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Volume 15 (2019) Article 7 pp. 1-36
Randomized Polynomial-Time Identity Testing for Noncommutative Circuits
Received: August 13, 2017
Revised: October 4, 2018
Published: October 13, 2019
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Keywords: algebraic complexity, non-commutative computation, Polynomial Identity Testing, randomized algorithm, Polynomial Indentity Lemma
ACM Classification: F.1.2
AMS Classification: 68W20

Abstract: [Plain Text Version]

$ \newcommand{\F}{\mathbb{F}} \newcommand{\poly}{\mathrm{poly}} $

In this paper we show that black-box polynomial identity testing (PIT) for $n$-variate noncommutative polynomials $f$ of degree $D$ with at most $t$ nonzero monomials can be done in randomized $\poly(n,\log t,\log D)$ time, and consequently in randomized $\poly(n,\log t, s)$ 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 double-exponential (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 identities, and an equivalence testing problem for words. Finally, we solve the black-box PIT problem for depth-3 $+$-regular circuits in randomized polynomial time. In particular, we show if $f$ is a nonzero noncommutative polynomial in $n$ variables over the field $\mathbb{F}$ computed by a depth-3 $+$-regular circuit of size $s$, then $f$ cannot be a polynomial identity for the matrix algebra $\mathbb{M}_s(\mathbb{F})$ when $\mathbb{F}$ is sufficiently large depending on the degree of $f$.

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A preliminary version of this paper appeared in the Proceedings of the 49th ACM Symp. on Theory of Computing (STOC'17).