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Characteristic Sets of Matroids

Dustin Cartwright, Dony Varghese

Abstract

We investigate possible linear, algebraic, and Frobenius flock characteristic sets of matroids. In particular, we classify possible combinations of linear and algebraic characteristic sets when the algebraic characteristic set is finite or cofinite. We also show that the natural density of an algebraic characteristic set in the set of primes may be arbitrarily close to any real number in the interval $[0,1]$. Frobenius flock realizations can be constructed from algebraic realizations, but the converse is not true. We show that the algebraic characteristic set may be an arbitrary cofinite set even for matroids whose Frobenius flock characteristic set is the set of all primes. In addition, we construct Frobenius flock realizations in all positive characteristics from linear realizations in characteristic 0, and also from Frobenius flock realizations of the dual matroid.

Characteristic Sets of Matroids

Abstract

We investigate possible linear, algebraic, and Frobenius flock characteristic sets of matroids. In particular, we classify possible combinations of linear and algebraic characteristic sets when the algebraic characteristic set is finite or cofinite. We also show that the natural density of an algebraic characteristic set in the set of primes may be arbitrarily close to any real number in the interval . Frobenius flock realizations can be constructed from algebraic realizations, but the converse is not true. We show that the algebraic characteristic set may be an arbitrary cofinite set even for matroids whose Frobenius flock characteristic set is the set of all primes. In addition, we construct Frobenius flock realizations in all positive characteristics from linear realizations in characteristic 0, and also from Frobenius flock realizations of the dual matroid.
Paper Structure (5 sections, 17 theorems, 30 equations)

This paper contains 5 sections, 17 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Specified characteristic sets
  3. Infinite algebraic characteristic sets
  4. Stretching Frobenius flocks
  5. Finite Frobenius flock characteristic sets

Key Result

Theorem 1

Let $C_{L}\subseteq C_{A}\subseteq\mathbb{P}\cup\left\{ 0\right\}$ be finite or cofinite subsets. Suppose either that $0\in C_{L}$ and $C_{L}$ is cofinite, or that $0\notin C_{A}$ and $C_{L}$ is finite. Then there exists a matroid $M$ such that $\chi_{L}(M)=C_{L}$ and $\chi_{A}(M)=C_{A}$.

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 7: Lem. 3.4.1 in EvansHrushovski
  • Lemma 8
  • proof
  • Proposition 9
  • ...and 23 more