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Representability of matroids by c-arrangements is undecidable

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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.

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We would like to thank Karim Adiprasito for his mentorship and for introducing us to the topic of c-arrangements. We are grateful to Rudi Pendavingh for helpful conversations on von Staudt constructions. Our application of them is inspired by joint work with him in [KPY20]. We would also like to thank Eran Nevo, and other participants of the 2020–21 mathematical writing workshop at the Hebrew University, for helpful comments. Lastly, we thank the anonymous referee for carefully reading an earlier version of this article and for giving many suggestions which significantly improved the paper.

An extended abstract of this paper appeared as “Undecidability of c-Arrangement Matroid Representations” in a proceedings volume of Séminaire Lotharingien de Combinatoire, among extended abstracts from the 2020 FPSAC conference [KY20].

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Correspondence to Lukas Kühne.

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L.K. was supported by a Minerva fellowship of the Max-Planck-Society, the Studienstiftung des deutschen Volkes and by ERC StG 716424 - CASe.

G.Y. was supported by ISF grant 1050/16.

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Kühne, L., Yashfe, G. Representability of matroids by c-arrangements is undecidable. Isr. J. Math. 252, 95–147 (2022).

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