3 research outputs found
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 is a
polynomial with monomials, with individual degrees of its variables bounded
by , then can be deterministically factored in time . Prior to our work, the only efficient factoring algorithms known for
this class of polynomials were randomized, and other than for the cases of
and , 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 is an -sparse polynomial in variables, with individual
degrees of its variables bounded by , then the sparsity of each factor of
is bounded by . This is the first nontrivial bound on
factor sparsity for . 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