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Abstract

In this paper we show that black-box polynomial identity testing (PIT) for nvariate noncommutative polynomials f of degree D with at most t nonzero monomials can be done in randomized poly(n, logt, log D) time, and consequently in randomized poly(n, logt, s) time if f is computable by a circuit of size s. This result makes progress on a question that has been open for over a decade. Our algorithm is based on efficiently isolating a monomial using nondeterministic automata. The above result does not yield an efficient randomized PIT for noncommutative circuits in general, as noncommutative circuits of size s can compute polynomials with a doubleexponential (in s) number of monomials. As a first step, we consider a natural class of homogeneous noncommutative circuits, that we call +-regular circuits, and give a white-box polynomial-time deterministic PIT for them. These circuits can compute noncommutative polynomials with number of monomials double-exponential in the circuit size. Our algorithm combines some new structural results for +-regular circuits with known PIT results for noncommutative algebraic branching programs, a rank bound for commutative depth-3 A preliminary version of this paper appeared in the Proceedings of the 49th ACM Symp. on Theory of Computing (STOC'17) [4]. * Part of the work was done while the author was a postdoctoral fellow at Chennai Mathematical Institute, India.

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“…See Section 4.5 for a formal definition. 2 Here, non-trivial means subexponential time, or quasipolynomial time, based on the hardness assumption.…”

“…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 and David Muller in 1954[30,25,36,6,42,37]. For a brief history, see[2] where the term "Polynomial Identity Lemma" is attributed to L. Babai.…”

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“…All these steps can be achieved in space polynomial in the input size, using the uniqueness of the annihilator for g [39, Theorem 47], Perron's degree bound [49] and linear algebra [15,9,43]. 5 Hitting set for VP: Proof of Theorem 1.4…”

“…"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]. For a brief history, see[5] where the term "Polynomial Identity Lemma" is attributed to L. Babai.…”

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“…2δ ] .By the Polynomial Identity Lemma,4 any non-constant polynomial of individual degree r (and therefore total degree 2r) can be equal to i on at most 2r • |A i | points in each A i × A i box, and hence agrees with both R and C on at most 2r ∑ |A i | ≤ 2r|F| points in total. Since for every i, it follows that |A i | 2 > 2r|F| for every i ∈ [N], and therefore the t polynomials Q i that maximize the probability Pr x,y [∃i ∈ [t] s.t.…”