Representability of matroids by c-arrangements is undecidable

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 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.