Deterministic Factorization of Sparse Polynomials with Bounded Individual Degree

In this article, we study the problem of deterministic factorization of sparse polynomials. We show that if f ∈ F[x1,x2,… ,xn] is a polynomial with s monomials, with individual degrees of its variables bounded by d, then f can be deterministically factored in time spoly(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 d2 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.