Resolution of singularities and geometric proofsof the Łojasiewicz inequalities

2017/08/31 by Paul M. N. Feehan, Paul Feehan · 3 citations
Computer Science · Mathematics · #Advanced Banach Space Theory #Analytic function #Euclidean space #Function (biology) #Gravitational singularity #Holomorphic and Operator Theory #Mathematical proof #Optimization and Variational Analysis #Resolution (logic) #Resolution of singularities #Simple (philosophy) #math.AG #math.AP #math.DG

paper · pdf · doi:10.2140/gt.2019.23.3273

published as Geom. Topol. 23 (2019) 3273-3313 · 28 pages, incorporating final galley proof corrections

openalex created_date 2017/09/15 · openalex publication_date 2019/12/30 · arxiv created 2020/01/06 · arxiv updated 2020/01/08 · openalex updated_date 2026/08/05

Abstract

The Łojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanisław Łojasiewicz (1959, 1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). Here we first give an elementary geometric, coordinate-based proof of the Łojasiewicz inequalities in the special case where the function is [math] with simple normal crossings. We then prove, partly following Bierstone and Milman (1997) and using resolution of singularities for (real or complex) analytic varieties, that the gradient inequality for an arbitrary analytic function follows from the special case where it has simple normal crossings. In addition, we prove the Łojasiewicz inequalities when a function is [math] and generalized Morse–Bott of order [math] ; we earlier gave an elementary proof of the Łojasiewicz inequalities when a function is [math] and Morse–Bott on a Banach space.

Citations

• Optimal Lojasiewicz-Simon inequalities and Morse-Bott Yang-Mills energy functions
• Coherent Analytic Sheaves
• Positivity in Algebraic Geometry I
• Theorems on Regularity and Singularity of Energy Minimizing Maps
• Simple Hironaka resolution in characteristic zero
• The Hironaka theorem on resolution of singularities (Or: A proof we always wanted to understand)
• Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I
• Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant
• The Łojasiewicz exponent of a set of weighted homogeneous ideals
• Good points and constructive resolution of singluarities
• The Stable Homotopy of the Classical Groups
• Sheaves on Manifolds
• Algebraic Geometry

Cited by

• Constructing Discontinuous but Locally Bounded Rational Functions using Łojasiewicz Inequalities
• On the Morse-Bott property of analytic functions on Banach spaces with Lojasiewicz exponent one half
• Emergence of Computational Structure in a Neural Network Physics Simulator

Related

• Lojasiewicz inequalities and applications
• Some resonances of Lojasiewicz inequalities
• Characterizations of Lojasiewicz inequalities and applications
• Lojasiewicz inequalities in a certain class of smooth functions
• A convex function satisfying the Lojasiewicz inequality but failing the gradient conjecture both at zero and infinity