${\mathbb{F}}_q$-zeros of sparse trivariate polynomials and toric 3-fold codes

Abstract

For a given lattice polytope $P$ in ${\mathbb{R}}^3$, consider the space ${\mathcal{L}}_P$ of trivariate polynomials over a finite field ${\mathbb{F}}_q$, whose Newton polytopes are contained in $P$. We give an upper bound for the maximum number of ${\mathbb{F}}_q$-zeros of polynomials in ${\mathcal{L}}_P$ 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 ${\mathcal{L}}_P$ at the points of the algebraic torus $({\mathbb{F}}_q^{\ast }{)}^3$. Our approach is based on understanding factorizations of polynomials in ${\mathcal{L}}_P$ 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.

31 pages; new section on 3-fold toric codes from width one polytopes; new examples of 3-fold toric codes whose parameters exceeds the Gilbert-Varshamov bound; 2 figures