Asymptotic for the number of star operations on one-dimensional Noetherian domains

TL;DR

This paper investigates the asymptotic behavior of the number of specific star operations on one-dimensional local Noetherian domains, linking it to closure operations in finite field extensions.

Contribution

It reduces the problem to studying multiplicative closure operations in finite extensions and analyzes how their count varies with the field size.

Findings

Reduction to closure operations in finite extensions

Analysis of how the number of star operations varies with field size

Establishment of asymptotic behavior for the set of star operations

Abstract

We study the set of star operations on local Noetherian domains $D$ of dimension $1$ such that the conductor $(D:T)$ (where $T$ is the integral closure of $D$) is equal to the maximal ideal of $D$. We reduce this problem to the study of a class of closure operations (more precisely, multiplicative operations) in a finite extension $k\subseteq B$, where $k$ is a field, and then we study how the cardinality of this set of closures vary as the size of $k$ varies while the structure of $B$ remains fixed.