Real Algebraic and Analytic Geometry |

On the topology of semi-algebraic functions on closed semi-algebraic sets.

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Submission: 2010, December 9.

*Abstract:
We consider a closed semi-algebraic set $X \subset \mathbb{R}^n$ and a $C^2$ semi-algebraic function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ such that $f_{\vert X}$ has a finite number of critical points. We relate the topology of $X$ to the topology of the sets
$\{ f * \alpha \}$, where $* \in \{\le,=,\ge \}$ and $\alpha \in \mathbb{R}$,
and the indices of the critical points of $f_{\vert X}$ and $-f_{\vert X}$. We also relate the topology of $X$ to the topology of the links at infinity of the sets
$\{ f * \alpha \}$ and the indices of these critical points. We give applications when $X=\mathbb{R}^n$ and when $f$ is a generic linear function.*

Mathematics Subject Classification (2010): 14P10, 14P25.

**Full text**, 25p.:
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pdf 380k.

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