  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.
