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On Zeros of a Polynomial in a Finite Grid

Part of: Finite geometry and special incidence structures Algebraic combinatorics Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  15 February 2018

ANURAG BISHNOI ,
PETE L. CLARK ,
ADITYA POTUKUCHI  and
JOHN R. SCHMITT
ANURAG BISHNOI
Affiliation:
Freie Universität Berlin, Institut für Mathematik, Arnimallee 3, 14195 Berlin, Germany. (e-mail: anurag.2357@gmail.com)
PETE L. CLARK
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30605, USA (e-mail: plclark@gmail.com)
ADITYA POTUKUCHI
Affiliation:
Department of Computer Science, Rutgers University, Piscataway, NJ 08854, USA (e-mail: apotu.57@gmail.com)
JOHN R. SCHMITT
Affiliation:
Department of Mathematics, Middlebury College, Middlebury, VT 05753, USA (e-mail: jschmitt@middlebury.edu)
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Abstract

A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain ‘Condition (D)’ on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further generalized Alon–Füredi theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon–Füredi. We then discuss the relationship between Alon–Füredi and results of DeMillo–Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon–Füredi theorem and its generalization in terms of Reed–Muller-type affine variety codes is shown, which gives us the minimum Hamming distance of these codes. Then we apply the Alon–Füredi theorem to quickly recover – and sometimes strengthen – old and new results in finite geometry, including the Jamison–Brouwer–Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.

MSC classification

Primary: 05E40: Combinatorial aspects of commutative algebra
Secondary: 11T06: Polynomials 11T71: Algebraic coding theory; cryptography 51E20: Combinatorial structures in finite projective spaces 51E21: Blocking sets, ovals, $k$-arcs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 3 , May 2018 , pp. 310 - 333
Copyright
Copyright © Cambridge University Press 2018 

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