Dieudonné Determinants in Skew Fields

• Dieudonné determinants are defined as invariants for matrices over discrete valuation skew fields, extending classical determinant properties to noncommutative settings.
• They leverage valuation properties and Bruhat decomposition to compute invariant quantities that inform about matrix invertibility and singularity.
• Efficient algorithms based on combinatorial relaxation and matrix expansion enable polynomial-time computations, impacting coding theory and system analysis.

The Dieudonné determinant is a generalization of the classical determinant to matrices over a noncommutative field, specifically discrete valuation skew fields (DVSFs). Given a matrix $A$ over such a field $F$, the Dieudonné determinant, $\operatorname{Det} A$, takes values in the abelianization $F^*/[F^*,F^*]$ and is well-defined up to commutators. The valuation of the Dieudonné determinant, denoted $(A) = v(\operatorname{Det} A)$, is a key invariant, extending properties of determinants to the noncommutative setting and providing essential data on invertibility and singularity. The theory of Dieudonné determinants underpins computational frameworks in both symbolic computation and noncommutative linear algebra, and is central in fields such as coding theory, systems theory, and module theory (Oki, 2019).

1. Dieudonné Determinants over Discrete Valuation Skew Fields

Let $F$ be a skew-field equipped with a discrete valuation $v: F^* \to \mathbb{Z}$ satisfying:

• $v(ab) = v(a) + v(b)$,
• $v(a+b) \ge \min\{v(a), v(b)\}$,
• $v(1) = 0$,
• $v(0) = +\infty$.

The valuation ring $R$ consists of elements $a \in F$ with $v(a) \ge 0$, with maximal ideal $J(R) = \{a \mid v(a)>0\}$, and a uniformizer $\pi \in R$ such that $v(\pi)=1$.

For $A \in GL_n(F)$, Bruhat decomposition (Cohn, 2003) yields $A = L D P U$, with $L, U$ unitriangular, $D$ diagonal with entries $e_1, \dots, e_n \in F^*$, and $P$ a permutation matrix. The Dieudonné determinant is

$$\operatorname{Det} A = \mathrm{sgn}(P)\cdot (e_1 e_2 \cdots e_n) \mod [F^*, F^*],$$

and its valuation

$$(A) := v(\operatorname{Det}A) \in \mathbb{Z} \cup \{+\infty\}, \quad (A)=+\infty \iff A\ \text{is singular}.$$

Basic properties include:

• (VD1) $(AB) = (A) + (B)$,
• (VD2) $(E) = 0$ for any elementary (unitriangular) matrix,
• (VD3) $$(\operatorname{diag}(d_1,\dots,d_n)) = \sum_i v(d_i)$$ (Oki, 2019).

2. Polynomial-Time Algorithms for Valuations

In the setting where $F$ is split—that is, $F$ admits a coefficient subfield $K \subset R$ and every $a \in R$ has a unique expansion $a = \sum_{d=0}^\infty a_d \pi^d$, $a_d \in K$—efficient algorithms for computing $(A) = v\operatorname{Det}A$ are available, assuming access to oracles for $K$-arithmetic and higher-derivations $\delta_d: K \to K$.

Given $A = \sum_{d=0}^l A_d \pi^d \in F^{n \times n}$ and an upper bound $M \ge (A)$, two frameworks are used (Oki, 2019):

2.1 Combinatorial Relaxation

This algorithm extends the combinatorial relaxation approaches of Murota (1995) and uses the following process:

• Build a bipartite graph $G(A)$ with edge weights $w(i,j) = v(A_{ij})$.
• The weight $\mu$ of a minimum-weight perfect matching gives a lower bound $\mu \le (A)$.
• Transform $A$ iteratively into a proper form ($C^k$), guided by LP duality and tightness of the associated coefficient matrix $B_0$.
• If $B_0$ is nonsingular in $K$, the process terminates, yielding $(A) = \mu$.
• Otherwise, further transformations ensure progress toward tightness; altogether, $O(M)$ iterations suffice, each involving weighted matching, $\delta$-updates, and matrix multiplication.

Total running time: $O(M^3 n^2 + M^2 n^\omega + M n^{2.5})$.

2.2 Matrix Expansion

Following work by Van Dooren et al. (1979) and Moriyama–Murota (2013), define the $\mu$-th order matrix expansion:

$$\Omega_\mu(A) = \begin{pmatrix} A_0^{(0)} & A_1^{(0)} & \cdots & A_{\mu-1}^{(0)} \ 0 & A_1^{(1)} & \cdots & A_{\mu-1}^{(1)} \ \vdots & \ddots & \ddots & \vdots \ 0 & \cdots & 0 & A_{\mu-1}^{(\mu-1)} \end{pmatrix} \in K^{\mu n \times \mu n}$$

where $A_d^{(i)}$ is the $\pi^d$-coefficient of $\pi^i A$.

