Towards Randomized Testing of $q$-Monomials in Multivariate Polynomials

TL;DR

This paper introduces a randomized algorithm that efficiently tests for the presence of $q$-monomials in multivariate polynomials, working for any fixed integer $q \,\geq 2$, regardless of whether $q$ is prime, and improves previous complexity bounds.

Contribution

It presents the first randomized algorithm with a complexity of $O^*(7.15^k)$ for testing $q$-monomials in polynomials represented by tree-like circuits, independent of $q$'s primality.

Findings

Algorithm runs in $O^*(7.15^k)$ time for degree $k$ monomials.

Works for any fixed integer $q \geq 2$, prime or not.

Improves previous algorithms for prime $q > 7$.

Abstract

Given any fixed integer $q\ge 2$, a $q$-monomial is of the format $\displaystyle x^{s_1}_{i_1}x^{s_2}_{i_2}...x_{i_t}^{s_t}$ such that $1\le s_j \le q-1$, $1\le j \le t$. $q$-monomials are natural generalizations of multilinear monomials. Recent research on testing multilinear monomials and $q$-monomails for prime $q$ in multivariate polynomials relies on the property that $Z_q$ is a field when $q\ge 2 $ is prime. When $q>2$ is not prime, it remains open whether the problem of testing $q$-monomials can be solved in some compatible complexity. In this paper, we present a randomized $O^*(7.15^k)$ algorithm for testing $q$-monomials of degree $k$ that are found in a multivariate polynomial that is represented by a tree-like circuit with a polynomial size, thus giving a positive, affirming answer to the above question. Our algorithm works regardless of the primality of $q$ and improves upon the time complexity of the previously known algorithm for testing $q$-monomials for prime $q>7$.