Characteristic Sets of Matroids

TL;DR

This paper explores the relationships between linear, algebraic, and Frobenius flock characteristic sets of matroids, classifying their possible combinations and densities, and constructing realizations across different characteristics.

Contribution

It classifies the possible combinations of characteristic sets for matroids and demonstrates the density and realization properties of these sets in various contexts.

Findings

Algebraic characteristic sets can be cofinite or finite.

The density of algebraic characteristic sets can approximate any real number in [0,1].

Frobenius flock realizations can be derived from algebraic realizations, but not vice versa.

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.