Computing sparse multiples of polynomials

TL;DR

This paper investigates the computational problem of finding sparse polynomial multiples, providing efficient algorithms over the rationals for fixed sparsity and establishing complexity bounds over finite fields.

Contribution

It introduces a polynomial-time algorithm for finding sparse multiples over Q with fixed sparsity and proves complexity bounds over finite fields.

Findings

Polynomial-time algorithm for fixed sparsity over Q

Complexity lower bounds over finite fields

Equivalence to multiplicative order problem in finite fields

Abstract

We consider the problem of finding a sparse multiple of a polynomial. Given f in F[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple h in F[x] of f such that h has at most t non-zero terms, and if so, to find such an h. When F=Q and t is constant, we give a polynomial-time algorithm in d and the size of coefficients in h. When F is a finite field, we show that the problem is at least as hard as determining the multiplicative order of elements in an extension field of F (a problem thought to have complexity similar to that of factoring integers), and this lower bound is tight when t=2.