Toric Ideals of Lattice Path Matroids and Polymatroids

TL;DR

This paper proves that the toric ideals of lattice path polymatroids and matroids are generated by quadratic binomials, and provides a Gr"obner basis under a specific monomial order, advancing algebraic understanding of these structures.

Contribution

It establishes that the toric ideals are generated by quadrics from symmetric exchanges and identifies a monomial order making these quadrics a Gr"obner basis, extending to lattice path matroids.

Findings

Toric ideals are generated by quadrics from symmetric exchanges.

A monomial order makes these quadrics form a Gr"obner basis.

Results apply to both lattice path polymatroids and matroids.

Abstract

We show that the toric ideal of a lattice path polymatroid is generated by quadrics corresponding to symmetric exchanges, and give a monomial order under which these quadrics form a Gr\"obner basis. We then obtain an analogous result for lattice path matroids.