Topological invariants of analytic sets associated with Noetherian families

Nowel, Aleksandra

Annales de l'Institut Fourier, Tome 55 (2005) no. 2, pp. 549-571.

Soit $\Omega \subset {ℝ}^n$ un ensemble semi-analytique compact et soit $ℱ$ une collection de fonctions analytiques réelles définies dans un voisinage de $\Omega$. Soit $Y_{\omega }$ le germe en $\omega \in \omega$ de l’ensemble ${\bigcap }_{f\in ℱ}{f}^{-1}\left(0\right)$. Alors il existe des fonctions analytiques $v_1,v_2,...,v_s$ définies dans un voisinage de $\Omega$ telles que $\frac{1}{2}\chi \left(\mathrm{lk}(\omega ,Y_{\omega })\right)={\sum }_{i=1}^s\mathrm{sgn}{v}_i\left(\omega \right)$, pour tout $\omega \in \Omega$.

Let $\Omega \subset {ℝ}^n$ be a compact semianalytic set and let $ℱ$ be a collection of real analytic functions defined in some neighbourhood of $\Omega$. Let $Y_{\omega }$ be the germ at $\omega$ of the set ${\bigcap }_{f\in ℱ}{f}^{-1}\left(0\right)$. Then there exist analytic functions $v_1,v_2,...,v_s$ defined in a neighbourhood of $\Omega$ such that $\frac{1}{2}\chi \left(\mathrm{lk}(\omega ,Y_{\omega })\right)={\sum }_{i=1}^s\mathrm{sgn}{v}_i\left(\omega \right)$, for all $\omega \in \Omega$.

Classification : 14P15, 32B20
Keywords: germs of semianalytic sets, Noetherian families, (sum of signs of) analytic functions, $\Omega $-Noetherian algebra
Mot clés : germes d’ensembles semi-analytiques, familles noethériennes, (somme des signes de) fonctions analytiques, algèbre $\Omega $-noethérienne.

[1] J. Bochnak; M. Coste; M.-F. Roy Real algebraic geometry, Springer-Verlag, Berlin, 1998 | MR | Zbl

[2] I. Bonnard; F. Pieroni Constructible functions on 2-dimensional analytic manifolds (preprint) | MR | Zbl

[3] M. Coste; K. Kurdyka On the link of a stratum in a real algebraic set, Topology, Volume 31 (1992) no. 2, pp. 323-336 | DOI | MR | Zbl

[4] M. Coste; K. Kurdyka Le discriminant d'un morphisme de variétés algébriques réelles, Topology, Volume 37 (1998) no. 2, pp. 393-399 | DOI | MR | Zbl

[5] A. El Khadiri; J.-C. Tougeron Familles noethériennes de modules sur $k\left[\right[x\left]\right]$ et applications, Bull. Sci. Math., Volume 120 (1996), pp. 253-292 | MR | Zbl

[6] A. El Khadiri; J.-C. Tougeron Familles noethériennes de modules sur $k\left[\right[x\left]\right]$ et applications (1984) (preprint) | MR

[7] J. Frisch Points de platitude d'un morphisme d'espaces analytiques complexes, Invent. Math., Volume 4 (1967), pp. 118-138 | DOI | MR | Zbl

[8] S. Lojasiewicz Introduction to complex analytic geometry, Birkhauser Verlag, Basel--Boston--Berlin, 1991 | MR | Zbl

[9] C. McCrory; A. Parusinski Algebraically constructible functions, Ann. Scient. École Norm. Sup., Volume 30 (1997) no. 4, pp. 527-552 | Numdam | MR | Zbl

[10] C. McCrory; A. Parusinski Topology of real algebraic sets of dimension 4: necessary conditions, Topology, Volume 39 (2000), pp. 495-523 | DOI | MR | Zbl

[11] J. Mather Stratifications and mappings. Dynamical systems (Proc. Sympos. Univ. Bahia, Salvador) (1973), pp. 195-232 | Zbl

[12] A. Parusinski; Z. Szafraniec Algebraically constructible functions and signs of polynomials, Manuscripta Math., Volume 93 (1997) no. 4, pp. 443-456 | MR | Zbl

[13] A. Parusinski; Z. Szafraniec On the Euler characteristic of fibres of real polynomial maps (Banach Center Publ.), Volume 44 (1996), pp. 175-182 | Zbl

[14] Z. Szafraniec On the Euler characteristic of analytic and algebraic sets, Topology, Volume 25 (1986) no. 4, pp. 411-414 | DOI | MR | Zbl

[15] Z. Szafraniec On the Euler characteristic of complex algebraic varieties, Math. Ann., Volume 280 (1988), pp. 177-183 | DOI | MR | Zbl

[16] B. Teissier Varietes polaires II. Multiplicités polaires, sections planes et conditions de Whitney, 169, Centre de Mathématiques de l'École Polytechnique, France, "Labo, 1980 | Zbl

[17] J.-C. Tougeron Idéaux de fonctions différentiables, Springer--Verlag, Berlin--Heidelberg--New York, 1972 | MR | Zbl

[18] H. Whitney Tangents to an analytic variety, Ann. of Math., Volume 81 (1965), pp. 96-549 | MR | Zbl