More results on the number of zeros of multiplicity at least r

TL;DR

This paper extends bounds on the number of zeros with multiplicity at least r for multivariate polynomials over finite sets, providing new closed-form formulas for multiple variables under certain conditions, generalizing previous bounds.

Contribution

It derives new closed-form bounds for zeros of multivariate polynomials with multiplicity at least r, generalizing previous results to multiple variables with specific leading monomial conditions.

Findings

Provides closed-form formulas for zeros in multiple variables

Generalizes the footprint bound to account for multiplicity

Improves estimates over previous recursive bounds

Abstract

We consider multivariate polynomials and investigate how many zeros of multiplicity at least $r$ they can have over a Cartesian product of finite subsets of a field. Here r is any prescribed positive integer and the definition of multiplicity that we use is the one related to Hasse derivatives. As a generalization of material in [2,5] a general version of the Schwartz-Zippel was presented in [8] which from the leading monomial -- with respect to a lexicographic ordering -- estimates the sum of zeros when counted with multiplicity. The corresponding corollary on the number of zeros of multiplicity at least $r$ is in general not sharp and therefore in [8] a recursively defined function $D$ was introduced using which one can derive improved information. The recursive function being rather complicated, the only known closed formula consequences of it are for the case of two variables [8]. In the present paper we derive closed formula consequences for arbitrary many variables, but for the powers in the leading monomial being not too large. Our bound can be viewed as a generalization of the footprint bound [10,6] -- the classical footprint bound taking not multiplicity into account.