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arXiv · 1912.06123

Representability of Matroids by c-Arrangements is Undecidable

Lukas Kühne·Geva Yashfe
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Abstract

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 network coding problems are also undecidable. In the proof, we introduce a generalized Dowling geometry to encode an instance of the uniform word problem for finite groups in matroids of rank three. The $c$-arrangement condition gives rise to some difficulties and their resolution is the main part of the paper.

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Lukas Kühne, Geva Yashfe. 2020-06-09. Representability of Matroids by c-Arrangements is Undecidable. https://doi.org/10.1007/s11856-022-2345-z

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