On multivariate polynomials with many roots over a finite grid

TL;DR

This paper investigates the maximum number of roots of multivariate polynomials over finite grids, revealing that only trivial polynomials attain the footprint bound within a large class, and identifying cases where non-trivial polynomials can also reach this maximum.

Contribution

It characterizes a large class of polynomials for which only trivial solutions attain the footprint bound, highlighting the need to explore beyond this class for maximal roots.

Findings

Footprint bound provides sharp upper limit on roots for certain polynomials.

Only trivial polynomials attain the bound within the characterized class.

Non-trivial polynomials, including irreducible ones, can also reach the bound outside this class.

Abstract

In this note we consider roots of multivariate polynomials over a finite grid. When given information on the leading monomial with respect to a fixed monomial ordering, the footprint bound [8, 5] provides us with an upper bound on the number of roots, and this bound is sharp in that it can always be attained by trivial polynomials being a constant times a product of an appropriate combination of terms consisting of a variable minus a constant. In contrast to the one variable case, there are multivariate polynomials attaining the footprint bound being not of the above form. This even includes irreducible polynomials. The purpose of the note is to determine a large class of polynomials for which only the mentioned trivial polynomials can attain the bound, implying that to search for other polynomials with the maximal number of roots one must look outside this class.