Deterministic Factorization of Sparse Polynomials with Bounded Individual Degree

In this paper we study the problem of deterministic factorization of sparse polynomials. We show that if $f\in \mathbb{F}[x_1,x_2,\dots ,x_n]$ is a polynomial with $s$ monomials, with individual degrees of its variables bounded by $d$, then $f$ can be deterministically factored in time $s^{$\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}$(d) \log n}$. 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 $d>2$. 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.