On Zeros of a Polynomial in a Finite Grid

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.