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Bounding the Number of Common Zeros of Multivariate Polynomials and Their Consecutive Derivatives

Part of: Finite fields and commutative rings (number-theoretic aspects) Real and complex fields Computational aspects and applications

Published online by Cambridge University Press:  03 August 2018

O. GEIL  and
U. MARTÍNEZ-PEÑAS
O. GEIL
Affiliation:
Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg, Denmark (e-mail: olav@math.aau.dk)
U. MARTÍNEZ-PEÑAS
Affiliation:
Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg, Denmark (e-mail: olav@math.aau.dk) Department of Electrical and Computer Engineering, University of Toronto, 10 King's College Road, Toronto, Ontario M5S 3G4, Canada (e-mail: umberto@comm.utoronto.ca)
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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.

MSC classification

Primary: 11T06: Polynomials
Secondary: 12D10: Polynomials 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 2 , March 2019 , pp. 253 - 279
Copyright
Copyright © Cambridge University Press 2018 

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