Markov bases and generalized Lawrence liftings

TL;DR

This paper characterizes when minimal Markov bases are contained within the universal Gröbner and Graver bases, and studies their properties in generalized Lawrence liftings for arbitrary matrices, revealing that their complexity is often consistent.

Contribution

It provides necessary and sufficient conditions for minimal Markov bases to lie within key algebraic bases and analyzes their properties in generalized Lawrence liftings for arbitrary matrices.

Findings

Conditions for Markov bases to be in universal Gröbner basis

Conditions for Markov bases to be in Graver basis

Complexity of Markov bases is consistent in generalized Lawrence liftings

Abstract

Minimal Markov bases of configurations of integer vectors correspond to minimal binomial generating sets of the assocciated lattice ideal. We give necessary and sufficient conditions for the elements of a minimal Markov basis to be (a) inside the universal Gr{\" o}bner basis and (b) inside the Graver basis. We study properties of Markov bases of generalized Lawrence liftings for arbitrary matrices $A\in\mathcal{M}_{m\times n}(\Bbb{Z})$ and $B\in\mathcal{M}_{p\times n}(\Bbb{Z})$ and show that in cases of interest the {\em complexity} of any two Markov bases is the same.