In this paper we study the computational complexity of computing the noncommutative determinant. We first consider the arithmetic circuit complexity of computing the noncommutative determinant polynomial. Then, more generally, we also examine the complexity of computing the determinant (as a function) over noncommutative domains. Our hardness results are summarized below:

... more >>>

In this work we give a randomized blackbox polynomial identity testing (PIT) algorithm for constant-depth homogeneous noncommutative circuits, with poly-logarithmic time complexity in the circuit size. In fact, we show that the polynomial computable by such a depth-$(\Delta-1)$ circuit of size $s$ cannot be a polynomial identity for the $O(\log ... more >>>

We prove that $\VPnc$ is closed under taking factors over every field. If a noncommutative circuit of size $s$ computes a nonconstant polynomial of degree at most $D$, then each left or right factor has a circuit of size $O(sD^3)$. A central factor $g$ in a decomposition $f=agb$, with nonconstant ... more >>>