Let $X=X_1\sqcup X_2\sqcup\ldots\sqcup X_k$ be a partitioned set of variables such that the variables in each part $X_i$ are noncommuting but for any $i\neq j$, the variables $x\in X_i$ commute with the variables $x'\in X_j$. 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)].

翻译：设 $X=X_1\sqcup X_2\sqcup\ldots\sqcup X_k$ 是一个分区的变量集合，其中每个部分 $X_i$ 内的变量是非交换的，但对任意 $i\neq j$，变量 $x\in X_i$ 与 $x'\in X_j$ 可交换。给定一个方阵 $T$，其条目为 $\mathbb{Q}\langle{X}\rangle$ 上的线性形式，我们考虑判断 $T$ 在部分交换多项式环 $\mathbb{Q}\langle{X}\rangle$ [Klep-Vinnikov-Volcic (2020)] 的泛型斜分式域上是否可逆的问题。在本文中，针对常数 $k$，我们设计了该问题的确定性多项式时间算法。特殊情况 $k=1$ 即非交换埃德蒙兹问题（NSINGULAR），已由近期结果 [Garg-Gurvits-Oliveira-Wigderson (2016), Ivanyos-Qiao-Subrahmanyam (2018), Hamada-Hirai (2021)] 给出确定性多项式时间算法。在此过程中，我们首次为 $k$ 带加权自动机（常数 $k$）的等价性检验问题提供了确定性多项式时间算法，解决了一个长期悬而未决的公开问题 [Harju and Karhumäki(1991), Worrell (2013)]。从代数角度看，等价性检验可归约为判断分区集合 $X$ 上的部分交换有理级数是否为零 [Worrell (2013)]。该问题的可判定性由 Harju 和 Karhumäki (1991) 建立。在本工作之前，Worrell (2013) 给出了该问题的随机多项式时间算法，随后 Arvind 等人 (2021) 又发展了确定性拟多项式时间算法。