Refining Multivariate Value Set Bounds

TL;DR

This paper introduces new geometric methods using Newton polytopes to derive tighter bounds on the size of polynomial value sets over finite fields, improving upon previous degree-based bounds.

Contribution

It develops a geometric approach with Newton polytopes for sharper bounds and proposes a method that surpasses existing degree and polytope-based bounds.

Findings

Tighter bounds on value set cardinality using Newton polytope properties

A new method providing stronger bounds independent of degree or polytope considerations

An alternative proof of Kosters' degree bound and improvements over existing bounds

Abstract

Over finite fields, if the image of a polynomial map is not the entire field, then its cardinality can be bounded above by a significantly smaller value. Earlier results bound the cardinality of the value set using the degree of the polynomial, but more recent results make use of the powers of all monomials. In this paper, we explore the geometric properties of the Newton polytope and show how they allow for tighter upper bounds on the cardinality of the multivariate value set. We then explore a method which allows for even stronger upper bounds, regardless of whether one uses the multivariate degree or the Newton polytope to bound the value set. Effectively, this provides an alternate proof of Kosters' degree bound, an improved Newton polytope-based bound, and an improvement of a degree matrix-based result given by Zan and Cao.