The Multivariate Schwartz-Zippel Lemma
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Motivated by applications in combinatorial geometry, we consider the following question: Let λ=(λ_1,λ_2,…,λ_m) be an m-partition of a positive integer n, S_i ⊆ℂ^λ_i be finite sets, and let S:=S_1 × S_2 ×…× S_m ⊂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 ∈ S so that p(t) ≠ 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 λ-reducible. This yields a simultaneous generalization of Szemerédi-Trotter theorem and Schwartz-Zippel lemma into higher dimensions, and has applications in incidence geometry. Finally, we develop a symbolic algorithm that identifies certain λ-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 λ-reducible polynomials, which we leave as an open problem.
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