THE MULTIVARIATE SCHWARTZ-ZIPPEL LEMMA

Motivated by applications in combinatorial geometry, we consider the following question: Let lambda = (lambda(1), lambda(2), ..,lambda(m)) be an m-partition of a positive integer n, Si subset of C-lambda i be finite sets, and let S := S-1 x S-2 x ... x S-m, subset of C-n be the multigrid 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 the combinatorial nullstellensatz that certifies existence of a point t is an element of S so that p(t) not equal 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 the Szemeredi-Trotter theorem and the 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.