restart; with(LinearAlgebra): # Define indeterminates QParam := [q1,q2,q3,q4,q5,q6,q7,q8,q9,q10,q11,q12,q13,q14,q15,q16]: PParam := Matrix([[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]]): # Special Fourier parameterization and inverse F := Matrix([[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [1,-1/3,1,-1/3,1,-1/3,-1/3,1,-1/3,1,-1/3,1,-1/3,-1/3,1,-1/3,1,-1/3,1,-1/3,-1/3,-1/3,1,-1/3,1], [1,1/2,0,0,1/3,1/6,0,0,-1/6,-1/3,-1/3,1/2,1/3,1/6,1/6,0,-1/6,-1/6,0,0,0,-1/6,0,-1/6,-1/3], [1,-1/3,-1/3,1/3,5/9,-1/3,-1/9,1/9,1/9,-1/3,1/3,5/9,-1/3,-1/9,1/9,1/9,-1/3,1/3,1/9,-1/9,-1/9,1/9,1/9,1/9,-1/3], [1,-1/3,1,-1/3,-1/3,1/9,1/9,-1/3,1/9,-1/3,1/9,1/3,-1/9,-1/9,1/3,-1/9,1/3,-1/9,-1/3,1/9,1/9,1/9,-1/3,1/9,-1/3], [1,1/3,-1/3,-1/3,-1/3,-1/3,0,-1/3,0,1/3,1/3,1/3,0,1/6,-1/3,-1/6,0,0,0,0,-1/6,1/6,-1/3,0,1/3], [1,-1/3,-1/3,1/3,-1/3,1/3,0,-1/3,0,1/3,-1/3,1/3,0,-1/6,-1/3,1/6,0,0,0,0,1/6,-1/6,-1/3,0,1/3], [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,-1/3,-1/3,-1/3], [1,-1/3,1,-1/3,1,-1/3,-1/3,1,-1/3,1,-1/3,-1/3,1/9,1/9,-1/3,1/9,-1/3,1/9,-1/3,1/9,1/9,1/9,-1/3,1/9,-1/3], [1,1/3,-1/3,-1/3,-1/3,1/3,-1/3,1,1/3,-1/3,-1/3,-1/3,-1/9,-1/9,1/9,1/9,1/9,1/9,1/9,1/9,-1/9,1/9,-1/3,-1/9,1/9], [1,-1/3,-1/3,1/3,-1/3,-1/3,1/3,1,-1/3,-1/3,1/3,-1/3,1/9,1/9,1/9,-1/9,1/9,-1/9,1/9,-1/9,1/9,-1/9,-1/3,1/9,1/9], [1,5/9,1/9,1/9,5/9,1/9,1/9,-1/3,-1/3,-1/3,-1/3,-1/3,-1/3,-1/9,-1/3,-1/9,1/9,1/9,-1/9,-1/9,1/9,1/9,1/3,1/3,1/3], [1,-1/3,-1/3,1/3,1,-1/3,-1/3,-1/3,1/3,-1/3,1/3,-1/3,1/3,0,-1/3,0,1/3,-1/3,-1/3,1/3,0,0,1/3,-1/3,1/3], [1,1/3,-1/3,-1/3,-1/3,-1/3,0,-1/3,0,1/3,1/3,-1/3,0,-1/6,1/3,1/6,0,0,0,0,1/6,-1/6,1/3,0,-1/3], [1,-1/3,-1/3,1/3,-1/3,1/3,0,-1/3,0,1/3,-1/3,-1/3,0,1/6,1/3,-1/6,0,0,0,0,-1/6,1/6,1/3,0,-1/3], [1,-1/3,1,-1/3,-1/3,1/9,1/9,-1/3,1/9,-1/3,1/9,-1/3,1/9,1/9,-1/3,1/9,-1/3,1/9,1/3,-1/9,-1/9,-1/9,1/3,-1/9,1/3]]): FI := Matrix([ [1/256,3/256,3/32,9/128,9/128,3/32,3/32,3/256,9/256,9/128,9/128,9/128,3/64,3/32,3/32,9/128], [3/128,-3/128,9/32,-9/64,-9/64,3/16,-3/16,9/128,-9/128,9/64,-9/64,15/64,-3/32,3/16,-3/16,-9/64], [3/256,9/256,0,-9/128,27/128,-3/32,-3/32,9/256,27/256,-9/128,-9/128,3/128,-3/64,-3/32,-3/32,27/128], [3/128,-3/128,0,9/64,-9/64,-3/16,3/16,9/128,-9/128,-9/64,9/64,3/64,3/32,-3/16,3/16,-9/64], [3/256,9/256,3/32,15/128,-9/128,-3/32,-3/32,9/256,27/256,-9/128,-9/128,15/128,9/64,-3/32,-3/32,-9/128], [3/128,-3/128,3/32,-9/64,3/64,-3/16,3/16,9/128,-9/128,9/64,-9/64,3/64,-3/32,-3/16,3/16,3/64], [3/64,-3/64,0,-3/32,3/32,0,0,9/64,-9/64,-9/32,9/32,3/32,-3/16,0,0,3/32], [3/256,9/256,0,3/128,-9/128,-3/32,-3/32,9/256,27/256,27/128,27/128,-9/128,-3/64,-3/32,-3/32,-9/128], [3/64,-3/64,-3/16,3/32,3/32,0,0,9/64,-9/64,9/32,-9/32,-9/32,3/16,0,0,3/32], [3/128,9/128,-3/16,-9/64,-9/64,3/16,3/16,9/128,27/128,-9/64,-9/64,-9/64,-3/32,3/16,3/16,-9/64], [3/128,-3/128,-3/16,9/64,3/64,3/16,-3/16,9/128,-9/128,-9/64,9/64,-9/64,3/32,3/16,-3/16,3/64], [3/128,9/128,9/32,15/64,9/64,3/16,3/16,-3/128,-9/128,-9/64,-9/64,-9/64,-3/32,-3/16,-3/16,-9/64], [3/64,-3/64,3/8,-9/32,-3/32,0,0,-3/64,3/64,-3/32,3/32,-9/32,3/16,0,0,3/32], [3/32,-3/32,3/8,-3/16,-3/16,3/8,-3/8,-3/32,3/32,-3/16,3/16,-3/16,0,-3/8,3/8,3/16], [3/128,9/128,3/32,3/64,9/64,-3/16,-3/16,-3/128,-9/128,3/64,3/64,-9/64,-3/32,3/16,3/16,-9/64], [3/32,-3/32,0,3/16,-3/16,-3/8,3/8,-3/32,3/32,3/16,-3/16,-3/16,0,3/8,-3/8,3/16], [3/64,9/64,-3/16,-9/32,9/32,0,0,-3/64,-9/64,3/32,3/32,3/32,3/16,0,0,-9/32], [3/64,-3/64,-3/16,9/32,-3/32,0,0,-3/64,3/64,3/32,-3/32,3/32,-3/16,0,0,3/32], [3/64,9/64,0,3/32,-9/32,0,0,-3/64,-9/64,3/32,3/32,-3/32,-3/16,0,0,9/32], [3/64,-3/64,0,-3/32,3/32,0,0,-3/64,3/64,3/32,-3/32,-3/32,3/16,0,0,-3/32], [3/32,-3/32,0,-3/16,3/16,-3/8,3/8,-3/32,3/32,-3/16,3/16,3/16,0,3/8,-3/8,-3/16], [3/32,-3/32,-3/8,3/16,3/16,3/8,-3/8,-3/32,3/32,3/16,-3/16,3/16,0,-3/8,3/8,-3/16], [3/128,9/128,0,3/64,-9/64,-3/16,-3/16,-3/128,-9/128,-9/64,-9/64,9/64,3/32,3/16,3/16,9/64], [3/64,-3/64,-3/16,3/32,3/32,0,0,-3/64,3/64,-3/32,3/32,9/32,-3/16,0,0,-3/32], [3/128,9/128,-3/16,-9/64,-9/64,3/16,3/16,-3/128,-9/128,3/64,3/64,9/64,3/32,-3/16,-3/16,9/64] ]): # List