arXiv subjects

Dawid E. Kędzierski

Publications and source records attributed to Dawid E. Kędzierski.

5 recordsLinked to original sources

arXiv2026

Compatible Bases and Actuator Scheduling

We study a constrained basis-selection problem for an ordered block matrix $B=[ B_1\mid\cdots\mid B_n ].$ The successive tail matrices determine a nested family of column spaces, and we consider bases of representatives which are compatible with this induced flag. Our main linear-algebraic result shows that, among bases of representatives, compatibility with the entire flag is characterized by the intrinsic block quotas $r_k=w_k-w_{k+1}$, where $w_k$ is the rank of the $k$-th tail matrix. We then show that the same structure arises naturally in finite-horizon linear control. For the reachability matrix $[A^{N-1}G\mid \cdots\mid G]$, requiring every actuator to be used exactly once turns an actuator schedule into a representative system. Preservation of the complete reachability profile is then equivalent to compatibility with the tail-space flag. This correspondence leads to an exact depth-first search with a future-rank pruning criterion. For an explicit structured family and a fixed search ordering, the baseline search visits $3\cdot2^m+m-2$ nodes before the first solution, whereas future-rank pruning reduces this number to $3m+1$. Exact-arithmetic experiments reproduce these formulas for all tested instances.

math.OC↗

arXiv2026

Homological Methods in the Generalization of Drinfeld Modules

We introduce and study a natural class of Anderson t- modules, called triangular t-modules, characterized by having Drinfeld modules as their $τ$-composition factors. They form a homologically meaningful generalization of Drinfeld modules and exhibit rich arithmetic structure.\smallskip We establish criteria for purity, strict and almost strict, and develop a reduction procedure that lowers the degrees of the defining biderivations. As a consequence, every almost strictly pure triangular t-module becomes strictly pure after a finite base extension. We then investigate morphisms and isogenies between triangular t-modules, provide a characterization of triangular isogenies, and describe the algebra of endomorphisms, including a criterion for commutativity. On the analytic side, we show that all triangular t- modules are uniformizable and establish finiteness and purity criteria with consequences for Taelman's conjecture. Finally, we develop a duality theory for triangular t- modules and their biderivations, proving compatibility with $τ$-composition series and establishing analogues of the Cartier-Nishi theorem and the Weil-Barsotti formula.

math.NT↗

arXiv2026

Algorithms for determination of t-module structures on some extension groups

In \cite{kk04} the second and third author extended the methods of \cite{pr} and determined the \tm module structure on $\Ext^1(Φ,Ψ)$ where $Φ$ and $Ψ$ were Anderson \tm modules over $A={\mathbf F}_q[t]$ of some specific types. This approach involved the concept of biderivation and certain reduction algorithm. In this paper we generalize the results of \cite{pr} and \cite{kk04} and present complete algorithm for computation of \tm module structure on $\Ext^1(Φ,Ψ)$ for \tm modules $Φ$ and $Ψ$ such that ${\mathrm{deg}}Φ> {\mathrm{deg}} Ψ.$ The last condition is not sufficient for our algorithm to be executable. We show that it can be applied when the matrix at the biggest power of $τ$ in $Φ_t$ is invertible. We also introduce a notion of $τ$-composition series which we find suitable for the additive category of \tm modules and show that under certain assumptions on the composition series of $Φ$ and $Ψ$ our algorithm is also executable.

math.NT↗

arXiv2025

Algorithmic Construction of Real Hyperfields from Minimal Axioms

We study real hyperfields, focusing in particular on those that are finite with cyclic positive cones. All real hyperfields have characteristic zero, although they can still be classified using the C-characteristic, an invariant that captures essential structural information. We present an algorithm to determine all such hyperfields up to isomorphism and compute their C-characteristic. The algorithm is optimal in the sense that the set of axioms used is minimal. We develop and implement this algorithm in software, enabling a complete classification of finite real hyperfields with cyclic positive cones of order up to 15, as well as identification of the C-characteristic that occur in such hyperfields of order up to 17. Restricting attention to finite hyperfields of cyclic positive cones enables substantial simplification of the algorithm, thereby enhancing its computational efficiency and allowing for the rapid generation of hyperfields of large order. Using a criterion that allows us to determine whether a given finite real hyperfield is a Krasner quotient hyperfield, we obtain many new examples of hyperfields that do not arise from Krasner's quotient construction.

math.RA↗

arXiv2025

Weil-Barsotti formula for $\mathbf{T}$-modules

In the work of M. A. Papanikolas and N. Ramachandran [A Weil-Barsotti formula for Drinfeld modules, Journal of Number Theory 98, (2003), 407-431] the Weil-Barsotti formula for the function field case concerning $\Ext_τ^1(E,C)$ where $E$ is a Drinfeld module and $C$ is the Carlitz module was proved. We generalize this formula to the case where $E$ is a strictly pure \tm module $Φ$ with the zero nilpotent matrix $N_Φ.$ For such a \tm module $Φ$ we explicitly compute its dual \tm module $Φ^{\vee}$ as well as its double dual $Φ^{{\vee}{\vee}}.$ This computation is done in a a subtle way by combination of the \tm reduction algorithm developed by F. Głoch, D.E. K{\k e}dzierski, P. Kraso{ń} [ Algorithms for determination of \tm module structures on some extension groups , arXiv:2408.08207] and the methods of the work of D.E. K{\k e}dzierski and P. Kraso{ń} [On $\Ext^1$ for Drinfeld modules, Journal of Number Theory 256 (2024) 97-135]. We also give a counterexample to the Weil-Barsotti formula if the nilpotent matrix $N_Φ$ is non-zero.

math.NT↗