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.

翻译：受组合几何应用的驱动,我们考虑以下问题: $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_m)$是正整数的美元部分, $S_i\subseteq\mathb{C ⁇ lambda_i}美元是有限的套件, 让 $S=_ 1\times S_ 2\timets\wards\ldots\ times\times S_m@subsetre=S_mathbs_mathb_c_C}( little) listalditions) listal- listaldalations=( littledalda) listal- materaldal- macal directrations), 然后我们将一个普通的直数数数的直方形的直方的直方的直方程式 。