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)].

中文翻译：

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埃德蒙兹问题中用决定论换取非交换性

令 $X=X_1\sqcup X_2\sqcup\ldots\sqcup X_k$ 为一组分区变量，使得每个部分 $X_i$ 中的变量都是非交换的，但对于任何 $i\neq j$，变量 $x\in X_i$ 与 X_j$ 中的变量 $x'\ 进行交换。给定一个方阵 $T$ 作为输入，其条目是 $\mathbb{Q}\langle{X}\rangle$ 上的线性形式，我们考虑检查 $T$ 在通用偏斜场上是否可逆的问题部分交换多项式环 $\mathbb{Q}\langle{X}\rangle$ 的分数 [Klep-Vinnikov-Volcic (2020)]。在本文中，我们针对常数 $k$ 为该问题设计了一种确定性多项式时间算法。特殊情况 $k=1$ 是非交换 Edmonds 问题 (NSINGULAR)，根据最近的结果，该问题具有确定性多项式时间算法 [Garg-Gurvits-Oliveira-Wigderson (2016)、Ivanyos-Qiao-Subrahmanyam (2018)、Hamada -平井（2021）]。途中，我们获得了第一个确定性多项式时间算法，用于 $k$-tape \emph{weighted} 自动机（对于常数 $k$）的等价测试问题，解决了长期存在的开放问题 [Harju 和 Karhum"{ a}ki(1991)，Worrell (2013)] 从代数上讲，等价问题简化为测试划分集 $X$ 上的部分交换有理级数是否为零 [Worrell (2013)]。由 Harju 和 Karhum\"{a}ki (1991) 建立。在这项工作之前，Worrell (2013) 给出了解决该问题的 \emph{randomized} 多项式时间算法，随后还开发了确定性拟多项式时间算法 [Arvind et al. 2013]。 （2021）]。