of polynomial parametrizations P0 := [ 1/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^5*g0^4+3/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^5*g1^4-3/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^4*e1*g0^3*g1-3/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^4*e1*g0*g1^3-6/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^4*e1*g1^4-3/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^3*e1^2*g0^4+9/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^3*e1^2*g0^3*g1+9/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^3*e1^2*g0*g1^3+9/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^3*e1^2*g1^4-1/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^2*e1^3*g0^4-9/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^2*e1^3*g0^3*g1-9/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^2*e1^3*g0*g1^3-21/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^2*e1^3*g1^4+6/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0*e1^4*g0^4+3/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0*e1^4*g0^3*g1+3/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0*e1^4*g0*g1^3+24/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0*e1^4*g1^4-3/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e1^5*g0^4-9/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e1^5*g1^4+6/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^5*g0^3*g1+6/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^5*g0^2*g1^2+6/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^5*g0*g1^3+18/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^5*g1^4-3/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^4*e1*g0^4-9/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^4*e1*g0^3*g1-18/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^4*e1*g0^2*g1^2-33/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^4*e1*g0*g1^3-45/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^4*e1*g1^4+9/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^3*e1^2*g0^4+9/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^3*e1^2*g0^3*g1+36/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^3*e1^2*g0^2*g1^2+81/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^3*e1^2*g0*g1^3+81/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^3*e1^2*g1^4-9/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^2*e1^3*g0^4-33/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^2*e1^3*g0^3*g1-60/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^2*e1^3*g0^2*g1^2-105/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^2*e1^3*g0*g1^3-153/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^2*e1^3*g1^4+3/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0*e1^4*g0^4+45/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0*e1^4*g0^3*g1+54/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0*e1^4*g0^2*g1^2+69/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0*e1^4*g0*g1^3+153/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0*e1^4*g1^4-18/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e1^5*g0^3*g1-18/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e1^5*g0^2*g1^2-18/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e1^5*g0*g1^3-54/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e1^5*g1^4+6/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^5*g0^3*g1+24/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^5*g0^2*g1^2+42/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^5*g0*g1^3+36/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^5*g1^4-3/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^4*e1*g0^4-27/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^4*e1*g0^3*g1-72/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^4*e1*g0^2*g1^2-87/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^4*e1*g0*g1^3-135/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^4*e1*g1^4+9/