Deterministic Factorization of Sparse Polynomials with Bounded Individual Degree

Abstract

In this paper we study the problem of deterministic factorization of sparse polynomials. We show that if f is an n-variate polynomial with s monomials, with individual degrees of its variables bounded by d, then f can be deterministically factored in time s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">poly(d) log n</sup> . 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 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(d2 log n)</sup> . This is the first nontrivial 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 Caratheodory's Theorem. Our work addresses and partially answers a question of von zur Gathen and Kaltofen (JCSS 1985) who asked whether a quasi-polynomial bound holds for the sparsity of factors of sparse polynomials.