2+4*c1*c3^2*e1*e2^2*e3+8*c2^3*e0*e1*e2*e3+4*c2^2*c3*e0^2*e2*e3+4*c2^2*c3*e0^2*e3^2+4*c2^2*c3*e0*e1*e2^2+4*c2^2*c3*e0*e1*e3^2+4*c2^2*c3*e1^2*e2^2+4*c2^2*c3*e1^2*e2*e3+4*c2*c3^2*e0^2*e2*e3+4*c2*c3^2*e0^2*e3^2+4*c2*c3^2*e0*e1*e2^2+4*c2*c3^2*e0*e1*e3^2+4*c2*c3^2*e1^2*e2^2+4*c2*c3^2*e1^2*e2*e3+8*c3^3*e0*e1*e2*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;
}