In this paper 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 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 O ( d 2 log n ) . This is the first nontrivial bound on factor sparsity for 2"> d 2 . Our sparsity bound uses techniques from convex geometry, such as the theory of Newton polytopes and an approximate version of the classical Carath\'eodory'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.