On some properties of the asymptotic Samuel function

TL;DR

This paper investigates properties of the asymptotic Samuel function in excellent, equidimensional rings with a focus on the Samuel slope, an invariant linked to singularity resolution, and examines its behavior under flat extensions.

Contribution

It extends understanding of the asymptotic Samuel function and Samuel slope in complex algebraic structures, including their behavior under flat extensions.

Findings

Properties of the asymptotic Samuel function in specific rings are characterized.

The behavior of the Samuel slope under certain extensions is analyzed.

Results contribute to the understanding of singularity resolution algorithms.

Abstract

The asymptotic Samuel function generalizes to arbitrary rings the usual order function of a regular local ring. Here we explore some natural properties in the context of excellent, equidimensional rings containing a field. In addition, we establish some results regarding the Samuel slope of a local ring. This is an invariant related with algorithmic resolution of singularities of algebraic varieties. Among other results, we study its behavior after certain faithfully flat extensions.