Publication: Stability index of closed semianalytic set germs.
Loading...
Files
Official URL
Full text at PDC
Publication Date
1998
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Springer
Citations
Exportar
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.
Description
UCM subjects
Unesco subjects
Keywords
Citation
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).