Small Groebner Fans of Ideals of Points

TL;DR

This paper investigates classes of point ideals in polynomial rings with unique or multiple reduced Groebner bases, providing methods to construct such ideals and analyze their Groebner fans, with applications in modeling biological systems.

Contribution

It identifies classes of ideals with unique Groebner bases and develops methods to construct them, also exploring ideals sharing the same number of Groebner bases.

Findings

Identified classes of ideals with unique reduced Groebner bases.

Developed methodologies for constructing ideals with specified Groebner fan properties.

Analyzed pairs of ideals with identical numbers of reduced Groebner bases.

Abstract

In the context of modeling biological systems, it is of interest to generate ideals of points with a unique reduced Groebner basis, and the first main goal of this paper is to identify classes of ideals in polynomial rings which share this property. Moreover, we provide methodologies for constructing such ideals. We then relax the condition of uniqueness. The second and most relevant topic discussed here is to consider and identify pairs of ideals with the same number of reduced Groebner bases, that is, with the same cardinality of their associated Groebner fan.