Before revision:  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.
After revision:  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 to detect \lambda-reducibility.