Abstract
This paper treats the theory of semianalytic function germs over real closed fields more general than ℝ. An ordered field is microbial if it has a non-zero element whose powers converge to zero. The fields we treat are direct limits of countable microbial subfields. We define local rings of analytic function germs algebraically and use the Weierstrass preparation theory to prove an Artin-Lang property. We end by relating seminash functions to abstract semialgebraic functions on the real spectrum of the local rings.
Similar content being viewed by others
References
S. Abhyankar:Local Analytic Geometry. Academic Press, New York (1964)
N. Alling:Foundations of Analysis over Surreal Number Fields, to appear, North Holland (1987)
N. Ailing: The ξ-Topology on ηξ with Applications to Real Algebraic Geometry, Math Reports Acad. Sci. Canada 3 (1984) 145–150
E. Becker:On the Real Spectrum of a Ring and its Applications to Semialgebraic Geometry, A.M.S. Bulletin 15 (1986) 19–60
Coste & Coste-Roy:Topologies for Real Algebraic Geometry, in Topos Theoretic Methods in Geometry (ed. A Koch) Various Publ. Series 30, Aarhus Universitet (1979)
H. Delfs:Kohomologie Affiner Semialgebraische Räume. Diss., Univ. Regensburg, Regensburg, West Germany (1980)
H. Delfs:The Homotopy Axiom in Semialgebraic Geometry. J. Reine u. Angew. Math. 355 (1985) 108–128
Dubois & Bukowski:Real Commutative Algebra II, Rev. Mat. Hisp. Amer. (4) 39 (1979) 149–161
T. Y. Lam:Orderings, Valuations, and Quadratic Forms, CBMS Regional Conf. Series in Math 52, A.M.S., Providence, R.I.
J. Merrien:Faisceaux analytiques semi-cohérents et fonctions différentiates. Thèse, Université de Rennes I (1980)
M. Nagata:Local Rings, Interscience, New York (1962)
S. Prieß-Crampe,Angeordnete Strukturen: Gruppen. Körper, projektive Ebenen. Ergebnisse der Mathematik 98, Springer Verlag, Berlin Heidelberg Tokyo New York (1983)
R. Robson,Spectra and Model Theory, submitted to J. Pure and Applied Algebra (1987)
M-F, Roy:Faisceau Structural sur le Spectre Reel et Fonctions de Nash, Springer Lecture Notes in Math 959 (1982) 406–432
N. Schwartz:Real Closed Spaces, Hablitationsschrift, München (1984)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Robson, R.O. Local semianalytic geometry. Manuscripta Math 63, 215–231 (1989). https://doi.org/10.1007/BF01168873
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01168873