Representability of matroids by c-arrangements is undecidable

For a natural number c , a c -arrangement is an arrangement of dimension c subspaces satisfying the following condition: the sum of any subset of the subspaces has dimension a multiple of c . Matroids arising as normalized rank functions of c -arrangements are also known as multilinear matroids. We...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Israel journal of mathematics 2022-12, Vol.252 (1), p.95-147
Hauptverfasser: Kühne, Lukas, Yashfe, Geva
Format: Artikel
Sprache:eng
Schlagworte:
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung:For a natural number c , a c -arrangement is an arrangement of dimension c subspaces satisfying the following condition: the sum of any subset of the subspaces has dimension a multiple of c . Matroids arising as normalized rank functions of c -arrangements are also known as multilinear matroids. We prove that it is algorithmically undecidable whether there exists a c such that a given matroid has a c -arrangement representation, or equivalently whether the matroid is multilinear. It follows that certain problems on network coding and secret sharing schemes are also undecidable. In the proof, we encode group presentations in frame matroids of rank three which we call generalized Dowling geometries: the construction is inspired by Dowling geometries of finite groups and by the von Staudt construction. The idea is to construct a reduction from the uniform word problem for finite groups to multilinear representability of matroids. The c -arrangement condition gives rise to some difficulties and their resolution is the main part of the paper.
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-022-2345-z