Maximal generating degrees of powers of homogeneous ideals

TL;DR

This paper investigates the degree excess function of powers of homogeneous ideals, showing its flexibility, providing bounds for monomial ideals, and relating it to the stability index of Castelnuovo-Mumford regularity.

Contribution

It demonstrates that any non-increasing numerical function can be realized as a degree excess function and establishes bounds and properties for monomial ideals.

Findings

Any non-increasing numerical function can be realized as a degree excess function.

An upper bound on the degree excess function for monomial ideals is provided.

The stability index of Castelnuovo-Mumford regularity can grow exponentially with the number of variables.

Abstract

The degree excess function $\epsilon(I;n)$ is the difference between the maximal generating degree $d(I^n)$ of a homogeneous ideal $I$ of a polynomial ring and $p(I)n$, where $p(I)$ is the leading coefficient of the asymptotically linear function $d(I^n)$. It is shown that any non-increasing numerical function can be realized as a degree excess function, and there is a monomial ideal $I$ whose $\epsilon(I;n)$ has exactly a given number of local maxima. In the case of monomial ideals, an upper bound on $\epsilon(I;n)$ is provided. As an application it is shown that in the worst case, the so-called stability index of the Castelnuovo-Mumford regularity of a monomial ideal $I$ must be at least an exponential function of the number of variables.