$\mathbb{F}_q$-Zeros of Sparse Trivariate Polynomials and Toric 3-Fold Codes

Abstract

For a given lattice polytope P ⊂ R, consider the space LP of trivariate polynomials over a finite field Fq, whose Newton polytopes are contained in P . We give upper bounds for the maximum number of Fq-zeros of polynomials in LP in terms of the Minkowski length of P and q, the size of the field. Consequently, this produces lower bounds for the minimum distance of toric codes defined by evaluating elements of LP at the points of the algebraic torus (Fq). Our approach is based on understanding factorizations of polynomials in LP with the largest possible number of non-unit factors. The related combinatorial result that we obtain is a description of Minkowski sums of lattice polytopes contained in P with the largest possible number of non-trivial summands.