Real Algebraic and Analytic Geometry Preprint Server

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Submission: 2004, April 6.

Abstract:
To understand how a trajectory of the gradient field of a real polynomial function P goes to infinity, and what this means on the asymptotic geometry of the levels of P, investigations begun in our joint work with D. D'ACUNTO (available on this server), we need to know the limit set of such a trajectory. So, given a semialgebraic function F two times continuously differentiable, defined on open set U of $\mathbb{R}^n$, we show that the limit set of any trajectory of the gradient vector field of F consists of at most two points. Following VERY CLOSELY the proof of the gradient conjecture of R. Thom, given by Kurdyka, Mostowski and Parusinski, there is just to consider the case of a trajectory leaving any compact of $\mathbb{R}^n$ and to show that the secants OM have a limit when M goes to infinity on the trajectory. .

Mathematics Subject Classification (2000): 34A26 34C08 32Bxx, 32Sxx, 14P10.

Keywords and Phrases: Gradient vector field, characteristic exponents, limit sets, trajectories of infinite length.

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