Closure Results for Polynomial Factorization

Abstract

$ \newcommand{\VNP}{\mathsf{VNP}} \newcommand{\VP}{\mathsf{VP}} \newcommand{\poly}{\mathsf{poly}} $ 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 Bürgisser'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 the polynomial factoring literature, and the depth-reduction results for arithmetic circuits, and hold over fields of characteristic zero or of sufficiently large characteristic. ------------- A preliminary version of this paper, titled ``Hardness vs Randomness for Bounded Depth Arithmetic Circuits,'' appeared in the Proceedings of the 33rd Computational Complexity Conference, 2018.