### 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, S_{1}, S_{2},...,S_{n} 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 = S_{1}×S_{×}×...S_{n}⊆F^{n}. 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.

Original language | English |
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Pages (from-to) | 589-605 |

Number of pages | 17 |

Journal | Combinatorica |

Volume | 32 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2012 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

### Cite this

*Combinatorica*,

*32*(5), 589-605. https://doi.org/10.1007/s00493-012-2758-0

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

Research output: Article

*Combinatorica*, vol. 32, no. 5, pp. 589-605. https://doi.org/10.1007/s00493-012-2758-0

}

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 -