Distance Reducing Markov Bases

TL;DR

This paper investigates the properties of distance reducing Markov bases, providing characterizations for monomial curves, especially complete intersections, and analyzing the fundamental moves common to all such bases.

Contribution

It offers a complete characterization of distance reduction for monomial curves, utilizing gluings of numerical semigroups, and examines distance irreducible elements.

Findings

Characterization of distance reduction for complete intersection monomial curves

Analysis of distance reducing property for non-complete intersection monomial curves in small dimensions

Identification of moves appearing in all distance reducing Markov bases

Abstract

The distance reducing property for Markov bases is an important property that provides a bound on the mixing time of the associated Markov chain. The goal of this project is to understand properties of distance-reducing Markov bases. We explore the distance reducing property for monomial curves and give a complete characterisation of distance reduction in the case of complete intersection monomial curves. Our characterisation carefully uses the notion of gluings for numerical semigroups. We also characterise the distance reducing property for non-complete intersection monomial curves in small dimensions. We also explore the distance irreducible elements: the moves that appear in all distance reducing Markov bases.