Asymptotic behavior of Markov complexity of matrices

TL;DR

This paper investigates the asymptotic properties of Markov complexity in matrices, revealing conditions under which complexities grow arbitrarily large or remain bounded based on matrix parameters.

Contribution

It introduces a matroid-based framework to analyze Markov complexity and establishes bounds related to matrix size and entries, advancing understanding of complexity behavior.

Findings

Markov and Graver complexities can be arbitrarily large for certain matrices.

Bouquets behave well under Lawrence liftings, aiding complexity analysis.

Complexities are bounded by matrix size and maximum entry value.

Abstract

To any integer matrix $A$ one can associate a matroid structure consisting of a graph and another integer matrix $A_B$. The connected components of this graph are called bouquets. We prove that bouquets behave well with respect to the $r$--th Lawrence liftings of matrices and we use it to prove that the Markov and Graver complexities of $m\times n$ matrices of rank $d$ may be arbitrarily large for $n\geq 4$ and $d\leq n-2$. In contrast, we show they are bounded in terms of $n$ and the largest absolute value $a$ of any entry of $A$.