The goal of this course is to give a quick backround in real analytic geometry, more precisely to semi-analytic geometry which was created by S.Lojasiewicz in his celebrated IHES lecture notes [5] from 1965 (now available on web). Actually the lecture will be based on a more recent paper [4] of S. Lojasiewicz, M-A. Zurro and myself.

Lecture 1. Holomorphic and analytic functions in several variables. Power series, holomorphic functions, Riemann extension theorem, separate analyticity. References: [3], [6], [8].

Lecture 2. