Some resonances of Lojasiewicz inequalities

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.