Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Von Staudt Constructions for Skew-Linear and Multilinear Matroids

Lukas Kühne, Rudi Pendavingh, Geva Yashfe

TL;DR

This work analyzes the relationship between skew-linear matroids (representable over division rings) and multilinear matroids (representable via invertible matrices) through a refined von Staudt framework that translates polynomial equations into matroid circuits. The authors establish undecidability results for skew-linear representations, construct a matroid with an infinite multilinear characteristic set excluding $0$, and present a skew-linear matroid that is not multilinear, illustrating fundamental separations between the two representation notions. The approach blends projective-plane coordinatization, von Staudt-type constructions, and classical algebraic results (e.g., the Weyl algebra and Baumslag–Solitar groups) to demonstrate universality phenomena and boundaries of representability. The findings have implications for understanding representability problems in matroid theory and their connections to computational complexity and noncommutative algebra, with explicit constructions linking algebraic solvability to combinatorial realizability. Overall, the paper advances the theory of matroid representations by explicitly tying algebraic undecidability and characteristic-set behavior to skew-linear and multilinear frameworks.

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: $\bullet$ Undecidability of several matroid representation problems over division rings. $\bullet$ An example of a matroid with an infinite multilinear characteristic set, but which is not multilinear in characteristic $0$. $\bullet$ An example of a skew-linear matroid that is not multilinear.

Von Staudt Constructions for Skew-Linear and Multilinear Matroids

TL;DR

This work analyzes the relationship between skew-linear matroids (representable over division rings) and multilinear matroids (representable via invertible matrices) through a refined von Staudt framework that translates polynomial equations into matroid circuits. The authors establish undecidability results for skew-linear representations, construct a matroid with an infinite multilinear characteristic set excluding , and present a skew-linear matroid that is not multilinear, illustrating fundamental separations between the two representation notions. The approach blends projective-plane coordinatization, von Staudt-type constructions, and classical algebraic results (e.g., the Weyl algebra and Baumslag–Solitar groups) to demonstrate universality phenomena and boundaries of representability. The findings have implications for understanding representability problems in matroid theory and their connections to computational complexity and noncommutative algebra, with explicit constructions linking algebraic solvability to combinatorial realizability. Overall, the paper advances the theory of matroid representations by explicitly tying algebraic undecidability and characteristic-set behavior to skew-linear and multilinear frameworks.

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 . An example of a skew-linear matroid that is not multilinear.

Paper Structure

This paper contains 21 sections, 14 theorems, 36 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Related work
  4. History and applications of the von Staudt construction
  5. Outline of the paper
  6. Multilinear and skew-linear representations of matroids
  7. Projective equivalence of representations
  8. A brief review of the projective plane
  9. Notation
  10. Matroid representations and the projective plane
  11. Von Staudt constructions in matrices and division rings
  12. The geometric idea
  13. The construction
  14. Undecidability in division rings
  15. Macintyre's results
  16. ...and 6 more sections

Key Result

Lemma 2.4

[lemma]lem:proj_equiv If $\ell_{1}$ and $\ell_{2}$ are two lines of $D\mathbb{P}^{2}$ intersecting at a point $\overline{O}$, and $\overline{a},\overline{b}\in\ell_{1}$, $\overline{c},\overline{d}\in\ell_{2}$ are points distinct from each other and $\overline{O}$, there exists a projective transform respectively.

Figures (3)

  • Figure 1: Two diagrams motivating the following von Staudt constructions.
  • Figure 2: The two building blocks of the von Staudt matroid $M_{\mathcal{P}}$. The elements and circuits corresponding to the polynomial $P$ are depicted in blue. These pictures correspond to the ones shown in \ref{['fig:staudt_explanation']} after adding the line at infinity.
  • Figure 3: A geometric representation of a part of the Weyl matroid $M_W$.

Theorems & Definitions (36)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Remark 2.5
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • ...and 26 more