We show that^{\lambda_i} be finite sets, and let S:=S_1 \times S_2 \times \ldots \times S_m \subset 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-, a natural multivariate generalization of Schwartz-Zippel-DeMillo-Lipton lemma holds. Moreover , we develop a symbolic algorithm to detect \lambda-reducibility. Our work is motivated by and has applications in combinatorial geometry. Along the way we also present a multivariate generalization of Combinatorial Nullstellensatz, which might be of independent interest .
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 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 .
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We show that
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
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-that
that
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-reducible-, a natural multivariate generalization of Schwartz-Zippel-DeMillo-Lipton lemma holds. Moreover
-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
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to detect \lambda-reducibility. Our work is motivated by and has applications in combinatorial geometry. Along the way we also present a multivariate generalization of Combinatorial Nullstellensatz, which might be of independent interest
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