Lower Bounds for the Number of Generic Initial Ideals

TL;DR

This paper establishes a linear lower bound on the number of generic initial ideals for certain graded ideals in three variables, showing that this complexity grows at least linearly with the degree of generators.

Contribution

It provides explicit examples of ideals with a linear lower bound on the number of generic initial ideals, and proves this bound applies broadly to generic ideals in three variables.

Findings

Number of generic initial ideals can grow at least linearly with degree d.

Constructs explicit family of ideals demonstrating this lower bound.

The bound applies to all generic graded ideals in three variables.

Abstract

Given a graded ideal $I$ in a polynomial ring over a field $K$ it is well known, that the number of distinct generic initial ideals of $I$ is finite. While it is known that for a given $d\in\N$ there is a global upper bound for the number of generic initial ideals of ideals generated in degree less than $d$, it is not clear how this bound has to grow with $d$. In this note we will explicitly give a family $(I(d))_{d\in\N}$ of ideals in $S=K[x,y,z]$, such that $I(d)$ is generated in degree $d$ and the number of generic initial ideals of $I(d)$ is bounded from below by a linear bound in $d$. Moreover, this bound holds for all graded ideals in $S$, which are generic in an appropriate sense.