Let $$\omega_\mu(A) = \operatorname{rank}_K \Omega_\mu(A)$$. Using Smith–McMillan invariants and Legendre duality:

$$(A) = \max_\mu (n\mu - \omega_\mu(A)),\quad \text{and if } (A) \le M, \ (A) = M n - \omega_M(A).$$

Thus, $(A)$ is computed by evaluating the rank of $\Omega_M(A)$—an $O(M^\omega n^\omega)$ operation—plus assembling $\Omega_M(A)$ in $O(M^3 n^2)$.

Total running time: $O(M^3 n^2 + M^\omega n^\omega)$ (Oki, 2019).

3. Estimation of Upper Bounds for Skew Polynomial Matrices

For $$A(s) = \sum_{d=0}^l A_d s^d \in K[s;\sigma, \delta]^{n \times n}$$, regarded as nonsingular in $F = K(s; \sigma, \delta)$ with $v = -\deg$, the inequality

$(A(s)) = -\deg \operatorname{Det} A(s) \le l n$

holds. This follows by multiplying $A$ by $s^l$ so all entries become polynomials of degree $\ge 0$, and examining the Jacobson normal form. This upper bound characterizes $F \cong K\langle s \rangle$, the skew-rational series field. The bound estimation is valid for skew polynomial matrices and forms the foundation for upper bounds required by the algorithms above (Oki, 2019).

4. Applications to Edmonds’ Problems

4.1 Noncommutative Weighted Edmonds’ Problem (nc-WEP)

Given a noncommutative-linear polynomial matrix $A(s)$ over $K\langle x_1,\ldots,x_m\rangle$, the nc-WEP is the problem of computing the degree of the Dieudonné determinant. Applying combinatorial relaxation, the problem reduces to solving the maximum-vanishing-subspace problem (MVSP) over $K$ to factor out subspaces and identify tightness at each step.

With deterministic polynomial-time MVSP oracles (e.g., Hamada–Hirai 2020, Ivanyos et al. 2018), the nc-WEP over $\mathbb{Q}$ is solvable in bit-length polynomial time; over any field $K$, the solution can be obtained in polynomially many arithmetic operations (Oki, 2019).

4.2 Matrix Expansion Reduction

Alternatively, matrix expansion assembles $\Omega_{ln}(A s^{-l})$ in $K[x,s]^{ln\cdot n \times ln\cdot n}$ or $K\langle x\rangle(s)$. The rank gives $\omega_{ln}(A s^{-l})$, yielding

$(A) = ln - \omega_{ln}(A s^{-l}).$

Therefore, WEP (and nc-WEP) reduces to the commutative (resp. noncommutative) Edmonds problem in one call at polynomial size and time.

5. Dieudonné Determinants in Linear Time-Varying Systems

Consider $(K,\sigma,\delta)$ a differential or difference field with constant field $C = \operatorname{Const}(K)$. The general linear system

$$A_0 y + A_1 \theta(y) + \cdots + A_l \theta^l(y) = f, \quad A_i \in K^{n \times n},$$

with $\theta = \delta$ (differential) or $\theta = \sigma$ (difference), is encoded by the skew-polynomial matrix $A(\theta) = \sum A_i \theta^i$. The solution space $V \subset K^n$ is an affine $C$-module. Over an “adequate” extension $R$ of $K$, dimension is related to Dieudonné determinant valuation:

• If $\theta = \delta$, $\dim_C V = \deg \operatorname{Det} A(\theta)$,
• If $\theta = \sigma$, $$\dim_C V = \deg \operatorname{Det} A(\theta) - \operatorname{ord} \operatorname{Det} A(\theta)$$.

Hence, the valuation algorithms for the Dieudonné determinant provide computational access to structural invariants and degrees of freedom in time-varying system analyses (Oki, 2019).

6. Computational and Theoretical Significance

The described algorithms and theoretical results extend classical determinant-based methodologies to matrices over noncommutative and valued fields. The valuation of the Dieudonné determinant preserves critical properties of the classical determinant (multiplicativity, invariance under elementary operations, decomposition over block-diagonal forms), while supporting practical computational (polynomial-time) solutions for structurally complex matrix problems. Notable applications include the resolution of the noncommutative weighted Edmonds' problem and the explicit computation of solution space dimensions for linear differential and difference equations, providing essential computational resources for symbolic computation, control theory, and noncommutative algebra (Oki, 2019).