Radical support for multigraded ideals

TL;DR

This paper introduces the concept of radical support in multigraded polynomial rings, providing a combinatorial characterization and linking it to Cartwright-Sturmfels ideals, thus advancing understanding of radical ideals.

Contribution

It defines radical support and characterizes it combinatorially, establishing a connection with Cartwright-Sturmfels ideals in multigraded settings.

Findings

Characterization of radical supports via labelled graph cycles

Radical supports correspond exactly to Cartwright-Sturmfels ideals

Every ideal generated by degrees forming a radical support is Cartwright-Sturmfels

Abstract

Can one tell if an ideal is radical just by looking at the degrees of the generators? In general, this is hopeless. However, there are special collections of degrees in multigraded polynomial rings, with the property that any multigraded ideal generated by elements of those degrees is radical. We call such a collection of degrees a radical support. In this paper, we give a combinatorial characterization of radical supports. Our characterization is in terms of properties of cycles in an associated labelled graph. We also show that the notion of radical support is closely related to that of Cartwright-Sturmfels ideals. In fact, any ideal generated by multigraded generators whose multidegrees form a radical support is a Cartwright-Sturmfels ideal. Conversely, a collection of degrees such that any multigraded ideal generated by elements of those degrees is Cartwright-Sturmfels is a radical support.