On the number of zeros of multiplicity r

TL;DR

This paper refines bounds on the number of zeros of multivariate polynomials with a given multiplicity over finite sets by incorporating the leading monomial, enabling more precise estimates than previous bounds.

Contribution

It introduces a method that considers the leading monomial to improve bounds on zeros of multiplicity r, extending the applicability to more general point ensembles.

Findings

Provides tighter upper bounds on zeros with multiplicity r.

Offers lower bounds complementing the upper bounds.

Enhances understanding of polynomial zeros over finite sets.

Abstract

Let S be a finite subset of a field. For multivariate polynomials the generalized Schwartz-Zippel bound [2], [4] estimates the number of zeros over Sx...xS counted with multiplicity. It does this in terms of the total degree, the number of variables and |S|. In the present work we take into account what is the leading monomial. This allows us to consider more general point ensembles and most importantly it allows us to produce much more detailed information about the number of zeros of multiplicity r than can be deduced from the generalized Schwartz-Zippel bound. We present both upper and lower bounds.