On testing for zero polynomials by a set of points with bounded precision

Abstract

We consider a general methodology proposed by Chen and Kao [4] for testing polynomial identities. We prove that the test cannot be completely derandomized by any specified set of rational approximations to algebraic numbers up to a polynomial number of bits. The proof is a direct application of Dirichlet’s box principle. We also give some number theoretic estimates for the likelihood of a multiplicatively independent sequence of integers which can be used in their algorithm.