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.

翻译：本文针对的是Dieudonn\'e 矩阵决定因素在离散估值柱形字段( DVSFs) 上的计算值问题。 在合理的计算模型下, 我们为一类DVSF提出两种算法, 称为分裂。 我们的算法是穆罗塔组合放松的延伸(1995年), 以及森山- 毛罗塔( 2013年) 矩阵扩展的扩展, 两者都是基于组合优化。 虽然我们的算法要求对输出进行上限限制, 我们估计了 skew 多边矩阵的界限, 并显示这一估计只适用于 skew 多边矩阵。 我们考虑这一问题的两个应用。 第一个是非对称加权Edmonds 问题( nc- WEP ), 这是对具有非组合符号的矩阵决定因素的度进行比较。 我们的算法将nc- WEP 降低到多元时空空间的未加权问题。 特别是, 我们显示, 在目前我们理性阵度阵度阵度阵列中, 的Nc- WEWEP 直线式阵式阵式阵式阵式阵式阵式阵列的阵式公式也可用于确定我们理性阵势阵势阵势的阵势阵势的阵列的阵列的阵列的阵列。