Resultant polytope f-vectors in four and five dimensions
Profili, Daniel A.
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For a system of polynomials with A = (A_1, ..., A_k) as supports, the Newton polytope of the resultant, or resultant polytope, is the convex hull of the resultant monomial exponent vectors in Z^n and encodes certain combinatorial properties of the resultant polynomial. Using tropical hypersurface fan traversals, we investigate the f-vectors (vectors of face cardinalities) of resultant polytopes in four and five dimensions. Using the software Gfan to perform tropicalization calculations, our experiments support the currently conjectured maximal f-vector (22, 66, 66, 22) for the 4-dimensional case after sampling 200,000 random point configurations with coordinates in the [0, 10] range. For the 5-dimensional case, we sample over 160,000 resultant polytopes and offer an experimental lower bound for the maximal f-vector of (58, 232, 330, 201, 47). Finally, we present some ideas for further computational and theoretical approaches to f-vector characterization using tropical geometry.