Let $T$ be a matrix whose entries are linear forms over the noncommutative variables $x_1, x_2, \ldots, x_n$. The noncommutative Edmonds' problem (NSINGULAR) aims to determine whether $T$ is invertible in the free skew field generated by $x_1,x_2,\ldots,x_n$. Currently, there are three different deterministic polynomial-time algorithms to solve this problem: using operator scaling [Garg, Gurvits, Oliveira, and Wigserdon (2016)], algebraic methods [Ivanyos, Qiao, and Subrahmanyam (2018)], and convex optimization [Hamada and Hirai (2021)]. In this paper, we present a simpler algorithm for the NSINGULAR problem. While our algorithmic template is similar to the one in Ivanyos et. al.(2018), it significantly differs in its implementation of the rank increment step. Instead of computing the limit of a second Wong sequence, we reduce the problem to the polynomial identity testing (PIT) of noncommutative algebraic branching programs (ABPs). This enables us to bound the bit-complexity of the algorithm over $\mathbb{Q}$ without requiring special care. Moreover, the rank increment step can be implemented in quasipolynomial-time even without an explicit description of the coefficient matrices in $T$. This is possible by exploiting the connection with the black-box PIT of noncommutative ABPs [Forbes and Shpilka (2013)].

翻译：设$T$是一个矩阵，其元素为关于非交换变量$x_1, x_2, \ldots, x_n$的线性形式。非交换Edmonds问题（NSINGULAR）旨在确定$T$在由$x_1,x_2,\ldots,x_n$生成的自由斜域中是否可逆。目前，有三种不同的确定性多项式时间算法可解决该问题：使用算子缩放[Garg, Gurvits, Oliveira, and Wigserdon (2016)]、代数方法[Ivanyos, Qiao, and Subrahmanyam (2018)]以及凸优化[Hamada and Hirai (2021)]。在本文中，我们提出了一种更简单的NSINGULAR问题算法。虽然我们的算法模板与Ivanyos等人(2018)的算法相似，但在秩递增步骤的实现上有显著差异。我们没有计算第二个Wong序列的极限，而是将问题归约到非交换代数分支程序（ABP）的多项式恒等判定（PIT）。这使得我们能够在不需特殊处理的情况下，限定算法在$\mathbb{Q}$上的比特复杂度。此外，即使没有$T$中系数矩阵的显式描述，秩递增步骤也可以在拟多项式时间内实现。这得益于与非交换ABP的黑盒PIT[Forbes and Shpilka (2013)]的关联。