Identity Testing for +-Regular Noncommutative Arithmetic Circuits

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Abstract

An efficient randomized polynomial identity test for noncommutative polynomials given by noncommutative arithmetic circuits remains an open problem. The main bottleneck to applying known techniques is that a noncommutative circuit of size $s$ can compute a polynomial of degree exponential in $s$ with a double-exponential number of nonzero monomials. In this paper, we report some progress by dealing with two natural subcases (both allow for polynomials of exponential degree and a double exponential number of monomials): (1) We consider \emph{$+$-regular} noncommutative circuits: these are homogeneous noncommutative circuits with the additional property that all the $+$-gates are layered, and in each $+$-layer all gates have the same syntactic degree. We give a \emph{white-box} polynomial-time deterministic polynomial identity test for such circuits. Our algorithm combines some new structural results for $+$-regular circuits with known results for noncommutative ABP identity testing [RS05PIT], rank bound of commutative depth three identities [SS13], and equivalence testing problem for words [Loh15, MSU97, Pla94]. (2) Next, we consider $\Sigma\Pi^*\Sigma$ noncommutative circuits: these are noncommutative circuits with layered $+$-gates such that there are only two layers of $+$-gates. These $+$-layers are the output $+$-gate and linear forms at the bottom layer; between the $+$-layers the circuit could have any number of $\times$ gates. We given an efficient randomized \emph{black-box} identity testing problem for $\Sigma\Pi^*\Sigma$ circuits. In particular, we show if $f\in F<Z>$ is a nonzero noncommutative polynomial computed by a $\Sigma\Pi^*\Sigma$ circuit of size $s$, then $f$ cannot be a polynomial identity for the matrix algebra $\mathbb{M}_s(F)$, where the field $F$ is a sufficiently large extension of $F$ depending on the degree of $f$.