subject

math.RA

math.RA: explore 4 source-linked works published from 2026 to 2026, with original documents and citations.

This collection is a preview while coverage and quality are evaluated.

Coverage and selection

Includes records with this source-supplied label or an explicit phrase match in their metadata. Matches indicate a mention, not proof that a paper uses a method or tests a material. Source versions are consolidated by DOI.

Sources: arxiv. Collection updated 2026-09-15. Counts describe this index, not the complete source archives.

arXiv2026

Representation Redundancy and Structural Complexity in Finite-Field Inversion

The representation chosen for a mathematical operation can affect both its algebraic form and its empirical learning difficulty. We study this phenomenon for inversion over $\mathbb F_{2^n}$, with field elements expressed in varying ordered $\mathbb F_2$-bases. We prove that two ordered bases induce the same coordinate inversion map if and only if they belong to the same Galois orbit. Since every orbit has size $n$, the correspondence between ordered bases and distinct inversion maps is exactly $n$-to-one. We then analyze three Boolean formulations of inversion. The reference formulation has algebraic degree $n-1$ and joint ANF leap $1$, the mixed representation formulation has degree $2(n-1)$ and joint ANF leap $2$, and the complete raw formulation has degree at most $3(n-1)$ and joint ANF leap at least $n$. Exhaustive computations agree with the theoretical results and bounds in the cases considered. Controlled experiments with multilayer perceptrons show the same ordering in learning difficulty, while Galois orbit redundancy provides only a limited generalization benefit under the tested conditions. These results show that exact redundancy among representations can coexist with changes in Boolean structure and learning behavior when the representation is exposed as part of the input.

cs.LG↗

arXiv2026

Circular Chromatic Numbers, Signability, Relation Algebras, and Network Satisfaction Problems

In this paper, we characterize finite graphs with circular chromatic number less than 3 in terms of the existence of certain signings ($\mathbb Z_2$-labellings studied in the context of signed graphs). In fact, we construct a signed graph which is universal for all such signings -- called anti-triangle-signings in this paper -- of finite $\overline{K_3}$-free graphs, and is closely related to the generic circular triangle-free graph studied by Bodirsky and Guzmán-Pro. Moreover, our universal structure gives rise to a representation of the relation algebra $56_{65}$. We then use this representation to show that the network satisfaction problem described by this relation algebra belongs to NP. This concludes the full classification of the existence of a universal square representation, as well as the complexity of the corresponding network satisfaction problem, for relation algebras with at most four atoms.

math.CO↗

arXiv2026

An Algebraic Framework for Data Systems: Classification and Optimization of Algebraic Structures for Data Organization and Coding

Modern data systems are commonly studied through computational and information-theoretic methods, while their algebraic properties remain largely unexplored. This paper introduces a mathematical framework for modelling data systems using algebraic structures drawn from group theory and coding theory. The central result is an axiomatic classifier: the coding-theoretic capability of a finite algebraic structure (group, ring, or field) is shown to be determined by its axiom signature (the set of algebraic axioms it satisfies), with each axiom acting as a gate that enables a specific capability (inverses enable the algebraic Hamming metric; commutativity enables syndrome decoding via quotient groups; field structure enables MDS codes via polynomial evaluation). Three types of results are presented. Proved: the axiomatic classification theorem; that ACID transactions form a monoid under sequential composition (not a group); that schema-preserving transformations form a finite group whose orbits, counted by Burnside's lemma, yield exact deduplication of equivalent configurations; and that every integrity-preserving bijection of a data system defines an algebraizable equivalence class under a finite group action. Demonstrated: explicit code constructions over $\mathbb{F}_5$, $\mathbb{Z}_4$ (yielding codes inaccessible to classical $\mathbb{F}_q$-linear theory), and $(2^U, \triangle)$ (yielding group-theoretic anomaly detection for set-valued data). Proposed: an encoding efficiency measure Ef and an optimization functional $Φ$ with tunable weights, whose induced ranking is verified computationally to be consistent with the axiomatic classification for the structures studied.

math.RA↗

arXiv2026

An algebraic proof of Colombo's difference-power determinant conjecture

Let $n\ge2$ be even, let $λ=(λ_1,\ldots,λ_n)\in\mathbb{R}^n$ have pairwise distinct coordinates, and define the difference-power matrix \[ A_d(λ) := \bigl[(λ_r-λ_s)^d\bigr]_{r,s=1}^n, \qquad d\in\mathbb{N}. \] In 1928, Colombo proved that $\det A_{n-1}(λ)\ne0$---and hence $\det A_{n-1}(λ)>0$---and that $\operatorname{rank} A_d(λ)=d+1$ for $0\le d<n-1$. He conjectured that \[ \det A_d(λ)\ne0 \qquad\text{for every } d\ge n-1. \] For even $d$, the conjectured nonsingularity follows from previously published results on distance-power matrices. The remaining open cases were therefore the supercritical odd exponents $d\ge n+1$. We prove nonsingularity for all these odd exponents, thereby completing Colombo's conjecture. Consequently, \[ \operatorname{rank} A_d(λ)=\min\{n,d+1\} \qquad(d\in\mathbb{N}). \] Our proof converts a hypothetical kernel vector into a real binary form having more projective real linear factors, counted with multiplicity, than its real Waring length permits.

cs.LG↗

Compare source metadata on this page

Work	Published	Source identifier	Source
Representation Redundancy and Structural Complexity in Finite-Field Inversion	2026-09-04	2609.04583	arxiv
Circular Chromatic Numbers, Signability, Relation Algebras, and Network Satisfaction Problems	2026-09-02	2512.06878	arxiv
An Algebraic Framework for Data Systems: Classification and Optimization of Algebraic Structures for Data Organization and Coding	2026-08-31	2609.00273	arxiv
An algebraic proof of Colombo's difference-power determinant conjecture	2026-08-28	2608.28274	arxiv

These are bibliographic comparisons, not experimental rankings. Follow the original document for methods and conditions.