Algorithms for Testing Monomials in Multivariate Polynomials

Abstract

This paper is our second step towards developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by an arithmetic circuit has some types of monomials in its sum-product expansion. The complexity aspects of this problem and its variants have been investigated in our first paper by Chen and Fu (2010), laying a foundation for further study. In this paper, we present two pairs of algorithms. First, we prove that there is a randomized O*(pk) time algorithm for testing p-monomials in an n-variate polynomial of degree k represented by an arithmetic circuit, while a deterministic O*((6.4p)k) time algorithm is devised when the circuit is a formula, here p is a given prime number. Second, we present a deterministic O*(2k) time algorithm for testing multilinear monomials in ΠmΣ2Πt×ΠkΣ3 polynomials, while a randomized O*(1.5k) algorithm is given for these polynomials. Finally, we prove that testing some special types of multilinear monomial is W[1]-hard, giving evidence that testing for specific monomials is not fixed-parameter tractable.