In this paper we study the problem of deterministic factorization of sparse
polynomials. We show that if fβF[x1β,x2β,β¦,xnβ] 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)logn. 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 sO(d2logn). 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