Real Algebraic and Analytic Geometry |

On gradient at infinity of real polynomials.

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Submission: 2004, February 26.

*Abstract:
Let $f:\mathbb{R}^n\to\mathbb{R}$ be a polynomial function. We discuss
on different conditions to trivialise the graph of $f$ by its level sets
in the neighbourhood of a critical value at infinity via the gradient
field of $f$. We also exhibit a \L ojasiewicz type inequality which is
useful to the present study. When $n=2$, we are able to relate this \L
ojasiewicz type inquality with generic polar curves.*

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

Keywords and Phrases: gradient trajectories, polynomials.

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