Andradas Heranz, CarlosDíaz-Cano Ocaña, Antonio2023-06-202023-06-202001C. Andradas, E. Becker, A note on the real spectrum of analytic functions on an analytic manifold of dimension one, Springer Lect. Notes Math. 1420 (1990), 1±21. C. Andradas, L. BroÈcker, J. Ruiz, Minimal generation of basic open semianalytic sets, Invent. Math. 92 (1988), 409±430. C. Andradas, L. BroÈcker, J. Ruiz, Constructible sets in real geometry, Springer-Verlag, 1996. E. Becker, On the real spectrum of a ring and its applications to semialgebraic geometry, Bull. Amer. Math. Soc. (N.S.) 15 (1986), 19±60. J. Bochnak, M. Coste, M. F. Roy, GeÂomeÂtrie algeÂbrique reÂelle, Springer-Verlag, 1987. J. Bochnak, W. Kucharz, M. Shiota, On equivalence of ideals of real global analytic functions and the 17th Hilbert problem, Invent. Math. 63 (1981), 403±421. J. Bochnak, J. Risler, Le theÂoreÁme des zeÂros pour les varieÂteÂs analytiques reÂelles de dimension 2, Ann. Sci. EÂ c. Norm. Sup. 8 (1975), 353±364. L. BroÈcker, On basic semialgebraic sets, Expo. Math. 9 (1991), 289±334. A. Castilla, Artin-Lang property for analytic manifolds of dimension two, Math. Z. 217 (1994), 5±14. A. Castilla, Sums of 2n-th powers of meromorphic functions with compact zero set, Springer Lect. Notes Math. 1524 (1991), 174±177. A. Castilla, C. Andradas, Connected components of global semianalytic subsets of 2-dimensional analytic manifolds, J. reine angew. Math. 475 (1996), 137±148. S. Coen, Sul rango dei fasci coerenti, Boll. Un. Mat. Ital. 22 (1967), 373±383. A. DõÂaz-Cano, IÂndice de estabilidad y descripcioÂn de conjuntos semianalõÂticos, Ph.D. Thesis, Universidad Complutense de Madrid, 1999. A. DõÂaz-Cano, The t-invariant of analytic set germs of dimension 2, J. Pure Appl. Algebra, to appear. A. DõÂaz-Cano, C. Andradas, Stability index of closed semianalytic set germs, Math. Z. 229 (1998), 743±751. H. Grauert, R. Remmert, Coherent Analytic Sheaves, Springer Grundl. math. Wiss. 265 (1984). F. Guaraldo, P. MacrõÁ, A. Tancredi, Topics on Real Analytic Spaces, Adv. Lect. Math., Vieweg, 1986. M. Hirsch, Di¨erential Topology, Springer-Verlag, 1976. P. Jaworski, The 17-th Hilbert problem for noncompact real analytic manifolds, Springer Lect. Notes Math. 1524 (1991), 289±295. M. Marshall, Spaces of orderings and abstract real spectra, Springer Lect. Notes Math. 1636 (1996). R. Narasimhan, Introduction to the theory of analytic spaces, Springer-Verlag, 1966. J. Ruiz, A note on a separation problem, Arch. Math. 43 (1984), 422±426. C. Scheiderer, Stability index of real varieties, Invent. Math. 97 (1989), 467±483.0075-410210.1515/crll.2001.042https://hdl.handle.net/20.500.14352/57153Let X subset of R-n be a real analytic manifold of dimension 2. We study the stability index of X, s(X), that is the smallest integer s such that any basic open subset of X can be written using s global analytic functions. We show that s(X) = 2 as it happens in the semialgebraic case. Also, we prove that the Hormander-Lojasiewicz inequality and the Finiteness Theorem hold true in this context. Finally, we compute the stability index for basic closed subsets, S, and the invariants t and (t) over bar for the number of unions of open (resp. closed) basic sets required to describe any open (resp. closed) global semianalytic set.engjournal articlehttp://www.degruyter.com/view/j/crllopen access512.7Geometria algebraica1201.01 Geometría Algebraica