  A 1993 result of Alon and F\"uredi 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-F\"uredi 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-F\"uredi. We then discuss the relationship between Alon-F\"uredi and
results of DeMillo-Lipton, Schwartz and Zippel. A direct coding theoretic
interpretation of Alon-F\"uredi 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-F\"uredi 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.
