Trading Determinism for Noncommutativity in Edmonds' Problem

Let $X=X1\sqcup X2\sqcup\ldots\sqcup Xk$ be a partitioned set of variables such that the variables in each part $Xi$ are noncommuting but for any $i\neq j$, the variables $x\in Xi$ commute with the variables $x'\in Xj$. Given as input a square matrix $T$ whose entries are linear forms over $\mathbb{Q}\langle{X}\rangle$, we consider the problem of checking if $T$ is invertible or not over the universal skew field of fractions of the partially commutative polynomial ring $\mathbb{Q}\langle{X}\rangle$ [Klep-Vinnikov-Volcic (2020)]. In this paper, we design a deterministic polynomial-time algorithm for this problem for constant $k$. The special case $k=1$ is the noncommutative Edmonds' problem (NSINGULAR) which has a deterministic polynomial-time algorithm by recent results [Garg-Gurvits-Oliveira-Wigderson (2016), Ivanyos-Qiao-Subrahmanyam (2018), Hamada-Hirai (2021)]. En-route, we obtain the first deterministic polynomial-time algorithm for the equivalence testing problem of $k$-tape \emph{weighted} automata (for constant $k$) resolving a long-standing open problem [Harju and Karhum"{a}ki(1991), Worrell (2013)]. Algebraically, the equivalence problem reduces to testing whether a partially commutative rational series over the partitioned set $X$ is zero or not [Worrell (2013)]. Decidability of this problem was established by Harju and Karhum\"{a}ki (1991). Prior to this work, a \emph{randomized} polynomial-time algorithm for this problem was given by Worrell (2013) and, subsequently, a deterministic quasipolynomial-time algorithm was also developed [Arvind et al. (2021)].