Computing Valuations of the Dieudonné Determinants
Authors: Taihei Oki
Summary: This paper addresses the issue of computing valuations of the Dieudonné determinants of matrices over discrete valuation skew fields (DVSFs). Beneath an inexpensive computational mannequin, we suggest two algorithms for a category of DVSFs, referred to as break up. Our algorithms are extensions of the combinatorial rest of Murota (1995) and the matrix growth by Moriyama — Murota (2013), each of that are primarily based on combinatorial optimization. Whereas our algorithms require an higher sure on the output, we give an estimation of the sure for skew polynomial matrices and present that the estimation is legitimate just for skew polynomial matrices. We contemplate two functions of this downside. The primary one is the noncommutative weighted Edmonds’ downside (nc-WEP), which is to compute the diploma of the Dieudonné determinants of matrices having noncommutative symbols. We present that the introduced algorithms cut back the nc-WEP to the unweighted downside in polynomial time. Specifically, we present that the nc-WEP over the rational discipline is solvable in time polynomial within the enter bit-length. We additionally current an software to analyses of levels of freedom of linear time-varying techniques by establishing formulation on the answer areas of linear differential/distinction equations.
