Computing Valuations of the Dieudonn\'e Determinants

TL;DR

This paper introduces algorithms to compute valuations of Dieudonné determinants over discrete valuation skew fields, enabling efficient solutions to noncommutative weighted Edmonds' problem and analysis of linear systems.

Contribution

It extends combinatorial optimization methods to compute valuations of Dieudonné determinants in DVSFs, including estimation bounds and applications to noncommutative problems.

Findings

Algorithms reduce nc-WEP to unweighted problem in polynomial time.

nc-WEP over rationals is solvable in polynomial time.

Provides formulas for degrees of freedom in linear time-varying systems.

Abstract

This paper addresses the problem of computing valuations of the Dieudonn\'e determinants of matrices over discrete valuation skew fields (DVSFs). Under a reasonable computational model, we propose two algorithms for a class of DVSFs, called split. Our algorithms are extensions of the combinatorial relaxation of Murota (1995) and the matrix expansion by Moriyama--Murota (2013), both of which are based on combinatorial optimization. While our algorithms require an upper bound on the output, we give an estimation of the bound for skew polynomial matrices and show that the estimation is valid only for skew polynomial matrices. We consider two applications of this problem. The first one is the noncommutative weighted Edmonds' problem (nc-WEP), which is to compute the degree of the Dieudonn\'e determinants of matrices having noncommutative symbols. We show that the presented algorithms reduce the nc-WEP to the unweighted problem in polynomial time. In particular, we show that the nc-WEP over the rational field is solvable in time polynomial in the input bit-length. We also present an application to analyses of degrees of freedom of linear time-varying systems by establishing formulas on the solution spaces of linear differential/difference equations.