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^3*e1^2*g0^4+63/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^3*e1^2*g0^3*g1+144/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^3*e1^2*g0^2*g1^2+135/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^3*e1^2*g0*g1^3+297/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^3*e1^2*g1^4-9/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^2*e1^3*g0^4-87/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^2*e1^3*g0^3*g1-240/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^2*e1^3*g0^2*g1^2-303/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^2*e1^3*g0*g1^3-441/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^2*e1^3*g1^4+3/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0*e1^4*g0^4+63/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0*e1^4*g0^3*g1+216/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0*e1^4*g0^2*g1^2+339/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0*e1^4*g0*g1^3+351/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0*e1^4*g1^4-18/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e1^5*g0^3*g1-72/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e1^5*g0^2*g1^2-126/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e1^5*g0*g1^3-108/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e1^5*g1^4+3/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^5*g0^4+12/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^5*g0^3*g1+18/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^5*g0^2*g1^2+24/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^5*g0*g1^3+51/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^5*g1^4-6/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^4*e1*g0^4-33/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^4*e1*g0^3*g1-54/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^4*e1*g0^2*g1^2-93/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^4*e1*g0*g1^3-138/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^4*e1*g1^4+9/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^3*e1^2*g0^4+63/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^3*e1^2*g0^3*g1+108/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^3*e1^2*g0^2*g1^2+207/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^3*e1^2*g0*g1^3+261/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^3*e1^2*g1^4-21/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^2*e1^3*g0^4-111/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^2*e1^3*g0^3*g1-180/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^2*e1^3*g0^2*g1^2-303/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^2*e1^3*g0*g1^3-465/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^2*e1^3*g1^4+24/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0*e1^4*g0^4+105/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0*e1^4*g0^3*g1+162/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0*e1^4*g0^2*g1^2+237/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0*e1^4*g0*g1^3+444/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0*e1^4*g1^4-9/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e1^5*g0^4-36/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e1^5*g0^3*g1-54/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e1^5*g0^2*g1^2-72/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e1^5*g0*g1^3-153/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e1^5*g1^4, 