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

TL;DR

This paper establishes bounds on the number of finite field zeros of sparse trivariate polynomials based on lattice polytope geometry, leading to insights into the minimum distance of associated toric codes.

Contribution

It introduces a new bound on the zeros of polynomials in lattice polytope spaces and relates this to the minimum distance of toric codes, using Minkowski length and polytope factorizations.

Findings

Upper bounds for zeros in terms of Minkowski length and field size

Lower bounds for toric code minimum distance

Characterization of Minkowski sums with maximal non-trivial summands

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 $(\mathbb{F}_q^*)^3$. 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.