Computing Valuations of the Dieudonné Determinants

Abstract

This paper addresses the problem of computing valuations of the Dieudonné determinants of matrices over discrete valuation skew fields (DVSFs). This problem is an extension of computing degrees of determinants of polynomial matrices and has an application to the analysis of degrees of freedom of linear time-varying systems. Under a reasonable computational model, we propose two algorithms for a kind 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 show that the bound is easily obtained if the input is skew polynomial matrices. CCS CONCEPTS• Computing methodologies → Combinatorial algorithms; Linear algebra algorithms; • Mathematics of computing → Differential equations.