Algorithms for Testing Monomials in Multivariate Polynomials

TL;DR

This paper develops algorithms for testing specific monomials in multivariate polynomials represented by arithmetic circuits, providing both randomized and deterministic methods with varying complexity, and explores the computational hardness of certain cases.

Contribution

It introduces new randomized and deterministic algorithms for monomial testing in multivariate polynomials, extending previous work and analyzing computational hardness.

Findings

Randomized $O^*(p^k)$ time algorithm for p-monomials.

Deterministic $O^*(6.4^k + p^k)$ time algorithm for p-monomials in formulas.

Deterministic $O^*(2^k)$ time algorithm for multilinear monomials.

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 $O^*(p^k)$ time algorithm for testing $p$-monomials in an $n$-variate polynomial of degree $k$ represented by an arithmetic circuit, while a deterministic $O^*(6.4^k + p^k)$ time algorithm is devised when the circuit is a formula, here $p$ is a given prime number. Second, we present a deterministic $O^*(2^k)$ time algorithm for testing multilinear monomials in $\Pi_m\Sigma_2\Pi_t\times \Pi_k\Pi_3$ polynomials, while a randomized $O^*(1.5^k)$ algorithm is given for these polynomials. The first algorithm extends the recent work by Koutis (2008) and Williams (2009) on testing multilinear monomials. Group algebra is exploited in the algorithm designs, in corporation with the randomized polynomial identity testing over a finite field by Agrawal and Biswas (2003), the deterministic noncommunicative polynomial identity testing by Raz and Shpilka (2005) and the perfect hashing functions by Chen {\em at el.} (2007). 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.