Tropical Implicitization Revisited

Authors: Kemal Rose, Bernd Sturmfels and Simon Telen

This page contains the code samples from the corresponding chapter in the OSCAR book. You can access the full chapter here.

using Pkg
Pkg.add(url = "https://github.com/kemalrose/TropicalImplicitization.jl", rev="cd4836b"; io=devnull)
using TropicalImplicitization

julia> using TropicalImplicitization, Oscar;

julia> R, (t,) = polynomial_ring(QQ,["t"]);

julia> f1 = 11*t^2 + 5*t^3 - 1*t^4;

julia> f2 = 11 + 11*t + 7*t^8;

julia> Q1 = newton_polytope(f1);

julia> Q2 = newton_polytope(f2);

julia> newton_pols = [Q1, Q2];

julia> cone_list, weight_list = get_tropical_cycle(newton_pols);

julia> Delta = get_polytope_from_cycle(cone_list, weight_list);

julia> B = lattice_points(Delta);

julia> n_samples = length(B)-1;

julia> P = sample([f1,f2], n_samples);

julia> M_BP = get_vandermonde_matrix(B,P);

julia> coeffs_F = nullspace(M_BP)[2];

julia> A = [1 1 1 1 1 1 1 1; 0 0 0 0 1 1 1 1; 
            0 0 1 1 0 0 1 1; 0 1 0 1 0 1 0 1];

julia> cone_list, weight_list = get_trop_A_disc(A);

julia> Oscar.randseed!(43);

julia> Delta = get_polytope_from_cycle(cone_list, weight_list);

julia> f_vec, lattice_pts = f_vector(Delta), lattice_points(Delta);

julia> print(f_vec);
ZZRingElem[6, 14, 16, 8]

julia> Delta = get_polytope_from_cycle(cone_list, weight_list);

julia> A = [1 1 1 1 1 1; 2 3 5 7 11 13; 13 8 5 3 2 1];

julia> cone_list, weight_list = get_trop_A_disc(A);

julia> Delta = get_polytope_from_cycle(cone_list, weight_list);

julia> @time mons, coeffs = compute_A_discriminant(A,Delta,GF(101));
Collect all lattice points.
Sample 2754.0 points from Discriminant
Construct Vandermonde matrix.
Compute coefficients of the Discriminant.
 97.247222 seconds (210.17 M allocations: 17.505 GiB, 6.51% gc time, 0.00% compilation time)

julia> verts1 = [898 -614; -570 817; 892 -594];

julia> verts2 = [-603 -481; -623 -127; -36 732];

julia> verts3 = [-548 -864; -151 873; 800 -861];

julia> pols = convex_hull.([verts1, verts2, verts3]);

julia> Delta = get_polytope_from_cycle(get_tropical_cycle(pols)...);

julia> print(f_vector(Delta))
ZZRingElem[25, 49, 26]

julia> cone_list = positive_hull.([[1, 1, 0],[1, 2, 3], [1,0,1], [-1, -1, -4//3]]);

julia> weight_list = ones(Int64, 4);

julia> cone_list, weight_list = get_chow_fan(cone_list,weight_list);

julia> C_translated = get_polytope_from_cycle(cone_list,weight_list);