Toric ideals and their circuits

TL;DR

This paper investigates the generation of toric ideals by circuits, providing conditions under which certain classes of toric ideals, including those from graphs and specific algebraic structures, are generated by circuits.

Contribution

It offers a new sufficient condition for toric ideals to be generated by circuits, especially for those with squarefree quadratic initial ideals, and characterizes graph classes with this property.

Findings

Toric ideals with squarefree quadratic initial ideals can be generated by circuits under certain conditions.

Squarefree Veronese subrings and root system configurations satisfy the circuit generation condition.

Identifies classes of graphs whose toric ideals are generated by circuits and are nonnormal.

Abstract

In this paper, we study toric ideals generated by circuits. For toric ideals which have squarefree quadratic initial ideals, a sufficient condition to be generated by circuits is given. In particular, squarefree Veronese subrings, the second Veronese subrings and configurations arising from root systems satisfy the condition. In addition, we study toric ideals of finite graphs and characterize the graphs whose toric ideals are generated by circuits u -v such that either u or v is squarefree. There exists several classes of graphs whose toric ideals satisfy this condition and whose toric rings are nonnormal.