Toric ideals generated by circuits

TL;DR

This paper characterizes when toric ideals of homogeneous normal configurations are generated by circuits, linking this property to the existence of connectors in unbalanced circuits, with applications to normal edge subrings of multigraphs.

Contribution

It provides a new criterion for toric ideals to be generated by circuits based on the properties of unbalanced circuits and connectors.

Findings

Toric ideal is generated by circuits if and only if each unbalanced circuit has a connector.

If all circuits with non-square-free terms are balanced, then the ideal is generated by circuits.

Normal edge subrings of multigraphs have toric ideals generated by circuits with square-free terms.

Abstract

Let I be the toric ideal of a homogeneous normal configuration. We prove that I is generated by circuits if and only if each unbalanced circuit of I has a "connector" which is a linear combination of circuits with a square-free term. In particular if each circuit of I with non-square-free terms is balanced, then I is generated by circuits. As a consequence we prove that the toric ideal of a normal edge subring of a multigraph is generated by circuits with a square-free term.