Some resonances of Lojasiewicz inequalities

TL;DR

This paper explores three interconnected aspects of Lojasiewicz inequalities, including geometric interpretations, limitations of rational exponents, and extensions to infinite-dimensional spaces.

Contribution

It provides new insights into the geometric interpretation of Lojasiewicz exponents, highlights restrictions on rational exponents, and discusses recent advances extending the inequality to infinite-dimensional contexts.

Findings

Interpretation of exponents via Newton polygons in complex geometry

Certain rational numbers cannot be Lojasiewicz exponents for gradients

Recent results suggest possible Lojasiewicz inequalities in infinite-dimensional spaces

Abstract

This note presents three resonances in commutative algebra and analytic geometry of the concept of Lojasiewicz inequality. The first is the interpretation in complex analytic geometry of the best possible exponent for a function g with respect to an ideal I at a point of a reduced complex space X as the inclination of a edge of a Newton polygon associated to the dicritical components of as log resolution of I. The second calls attention to recent results which show that some rational numbers connot be Lojasiewicz exponents for the gradient inequality of a holomorphic function of two variables. The last one reports on a recent result of Moret-Bailly which opens perspectives for a Lojasiewicz inequality in infinite dimensional spaces.