Theory of Computing
-------------------
Title     : Closure Results for Polynomial Factorization
Authors   : Chi-Ning Chou, Mrinal Kumar, and Noam Solomon
Volume    : 15
Number    : 13
Pages     : 1-34
URL       : https://theoryofcomputing.org/articles/v015a013

Abstract
--------

In a sequence of fundamental results in the 1980s, Kaltofen (SICOMP
1985, STOC'86, STOC'87, RANDOM'89) showed that factors of multivariate
polynomials with small arithmetic circuits have small arithmetic
circuits. In other words, the complexity class VP is closed under
taking factors. A natural question in this context is to understand if
other natural classes of multivariate polynomials, for instance,
arithmetic formulas, algebraic branching programs, bounded-depth
arithmetic circuits or the class VNP, are closed under taking
factors.

In this paper, we show that all factors of degree at most $\log^a n$
of polynomials with $\poly(n)$-size depth-$k$ circuits have $\poly(n)$-
size circuits of depth at most $O(k + a)$. This partially answers a
question of Shpilka--Yehudayoff (Found. Trends in TCS, 2010) and has
applications to hardness--randomness tradeoffs for bounded-depth
arithmetic circuits.

As direct applications of our techniques, we also obtain simple proofs
of the following results.

  * The complexity class VNP is closed under taking factors. This 
confirms Conjecture 2.1 in Burgisser's monograph (2000) and improves upon 
a recent result of Dutta, Saxena and Sinhababu (STOC'18) who showed a 
quasipolynomial upper bound on the number of auxiliary variables and 
the complexity of the verifier circuit of factors of polynomials in VNP. 

  * A factor of degree at most $d$ of a polynomial $P$ which can be
computed by an arithmetic formula (or an algebraic branching program)
of size $s$ has a formula (an algebraic branching program, resp.) of
size at most $\poly(s, d^{\log d}, \deg(P))$. This result was first
shown by Dutta et al. (STOC'18) and we obtain a slightly different
proof as an easy consequence of our techniques.  Our proofs rely on a
combination of the ideas, based on _Hensel lifting_, developed in
polynomial factoring literature, and the depth-reduction results for
arithmetic circuits, and hold over fields of characteristic zero or 
of sufficiently large characteristic.