On zeros of a polynomial in a finite grid

TL;DR

This paper extends the Alon-F"uredi theorem to provide sharp bounds on the zeros of multivariate polynomials over finite grids, incorporating degrees in each variable, and explores applications in coding theory and finite geometry.

Contribution

It generalizes the Alon-F"uredi theorem to include variable degrees, offers a new proof, and connects the results to coding theory and finite geometry applications.

Findings

Provides a sharp upper bound considering degrees in each variable.

Establishes a coding theoretic interpretation related to Reed--Muller codes.

Recovers and strengthens bounds in finite geometry, such as affine blocking sets.

Abstract

A 1993 result of Alon and F\"uredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain "Condition (D)" on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further Generalized Alon-F\"uredi Theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon-F\"uredi. We then discuss the relationship between Alon-F\"uredi and results of DeMillo-Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon-F\"uredi Theorem and its generalization in terms of Reed--Muller type affine variety codes is shown which gives us the minimum Hamming distance of these codes. Then we apply the Alon-F\"uredi Theorem to quickly recover (and sometimes strengthen) old and new results in finite geometry, including the Jamison/Brouwer-Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.