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Zbl 0934.32009
Kurdyka, Krzysztof
On gradients of functions definable in o-minimal structures. (English)
[J] Ann. Inst. Fourier 48, No.3, 769-783 (1998). ISSN 0373-0956; ISSN 1777-5310/e

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 $\gamma'(t)=-\text{grad} f(\gamma(s))$.\par Lojasiewicz proved that all trajectories of $-\text{grad} f$ are of finite length, when $f$ is real analytic in a neighborhood of a compact $U$.\par The notion of ``$o$-minimal structure on the real field'' describes abstractly [see {\it L. Van den Dries} and {\it 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).\par 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 $g^{-1}(0)$.
[S.Dimiev (Sofia)]

MSC 2000:: *32B20 Semi-analytic sets, etc.
32B05 Analytic algebras and generalizations
14P15 Real analytic and semianalytic sets
26E05 Real-analytic functions
26E10 Infinitely differentiable real functions, etc.
03C99 Model theory (logic)

Keywords: flows of gradient; $o$-minimal structure; subanalytic sets; Łojasiewicz inequalities; trajectories of gradient

Citations: Zbl 0889.03025

Cited in: Zbl 1142.49006

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