Real Algebraic and Analytic Geometry

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160. Riccardo Ghiloni:
Globalization and compactness of McCrory-Parusiński conditions.

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Submission: 2005, April 27.

Abstract:
Let $X \subset \mathbb{R}^n$ be a closed semialgebraic set and let $\widetilde{\mathscr{P}}(X)$ be the ring obtained from the characteristic function of $X$ by the operations $+$, $-$, $\ast$, the half link operator and by the polynomial operations with rational coefficients which preserve finite formal sum of signs. McCrory and Parusi\'nski proved that necessary conditions for $X$ to be semialgebraically homeomorphic to a real algebraic set is that $X$ is $\mathscr{P}$--euler, i.e, all the functions in $\widetilde{\mathscr{P}}(X)$ are integer--valued. These conditions are local. In this paper, we give global versions of these conditions. For example, we show that, if $X$ is $\mathscr{P}$--euler and $M$ is a compact Nash submani\-fold of $\mathbb{R}^n$ transverse to some semialgebraic Whitney stratification of $X$, then $X \cap M$ is $\mathscr{P}$--euler and, for each $\varphi \in \widetilde{\mathscr{P}}(X \cap M)$, the Euler integral of $\varphi$ is even. Moreover, we have the following result: Let $\mathcal{F}$ be a family of closed semialgebraic sets and let $\mathcal{F}_c$ be the family of all $X \in \mathcal{F}$ such that $X$ is compact. Suppose $\mathcal{F}$ is closed under the inverse images of regular maps. Then all the elements of $\mathcal{F}$ are $\mathscr{P}$--euler if and only if all the elements of $\mathcal{F}_c$ are. Since it is known that every compact arc--symmetric semialgebraic set is $\mathscr{P}$--euler, we infer that all the arc--symmetric semialgebraic sets are $\mathscr{P}$--euler, answering affirmatively to a question by Kurdyka, McCrory and Parusiński.

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