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12/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^4*e1*g0^2*g1^2+24/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^4*e1*g0*g1^3+12/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^4*e1*g1^4-6/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^3*e1^2*g0^3*g1-24/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^3*e1^2*g0^2*g1^2-30/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^3*e1^2*g0*g1^3-36/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^3*e1^2*g1^4+18/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^2*e1^3*g0^3*g1-54/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^2*e1^3*g0*g1^3+36/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0^2*e1^3*g1^4-18/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0*e1^4*g0^3*g1+24/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0*e1^4*g0^2*g1^2+102/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0*e1^4*g0*g1^3-12/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e0*e1^4*g1^4+6/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e1^5*g0^3*g1-12/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e1^5*g0^2*g1^2-42/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^3*e1^5*g0*g1^3+48/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^4*e1*g0^3*g1+84/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^4*e1*g0^2*g1^2+120/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^4*e1*g0*g1^3+180/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^4*e1*g1^4-12/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^3*e1^2*g0^4-78/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^3*e1^2*g0^3*g1-168/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^3*e1^2*g0^2*g1^2-246/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^3*e1^2*g0*g1^3-360/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^3*e1^2*g1^4+36/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^2*e1^3*g0^4-54/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^2*e1^3*g0^3*g1+18/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0^2*e1^3*g0*g1^3-36/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0*e1^4*g0^4+150/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0*e1^4*g0^3*g1+168/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0*e1^4*g0^2*g1^2+222/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0*e1^4*g0*g1^3+360/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e0*e1^4*g1^4+12/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e1^5*g0^4-66/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e1^5*g0^3*g1-84/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e1^5*g0^2*g1^2-114/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e1^5*g0*g1^3-180/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0^2*c1*e1^5*g1^4+36/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^4*e1*g0^4+120/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^4*e1*g0^3*g1+228/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^4*e1*g0^2*g1^2+336/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^4*e1*g0*g1^3+576/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^4*e1*g1^4-48/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^3*e1^2*g0^4-258/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^3*e1^2*g0^3*g1-456/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^3*e1^2*g0^2*g1^2-714/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^3*e1^2*g0*g1^3-1116/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^3*e1^2*g1^4-72/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^2*e1^3*g0^4+54/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^2*e1^3*g0^3*g1+126/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^2*e1^3*g0*g1^3-108/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0^2*e1^3*g1^4+144/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0*e1^4*g0^4+186/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0*e1^4*g0^3*g1+456/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0*e1^4*g0^2*g1^2+546/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0*e1^4*g0*g1^3+1260/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e0*e1^4*g1^4-60/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e1^5*g0^4-102/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e1^5*g0^3*g1-228/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e1^5*g0^2*g1^2-294/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e1^5*g0*g1^3-612/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