On testing monomials in multivariate polynomials

This paper presents a summary of our initial work on developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by certain economically compact structure has a multilinear monomial in its sum-product expansion. The complexity aspects of this problem and its variants are investigated with two objectives. One is to understand how this problem relates to critical problems in complexity, and if so to what extent. The other is to exploit possibilities of applying algebraic properties of polynomials to the study of those problems. A series of results about ΠΣΠ and ΠΣ polynomials is obtained in this paper, laying a basis for further study along this line. Several randomized and deterministic algorithms are devised for testing multilinear monomials or p-monomials in certain respective types of polynomials, where p is prime.