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

Hans Olav Geil, Umberto Martinez Peñas

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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 Gröbner 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 [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.

Original languageEnglish
JournalCombinatorics, Probability & Computing
Volume28
Issue number2
Pages (from-to)253-279
Number of pages27
ISSN0963-5483
DOIs
Publication statusPublished - 1 Mar 2019

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