## 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.

Original language | English |
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Pages (from-to) | 310-333 |

Number of pages | 24 |

Journal | Combinatorics Probability and Computing |

Volume | 27 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 May 2018 |

Externally published | Yes |