Real Algebraic and Analytic Geometry |

Globalization and compactness of McCrory-Parusiński conditions.

e-mail:

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.*

**Full text**, 8p.:
dvi 50k,
ps.gz 161k,
pdf 202k.

Server Home Page