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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 log a 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.

Ideals, Determinants, and Straightening: Proving and Using Lower Bounds for Polynomial Ideals

“…Taking the contrapositive, if f (x) cannot be computed by small circuits, then neither can any polynomial g ∈ f which has f as a factor of low multiplicity. Polynomial factorization has since been studied in restricted algebraic circuit classes, including low-depth circuits [DSY09;CKS19b], formulas [Oli16;DSS18], algebraic branching programs [DSS18; ST20], and sparse polynomials [BSV20]. This is motivated in part by the use of Kaltofen's theorem to establish hardness-to-pseudorandomness results for polynomial identity testing, as done in the work of Kabanets and Impagliazzo [KI04].…”

“…Prior to this, the best-known hitting set generator for low-depth circuits was given by Limaye, Srinivasan, and Tavenas [LST21], using the hardness-randomness results of Chou, Kumar, and Solomon [CKS19b]. They obtained, for all fixed ε > 0, a generator with seed length O(n ε ) and degree O(log n/ log log n).…”

“…For formulas, the best-known (conditional) constructions of hitting set generators prior to our work are due to Dvir, Shpilka, and Yehudayoff [DSY09] and Chou, Kumar, and Solomon [CKS19b]. Both works yield generators with parameters similar to the low-depth generator of Limaye, Srinivasan, and Tavenas [LST21] mentioned above (although the generator of [DSY09] can only hit formulas of small individual degree).…”

“…While our construction has better parameters, we use a stronger hardness assumption than what is needed by prior work. The constructions of Dvir, Shpilka, and Yehudayoff [DSY09] and Chou, Kumar, and Solomon [CKS19b] can be instantiated with any explicit family of polynomials that requires formulas of super-polynomial size. In contrast, our construction depends crucially on super-polynomial lower bounds on the border formula complexity of the determinant.…”

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“…Regarding the Polynomial Identity Lemma, the following footnote appears in[14]. "Variants of this lemma, often referred to as the Schwartz-Zippel Lemma, or the DeMillo-Lipton-Schwartz-Zippel Lemma, were discovered at least six times, starting with Øystein Ore in 1922 andDavid Muller in 1954 [46, 42, 56, 16, 60, 57].…”