Díaz-Cano Ocaña, Antonio and Andradas Heranz, Carlos
(1998)
*Stability index of closed semianalytic set germs.*
Mathematische Zeitschrift, 229
(4).
pp. 743-751.
ISSN 0025-5874

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Official URL: http://www.springerlink.com/content/100443/

## Abstract

Let X0 be an irreducible set germ at the origin 0 2 Rn, and let O(X0) denote the ring of analytic function germs at X0.

A basic closed semianalytic germ of X0 is a set germ of the form S0 = {g1 0, · · · , gs 0} X0 where gi 2 O(X0). The integer s(X0) is the minimum of all s 2 Z such that any basic closed semianalytic set germ ofX0 can be written with s elements of O(X0), the integer s(d) is the maximum of s(X0) for all d-dimensional analytic germsX0. In [C. Andradas, L. Br¨ocker and J. M. Ruiz, Constructible sets in real geometry, Springer, Berlin, 1996; MR1393194 (98e:14056)] it is shown that 12 d(d+1)−1 s(X0) 12 d(d+1), where d = dimX0, but, unlike the semialgebraic case, where it is known that s(X) = 12 d(d+1) for any d-dimensional algebraic variety X, it was still open whether for semianalytic germs this is also true.

The authors prove that s(X0) = 2 for any two-dimensional normal analytic germ, and provide examples of surface germ with s = 3. Pulling these examples to higher dimension they show that s(d) = 12 d(d+1) for d > 2, so that they obtain the same bound as in the semialgebraic case.

Item Type: | Article |
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Uncontrolled Keywords: | stability index; semianalytic set germs; normal analytic set germs |

Subjects: | Sciences > Mathematics > Algebraic geometry |

ID Code: | 14806 |

References: | Andradas, C., Br¨ocker, L., Ruiz, J.: Constructible sets in Real Geometry (Ergeb.der Math. 33, 3. folge) Berlin Heidelberg New York: Springer 1996. Bochnak, J., Coste, M., Roy, M.F.:G´eom´etrie Alg´ebriqueR´eelle (Ergeb. der Math. 12, 3. folge) Berlin Heidelberg New York: Springer 1987. Br¨ocker, L.: On basic semialgebraic sets. Expo. Math. 9, 289-334 (1991). D´ıaz-Cano, A.: Ph.D. dissertation. In preparation, U.C.M. Gunning, R., Rossi, H.: Analytic functions of several complex variables. New-Jersey: Prentice-Hall 1965. Ruiz, J.: The basic theory of power series (Advanced Lectures in Mathematics): Vieweg 1993. Ruiz, J.: A note on a separation problem. Arch. Math. 43, 422-426 (1984). |

Deposited On: | 18 Apr 2012 08:56 |

Last Modified: | 06 Feb 2014 10:09 |

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