The Complexity of Testing Monomials in Multivariate Polynomials

TL;DR

This paper explores the computational complexity of detecting multilinear monomials in multivariate polynomials, linking algebraic properties to complexity theory and establishing foundational results for specific polynomial classes.

Contribution

It introduces a theoretical framework for testing monomials in multivariate polynomials and provides initial complexity results for $\

Findings

Results on $\

Insights into algebraic properties and their impact on complexity

Abstract

The work in this paper is to initiate 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 folds of 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 $\Pi\Sigma\Pi$ and $\Pi\Sigma$ polynomials are obtained in this paper, laying a basis for further study along this line.