Von Staudt Constructions for Skew-Linear and Multilinear Matroids

Abstract

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.

Incidence geometry in the projective plane via almost-principal minors of symmetric matrices

“…This last affine operation can be performed by the projective ruler using the line at infinity not pictured here. Full descriptions of these classical constructions are given in [RG11, Section 5.6] and with emphasis on matroids (over skew fields) in [KPY20]. Here we give the algorithms with indeterminates x = [x : 0 : 1] and y = [y : 0 : 1] using cross products and using the same notation as in Figure 2:…”

“…Future work. Given the encoding of projective plane incidence relations over fields presented here, a natural direction for future work concerns skew fields with the same ambitions as in [KPY20]. The more fine-grained universality theorems mentioned in the introduction are an attractive target as well.…”

On entropic and almost multilinear representability of matroids

“…6.8.9]. Generalized matrix representability over a division ring and multilinear representability are also undecidable [KPY20,KY19].…”

“…This work was extended by Ben-Efraim and Matúš to entropic matroids [MBE20] building on Matúš' earlier work on these matroids in [Mat99]. We previously used generalized Dowling geometries to prove that the representability problem of multilinear matroids is undecidable [KY19] and, with Rudi Pendavingh, we used more general von Staudt constructions to compare the multilinear matroid representations with representations over division rings [KPY20]. Almost entropic matroids featured prominently in Matúš' recent article where he proved that algebraic matroids are almost entropic [Mat19].…”

“…We call the resulting matroids generalized Dowling geometries (GDG). We first used them in [KY19,KPY20]. They are special cases of frame matroids as studied by Zaslavsky in [Zas03].…”