c0*c1^2*e1^5*g1^4+12/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^4*e1*g0^4+120/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^4*e1*g0^3*g1+252/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^4*e1*g0^2*g1^2+384/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^4*e1*g0*g1^3+528/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^4*e1*g1^4-36/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^3*e1^2*g0^4-234/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^3*e1^2*g0^3*g1-504/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^3*e1^2*g0^2*g1^2-738/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^3*e1^2*g0*g1^3-1080/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^3*e1^2*g1^4+36/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^2*e1^3*g0^4-18/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^2*e1^3*g0^3*g1-90/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^2*e1^3*g0*g1^3+72/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0^2*e1^3*g1^4-12/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0*e1^4*g0^4+258/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0*e1^4*g0^3*g1+504/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0*e1^4*g0^2*g1^2+858/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0*e1^4*g0*g1^3+984/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e0*e1^4*g1^4-126/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e1^5*g0^3*g1-252/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e1^5*g0^2*g1^2-414/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e1^5*g0*g1^3-504/(e0^4-6*e0^2*e1^2+8*e0*e1^3-3*e1^4)*c1^3*e1^5*g1^4]: # Substitutions based on the model P := P0: P := subs(c0 = 1-3*c1, P): P := subs(e0 = 1-3*e1, P): P := subs(g0 = 1-3*g1, P): # Check that the polynomial parametrization lies in the probability simplex suma := 0: for i from 1 to nops(P) do suma := suma + P[i]: od: normal(expand(suma)); # Ideal of Invariants in Fourier coordinates Invariants := Matrix([ q13*q15-q14*q16, q5*q15-q7*q16, q4*q15-q6*q16, q13*q14-q12*q15, q11*q14-q10*q15, q7*q14-q6*q15, q5*q14-q6*q16, q4*q14-q3*q15, q13^2-q12*q16, q11*q13-q10*q16, q7*q13-q6*q16, q6*q13-q3*q15, q5*q13-q4*q16, q4*q13-q3*q16, q11*q12-q10*q13, q9*q12-q8*q16, q7*q12-q3*q15, q6*q12-q3*q14, q5*q12-q3*q16, q4*q12-q3*q13, q2*q12-q1*q16, q7*q11-q14*q16, q6*q11-q12*q15, q5*q11-q13*q16, q4*q11-q12*q16, q3*q11-q12*q13, q7*q10-q12*q15, q6*q10-q12*q14, q5*q10-q12*q16, q4*q10-q12*q13, q3*q10-q12^2, q4*q9-q2*q11, q3*q9-q2*q10, q5*q8-q2*q10, q4*q8-q1*q11, q3*q8-q1*q10, q2*q8-q1*q9, q5*q6-q4*q7, q4*q6-q3*q7, q4^2-q3*q5, q2*q3-q1*q5, q9*q15^2-q10*q16^2, q8*q15^2-q10*q12*q16, q2*q15^2-q3*q16^2, q1*q15^2-q3*q12*q16, q9*q14*q15-q10*q13*q16, q2*q14*q15-q3*q13*q16, q7*q9*q15-q12*q16^2, q6*q9*q15-q12*q13*q16, q7*q8*q15-q12^2*q16, q2*q7*q15-q3*q5*q16, q1*q7*q15-q3^2*q16, q2*q6*q15-q3*q4*q16, q9*q14^2-q10*q12*q16, q2*q14^2-q3*q12*q16, q9*q10*q14-q8*q11*q15, q6*q9*q14-q12^2*q16, q2*q6*q14-q3^2*q16, q12^2*q13-q6*q8*q15, q10*q12*q13-q8*q14*q15, q3*q12*q13-q1*q14*q15, q3^2*q13-q1*q6*q15, q12^3-q6*q8*q14, q10*q12^2-q8*q14^2, q3*q12^2-q1*q14^2, q3^2*q12-q1*q6*q14, q9*q10^2-q8*q11^2, q7^2*q9-q3*q16^2, q6*q7*q9-q3*q13*q16, q5*q7*q9-q2*q15*q16, q6^2*q9-q3*q12*q16, q5^2*q9-q2*q16^2, q7^2*q8-q3*q12*q16, q6*q7*q8-q1*q14*q15, q6^2*q8-q1*q14^2, q2*q6^2-q1*q7^2, q3*q5^2-q2*q7^2, q3*q4*q5-q2*q6*q7, q3^2*q5-q1*q7^2, q3^2*q4-q1*q6*q7, q3^3-q1*q6^2]): # Ideal of Invariants in probability coordinates Fourier := MatrixMatrixMultiply(F,PParam): PInvariants := Invariants: for i from 1 to nops(QParam) do PInvariants := subs(QParam[i] = Fourier[i, 1], PInvariants): od: # Evaluation of Invariants at the polynomial/rational parametrization num := op(PInvariants[1,1..-1])[1]: for j from 1 to num do coordpoly := PInvariants[1, j]: for i from 1 to op(PParam[1..-1,1])[1] do coordpoly := subs(PParam[i, 1] = P0[i], coordpoly): od: coordpoly :=expand(coordpoly): lprint(j,coordpoly); od: