Robust Toric Ideals

TL;DR

This paper characterizes robust toric ideals, especially those generated by quadrics, showing they are mainly determinantal, and explores their generalizations and properties related to higher Betti numbers.

Contribution

It provides a classification of robust toric ideals generated by quadrics as determinantal and discusses extensions to higher degrees and Betti numbers.

Findings

Robust toric ideals generated by quadrics are essentially determinantal.

A classification for determinantal ideals is established.

Counterexample provided for a natural extension to Lawrence ideals.

Abstract

We call an ideal in a polynomial ring robust if it can be minimally generated by a universal Gr\"obner basis. In this paper we show that robust toric ideals generated by quadrics are essentially determinantal. We then discuss two possible generalizations to higher degree, providing a tight classification for determinantal ideals, and a counterexample to a natural extension for Lawrence ideals. We close with a discussion of robustness of higher Betti numbers.