Resolution of singularities and geometric proofs of the Lojasiewicz inequalities

Feehan, Paul M. N. (2017) Resolution of singularities and geometric proofs of the Lojasiewicz inequalities. (Preprint)

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Abstract

The Łojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanisław Łojasiewicz in [83, 84, 87] using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman [7]. In this article, we first give an elementary geometric, coordinate-based proof of the Łojasiewicz inequalities in the special case where the function is C^1 with simple normal crossings. We then prove, partly following Bierstone and Milman [9, Section 2] and using resolution of singularities for real analytic varieties, that the gradient inequality for an arbitrary real analytic function follows from the special case where it has simple normal crossings. In addition, we give elementary proofs of the the Łojasiewicz inequalities when the function is C^2 and Morse–Bott or C^N and Morse–Bott of order N ≥ 2.

Item Type: Article
Divisions: School of Theoretical Physics > Preprints
Date Deposited: 19 Jun 2018 13:50
Last Modified: 12 Jul 2018 10:13
URI: http://dair.dias.ie/id/eprint/555

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