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Stability index of closed semianalytic set germs.

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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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
Uncontrolled Keywords:stability index; semianalytic set germs; normal analytic set germs
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:14806

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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
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