The Multivariate Schwartz--Zippel Lemma

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 $n$, $S_i \subseteq \mathbb{C}^{\lambda_i}$ finite sets, and let $S:=S_1 \times S_2 \cdots S_m \subset \mathbb{C}^n$ multigrid defined $S_i$. Suppose $p$ is $n$-variate degree $d$ polynomial. How many zeros does have on $S$? We first develop multivariate generalization nullstellensatz that certifies existence point $t \in S$ so $p(t) \neq 0$. Then show natural DeMillo--Lipton--Schwartz--Zippel lemma holds, except for special family polynomials call $\lambda$-reducible. This yields simultaneous Szemerédi--Trotter theorem Schwartz--Zippel into higher dimensions, has incidence geometry. Finally, symbolic algorithm identifies certain $\lambda$-reducible polynomials. More precisely, our detects include Cartesian product hypersurfaces their zero set. It likely using Chow forms can generalized to handle arbitrary polynomials, which leave as open problem.