Resolution of singularities and geometric proofs of the Lojasiewicz inequalities

TL;DR

This paper provides geometric and resolution of singularities-based proofs of the Lojasiewicz inequalities for real analytic functions, extending results to functions with simple normal crossings, Morse-Bott, and higher smoothness classes.

Contribution

It introduces an elementary geometric proof for $C^1$ functions with simple normal crossings and extends the inequalities to more general classes using resolution of singularities.

Findings

Elementary proof for $C^1$ functions with simple normal crossings.

Extension of inequalities to $C^N$ and Morse-Bott functions.

Resolution of singularities used to generalize gradient inequalities.

Abstract

The Lojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanislaw Lojasiewicz (1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). In this article, we first give an elementary geometric, coordinate-based proof of the Lojasiewicz inequalities in the special case where the function is $C^1$ with simple normal crossings. We then prove, partly following Bierstone and Milman (1997) and using resolution of singularities for real analytic varieties, that the gradient inequality for an arbitrary real or complex analytic function follows from the special case where it has simple normal crossings. In addition, we prove the Lojasiewicz inequalities when a function is $C^N$ and generalized Morse-Bott of order $N \geq 3$; we gave an elementary proof of the Lojasiewicz inequalities when a function is $C^2$ and Morse-Bott in arXiv:1708.09775v4 (finite-dimensional case) and arXiv:1706.09349 (infinite-dimensional case).