  In this paper we study the problem of deterministic factorization of sparse
polynomials. We show that if $f \in \mathbb{F}[x_{1},x_{2},\ldots ,x_{n}]$ is a
polynomial with $s$ monomials, with individual degrees of its variables bounded
by $d$, then $f$ can be deterministically factored in time $s^{\mathrm{poly}(d)
\log n}$. 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 $d>2$. 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.
