Effective Lojasiewicz gradient inequality for Nash functions with application to finite determinacy of germs
type: Item
instance of: scholarly article
title: Effective Lojasiewicz gradient inequality for Nash functions with application to finite determinacy of germs
zbMATH Open document ID: 1464.14061
author: Beata Osińska-Ulrych
author: Grzegorz Skalski
author: Stanisław Spodzieja
published in: Journal of the Mathematical Society of Japan
publication date: 2021-03-03T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jmsj/1606813225
review text: The main result is an effective estimation of the best exponent \(\rho \in \lbrack 0,1)\) in the Lojasiewicz gradient inequality \[ \left\vert \nabla f(x)\right\vert \geq C\left\vert f(x)\right\vert ^{\rho } \] in a neighbourhood of \(a\in U\subset \mathbb{R}^{n}\) for some constant \(C>0,\) for a Nash function \(f:U\rightarrow \mathbb{R},\) where \(f(a)=0.\) Namely, if \(d\) is the degree of a nonzero polynomial \(P\) such that \( P(x,f(x))=0,\) \(x\in U,\) then the above inequality holds for \[ \rho =1-\frac{1}{2(2d-1)^{3n+1}}. \] This is a generalization of the \textit{P. Solernó} [Appl. Algebra Eng. Commun. Comput. 2, No. 1, 1--14 (1991; Zbl 0754.14035)] and \textit{D. D'Acunto} and \textit{K. Kurdyka} [Ann. Pol. Math. 87, 51--61 (2005; Zbl 1093.32011)] type estimation for polynomials. As a corollary the authors obtain an estimation of the degree of sufficiency of non-isolated Nash function singularities.
Mathematics Subject Classification ID: 14P20
Mathematics Subject Classification ID: 14R99
zbMATH DE Number: 7317371
zbMATH Keywords: semialgebraic function
zbMATH Keywords: Nash function
zbMATH Keywords: Lojasiewicz gradient inequality
zbMATH Keywords: Lojasiewicz exponent
reviewed by: Tadeusz Krasiński
MaRDI profile type: Publication
OpenAlex ID: W3108974944
cites work: Sur Les Germes D'Applications Differentiables a Singularites Isolees
cites work: Q5625979
cites work: Sur les exposants de Lojasiewicz
cites work: The theorem of Kuiper-Kuo-Bochnak-Lojasiewicz at infinity
cites work: On C0-Sufficiency of Complex Jets
cites work: Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials
cites work: BOUNDS FOR GRADIENT TRAJECTORIES AND GEODESIC DIAMETER OF REAL ALGEBRAIC SETS
cites work: Topological triviality of deformations of functions and Newton filtrations
cites work: The modified analytic trivialization of family of real analytic functions
cites work: Multiplicities of Pfaffian intersections, and the Łojasiewicz inequality
cites work: Finite determinacy relative to closed and finitely generated ideals
cites work: The Łojasiewicz exponent of an analytic function at an isolated zero
cites work: An effective Łojasiewicz inequality for real polynomials
cites work: Q5646152
cites work: On \(C^0\)-sufficiency of jets of potential functions
cites work: Proof of the gradient conjecture of R. Thom.
cites work: Q5267610
cites work: Separation of real algebraic sets and the Łojasiewicz exponent
cites work: Convexifying Positive Polynomials and Sums of Squares Approximation
cites work: Metric properties of semialgebraic mappings
cites work: Finite Determination on Algebraic Sets
cites work: Integral closure of ideals and equisingularity
cites work: Q3744948
cites work: Q4001976
cites work: Invariants polaires des courbes planes
cites work: Finite determinacy of non-isolated singularities
cites work: Q4128900
cites work: Q3192358
cites work: Finite Determinacy of Functions with Non-Isolated Singularities
cites work: An explicit bound for the Łojasiewicz exponent of real polynomials
cites work: Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds
cites work: Effective formulas for the local Łojasiewicz exponent
cites work: Łojasiewicz exponent near the fibre of a mapping
cites work: Equivalence of mappings at infinity
cites work: Singularities with critical locus a 1-dimensional complete intersection and transversal type \(A_ 1\)
cites work: On analytic equivalence of functions at infinity
cites work: Effective Łojasiewicz inequalities in semialgebraic geometry
cites work: The Łojasiewicz exponent of subanalytic sets
cites work: Variétés polaires. I: Invariants polaires des singularites d'hypersurfaces
cites work: \(C^{0}\)-sufficiency, Kuiper-Kuo and Thom conditions for non-isolated singularity
DOI: 10.2969/JMSJ/83378337
description: scientific article; zbMATH DE number 7317371