  In this paper, we consider the computation of the degree of the Dieudonn\'e
determinant of a linear symbolic matrix $A = A_0 + A_1 x_1 + \cdots + A_m x_m$,
where each $A_i$ is an $n \times n$ polynomial matrix over $\mathbb{K}[t]$ and
$x_1,x_2,\ldots,x_m$ are pairwise "non-commutative" variables. This quantity is
regarded as a weighted generalization of the non-commutative rank (nc-rank) of
a linear symbolic matrix, and its computation is shown to be a generalization
of several basic combinatorial optimization problems, such as weighted
bipartite matching and weighted linear matroid intersection problems. Based on
the work on nc-rank by Fortin and Rautenauer (2004), and Ivanyos, Qiao, and
Subrahmanyam (2018), we develop a framework to compute the degree of the
Dieudonn\'e determinant of a linear symbolic matrix. We show that the deg-det
computation reduces to a discrete convex optimization problem on the Euclidean
building for ${\rm SL}(\mathbb{K}(t)^n)$. To deal with this optimization
problem, we introduce a class of discrete convex functions on the building.
This class is a natural generalization of L-convex functions in discrete convex
analysis (DCA). We develop a DCA-oriented algorithm (steepest descent
algorithm) to compute the degree of determinants. Our algorithm works with
matrix computation on $\mathbb{K}$, and uses a subroutine to compute a
certificate vector subspace for the nc-rank, where the number of calls of the
subroutine is sharply estimated. Our algorithm enhances some classical
combinatorial optimization algorithms with new insights, and is also understood
as a variant of the combinatorial relaxation algorithm, which was developed
earlier by Murota for computing the degree of the (ordinary) determinant.
