Deterministic Factorization of Sparse Polynomials with Bounded Individual Degree

Abstract

In this article, we study the problem of deterministic factorization of sparse polynomials. We show that if f ∈ F[ x 1 , x 2 ,… , x n ] is a polynomial with s monomials, with individual degrees of its variables bounded by d , then f can be deterministically factored in time s poly( d )log n . Prior to our work, the only efficient factoring algorithms known for this class of polynomials were randomized, and other than for the cases of d =1 and d =2, only exponential time-deterministic factoring algorithms were known. A crucial ingredient in our proof is a quasi-polynomial sparsity bound for factors of sparse polynomials of bounded individual degree. In particular, we show that if f is an s -sparse polynomial in n variables, with individual degrees of its variables bounded by d , then the sparsity of each factor of f is bounded by s (9 d 2 log n ) . This is the first non-trivial bound on factor sparsity for d > 2. Our sparsity bound uses techniques from convex geometry, such as the theory of Newton polytopes and an approximate version of the classical Carathéodory’s Theorem. Our work addresses and partially answers a question of von zur Gathen and Kaltofen [1985] who asked whether a quasi-polynomial bound holds for the sparsity of factors of sparse polynomials.