Cl\^oture int\'egrale des id\'eaux et \'equisingularit\'e

TL;DR

This paper revisits classical results on integral dependence in complex analytic geometry, connecting asymptotic order functions with Lojasiewicz exponents and exploring algebraic finiteness properties, supplemented by recent survey complements.

Contribution

It provides a detailed analysis of integral dependence and asymptotic order functions, linking them to Lojasiewicz exponents, and includes recent developments in the field.

Findings

Connection between asymptotic order functions and Lojasiewicz exponents.

Finiteness properties of graded algebras associated with ideals.

Survey of recent results related to classical integral dependence theory.

Abstract

This text has two parts; the first is the essentially unmodified text of the 1973-74 seminar of M. Lejeune-Jalabert and B. Teissier on integral dependence in complex analytic geometry with J-J. Risler's appendix on the Lojasiewicz exponents in the real analytic framework. The second part consists of seven complements written in 2007 surveying recent results directly connected to the content of the seminar. The main results of the first part concern the asymptotic order function with respect to an ideal and in particular its connection with the Lojasiewicz exponent. Another aspect concerns the finiteness properties of the graded algebra associated with the filtration by the asymptotic order function.