Bounding the number of common zeros of multivariate polynomials and their consecutive derivatives

TL;DR

This paper develops bounds on the number of common zeros of multivariate polynomials and their derivatives over finite grids, extending classical results using Gröbner basis tools and exploring their implications in algebra and combinatorics.

Contribution

It introduces new bounds on zeros of polynomials and derivatives, extending classical theorems with Gröbner basis methods and weighted multiplicity considerations.

Findings

Extended the Schwartz-Zippel bound to weighted multiplicities.

Established bounds on zeros of polynomials with derivatives using footprint techniques.

Connected new bounds with classical combinatorial and algebraic results.

Abstract

We upper bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gr\"obner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz, existence and uniqueness of Hermite interpolating polynomials over a grid, estimations on the parameters of evaluation codes with consecutive derivatives, and bounds on the number of zeros of a polynomial by DeMillo and Lipton, Schwartz, Zippel, and Alon and F\"uredi. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection with our extension of the footprint bound.