On gradients of functions definable in o-minimal structures. (English) Zbl 0934.32009

Kurdyka, Krzysztof

Ann. Inst. Fourier 48, No. 3, 769-783 (1998).

The well known Lojasiewicz inequality \((\|\text{grad} f\|>f^\alpha\), \(\alpha<1\), in short \(L\)-inequality) concerns real analytic functions \(f\) in a neighborhood of a point \(a\in R^n\), \(f(a)=0\). A trajectory of the vector field \(-\text{grad} f\) is defined as a maximal differentiable curve \(\gamma\) verifying \(\gamma'(t)=-\text{grad} f(\gamma(s))\).
Lojasiewicz proved that all trajectories of \(-\text{grad} f\) are of finite length, when \(f\) is real analytic in a neighborhood of a compact \(U\).
The notion of “\(o\)-minimal structure on the real field” describes abstractly [see L. Van den Dries and C. Miller, Duke Math. J. 84, No. 2, 497-540 (1996; Zbl 0889.03025)] different kinds of geometric categories of sets which appear in semialgebraic and subanalytic geometries. For instance \((R\), exp)-definable sets define a “\(o\)-minimal structure” (theorem of Wilkie).
Actually the generalizations of the \(L\)-inequality for “\(o\)-minimal structures” is of great interest. In this paper a new \(o\)-minimal version of the \(L\)-inequality is obtained. A study of all trajectories of \(-\text{grad} f\) is given too, proving the theorem that the length of mentioned trajectories is bounded by a constant independent of the trajectory. Another interesting author’s result says that the flow of \(-\text{grad} g\) with a non negative definable \(g\), determines a deformation retract onto \(g^{-1}(0)\).

Reviewer: S.Dimiev (Sofia)

MSC:

32B20	Semi-analytic sets, subanalytic sets, and generalizations
32B05	Analytic algebras and generalizations, preparation theorems
14P15	Real-analytic and semi-analytic sets
26E05	Real-analytic functions
26E10	\(C^\infty\)-functions, quasi-analytic functions
03C99	Model theory

Keywords:

flows of gradient; \(o\)-minimal structure; subanalytic sets; Łojasiewicz inequalities; trajectories of gradient

Citations:

Zbl 0889.03025

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