# Alon's Nullstellensatz for multisets

Kós, Géza and Rónyai, Lajos
(2012)
*Alon's Nullstellensatz for multisets.*
COMBINATORICA, 32 (5).
pp. 589-605.
ISSN 0209-9683
10.1007/s00493-012-2758-0

## 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 {Mathematical expression} be a field, S 1, S 2,..., S n be finite nonempty subsets of {Mathematical expression}. Alon's theorem is a specialized, precise version of the Hilbertsche Nullstellensatz for the ideal of all polynomial functions vanishing on the set {Mathematical expression}. 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(x 1,..., x n) 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. © 2012 János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.

Item Type: | ISI Article |
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Subjects: | Q Science > QA Mathematics and Computer Science > QA75 Electronic computers. Computer science / számítástechnika, számítógéptudomány |

Depositing User: | EPrints Admin |

Date Deposited: | 16 Jan 2014 10:31 |

Last Modified: | 05 Feb 2014 12:27 |

URI: | http://eprints.sztaki.hu/id/eprint/7408 |

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