  Motivated by applications in combinatorial geometry, we consider the
following question: Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_m)$ be an
$m$-partition of a positive integer $n$, $S_i \subseteq \mathbb{C}^{\lambda_i}$
be finite sets, and let $S:=S_1 \times S_2 \times \ldots \times S_m \subset
\mathbb{C}^n$ be the multi-grid defined by $S_i$. Suppose $p$ is an $n$-variate
degree $d$ polynomial. How many zeros does $p$ have on $S$?
  We first develop a multivariate generalization of Combinatorial
Nullstellensatz that certifies existence of a point $t \in S$ so that $p(t)
\neq 0$. Then we show that a natural multivariate generalization of the
DeMillo-Lipton-Schwartz-Zippel lemma holds, except for a special family of
polynomials that we call $\lambda$-reducible. This yields a simultaneous
generalization of Szemer\'edi-Trotter theorem and Schwartz-Zippel lemma into
higher dimensions, and has applications in incidence geometry. Finally, we
develop a symbolic algorithm that identifies certain $\lambda$-reducible
polynomials. More precisely, our symbolic algorithm detects polynomials that
include a cartesian product of hypersurfaces in their zero set. It is likely
that using Chow forms the algorithm can be generalized to handle arbitrary
$\lambda$-reducible polynomials, which we leave as an open problem.

Combinatorics;Computational Geometry;Symbolic Computation;Algebraic Geometry