Closure Results for Polynomial Factorization

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 loga n of polynomials with poly(n)size depth-k circuits have poly(n)-size circuits of depth 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 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. ∗Supported by Boaz Barak’s NSF awards CCF 1565264 and CNS 1618026. ACM Classification: F.1.3 AMS Classification: 68Q15, 68Q17