Alon's Nullstellensatz for multisets

Géza Kós, L. Rónyai

Research output: Article

5 Citations (Scopus)

Abstract

Alon's combinatorial Nullstellensatz (Theorem 1.1 from [2]) is one of the most powerful algebraic tools in combinatorics, with a diverse array of applications. Let F be a field, S1, S2,...,Sn be finite nonempty subsets of F. Alon's theorem is a specialized, precise version of the Hilbertsche Nullstellensatz for the ideal of all polynomial functions vanishing on the set S = S1×S××...Sn⊆Fn. From this Alon deduces a simple and amazingly widely applicable nonvanishing criterion (Theorem 1.2 in [2]). It provides a sufficient condition for a polynomial f(x1,...,xn) which guarantees that f is not identically zero on the set S. In this paper we extend these two results from sets of points to multisets. We give two different proofs of the generalized nonvanishing theorem. We extend some of the known applications of the original nonvanishing theorem to a setting allowing multiplicities, including the theorem of Alon and Füredi on the hyperplane coverings of discrete cubes.

Original languageEnglish
Pages (from-to)589-605
Number of pages17
JournalCombinatorica
Volume32
Issue number5
DOIs
Publication statusPublished - 2012

Fingerprint

Multiset
Polynomials
Theorem
Polynomial function
Combinatorics
Hyperplane
Set of points
Regular hexahedron
Deduce
Multiplicity
Covering
Polynomial
Subset
Sufficient Conditions
Zero

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

Cite this

Alon's Nullstellensatz for multisets. / Kós, Géza; Rónyai, L.

In: Combinatorica, Vol. 32, No. 5, 2012, p. 589-605.

Research output: Article

Kós, Géza ; Rónyai, L. / Alon's Nullstellensatz for multisets. In: Combinatorica. 2012 ; Vol. 32, No. 5. pp. 589-605.
@article{6d39ef0b16b644518485d1b4bebbea6b,
title = "Alon's Nullstellensatz for multisets",
abstract = "Alon's combinatorial Nullstellensatz (Theorem 1.1 from [2]) is one of the most powerful algebraic tools in combinatorics, with a diverse array of applications. Let F be a field, S1, S2,...,Sn be finite nonempty subsets of F. Alon's theorem is a specialized, precise version of the Hilbertsche Nullstellensatz for the ideal of all polynomial functions vanishing on the set S = S1×S××...Sn⊆Fn. From this Alon deduces a simple and amazingly widely applicable nonvanishing criterion (Theorem 1.2 in [2]). It provides a sufficient condition for a polynomial f(x1,...,xn) which guarantees that f is not identically zero on the set S. In this paper we extend these two results from sets of points to multisets. We give two different proofs of the generalized nonvanishing theorem. We extend some of the known applications of the original nonvanishing theorem to a setting allowing multiplicities, including the theorem of Alon and F{\"u}redi on the hyperplane coverings of discrete cubes.",
author = "G{\'e}za K{\'o}s and L. R{\'o}nyai",
year = "2012",
doi = "10.1007/s00493-012-2758-0",
language = "English",
volume = "32",
pages = "589--605",
journal = "Combinatorica",
issn = "0209-9683",
publisher = "Janos Bolyai Mathematical Society",
number = "5",

}

TY - JOUR

T1 - Alon's Nullstellensatz for multisets

AU - Kós, Géza

AU - Rónyai, L.

PY - 2012

Y1 - 2012

N2 - Alon's combinatorial Nullstellensatz (Theorem 1.1 from [2]) is one of the most powerful algebraic tools in combinatorics, with a diverse array of applications. Let F be a field, S1, S2,...,Sn be finite nonempty subsets of F. Alon's theorem is a specialized, precise version of the Hilbertsche Nullstellensatz for the ideal of all polynomial functions vanishing on the set S = S1×S××...Sn⊆Fn. From this Alon deduces a simple and amazingly widely applicable nonvanishing criterion (Theorem 1.2 in [2]). It provides a sufficient condition for a polynomial f(x1,...,xn) which guarantees that f is not identically zero on the set S. In this paper we extend these two results from sets of points to multisets. We give two different proofs of the generalized nonvanishing theorem. We extend some of the known applications of the original nonvanishing theorem to a setting allowing multiplicities, including the theorem of Alon and Füredi on the hyperplane coverings of discrete cubes.

AB - Alon's combinatorial Nullstellensatz (Theorem 1.1 from [2]) is one of the most powerful algebraic tools in combinatorics, with a diverse array of applications. Let F be a field, S1, S2,...,Sn be finite nonempty subsets of F. Alon's theorem is a specialized, precise version of the Hilbertsche Nullstellensatz for the ideal of all polynomial functions vanishing on the set S = S1×S××...Sn⊆Fn. From this Alon deduces a simple and amazingly widely applicable nonvanishing criterion (Theorem 1.2 in [2]). It provides a sufficient condition for a polynomial f(x1,...,xn) which guarantees that f is not identically zero on the set S. In this paper we extend these two results from sets of points to multisets. We give two different proofs of the generalized nonvanishing theorem. We extend some of the known applications of the original nonvanishing theorem to a setting allowing multiplicities, including the theorem of Alon and Füredi on the hyperplane coverings of discrete cubes.

UR - http://www.scopus.com/inward/record.url?scp=84871940489&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871940489&partnerID=8YFLogxK

U2 - 10.1007/s00493-012-2758-0

DO - 10.1007/s00493-012-2758-0

M3 - Article

AN - SCOPUS:84871940489

VL - 32

SP - 589

EP - 605

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 5

ER -