Trading Determinism for Noncommutativity in Edmonds' Problem

Let X=X₁ X₂… Xₖ be a partitioned set of variables such that the variables in each part Xᵢ are noncommuting but for any i≠ j, the variables x∈ Xᵢ commute with the variables x'∈ Xⱼ. Given as input a square matrix T whose entries are linear forms over QX, 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 QX [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 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 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)].