In [A. J. Diesl, Classes of strongly clean rings, Ph.D. Dissertation, University of California, Berkely (2006); Nil clean rings, J. Algebra383 (2013) 197–211], a nil clean ring was defined as a ring for which every element is the sum of a nilpotent and an idempotent. In this short paper, we characterize nil clean commutative group rings.

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.

In this paper we show that the black-box polynomial identity testing for noncommutative polynomials f ∈ F z 1 , z 2 , · · · , z n of degree D and sparsity t, can be done in randomized poly(n, log t, log D) time. As a consequence, if the black-box contains a circuit C of size s computing f ∈ F z 1 , z 2 , · · · , z n which has at most t non-zero monomials, then the identity testing can be done by a randomized algorithm with running time polynomial in s and n and log t. This makes significant progress on a question that has been open for over ten years.The earlier result by Bogdanov and Wee [BW05], using the classical Amitsur-Levitski theorem, gives a randomized polynomial-time algorithm only for circuits of polynomially bounded syntactic degree. In our result, we place no restriction on the degree of the circuit.Our algorithm is based on automata-theoretic ideas introduced in [AMS08, AM08]. In those papers, the main idea was to construct deterministic finite automata that isolate a single monomial from the set of nonzero monomials of a polynomial f in F z 1 , z 2 , · · · , z n . In the present paper, since we need to deal with exponential degree monomials, we carry out a different kind of monomial isolation using nondeterministic automata.

We revisit the main result of Carmosino et al [CILM18] which shows that an Ω( /2+ ) size noncommutative arithmetic circuit size lower bound (where is the matrix multiplication exponent) for a constantdegree -variate polynomial family ( ) , where each is a noncommutative polynomial, can be "lifted" to an exponential size circuit size lower bound for another polynomial family ( ) obtained from ( ) by a lifting process. In this paper, we present a simpler and more conceptual automata-theoretic proof of their result.

No abstract