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. In the proof, we introduce a non-commutative von Staudt construction 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.
We provide a new foundation for combinatorial commutative algebra and Stanley-Reisner theory using the partition complex introduced in [Adi18]. One of the main advantages is that it is entirely self-contained, using only a minimal knowledge of algebra and topology. On the other hand, we also develop new techniques and results using this approach. In particular, we provide 1. A novel, self-contained method of establishing Reisner's theorem and Schenzel's formula for Buchsbaum complexes. 2. A simple new way to establish Poincaré duality for face rings of manifolds, in much greater generality and precision than previous treatments. 3. A "master-theorem" to generalize several previous results concerning the Lefschetz theorem on subdivisions. 4. Proof for a conjecture of K ühnel concerning triangulated manifolds with boundary.
This paper compares skew-linear and multilinear matroid representations. These are matroids that are representable over division rings and (roughly speaking) invertible matrices, respectively. The main tool is the von Staudt construction, by which we translate our problems to algebra. After giving an exposition of a simple variant of the von Staudt construction we present the following results:• Undecidability of several matroid representation problems over division rings.• An example of a matroid with an infinite multilinear characteristic set, but which is not multilinear in characteristic 0.• An example of a skew-linear matroid that is not multilinear.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.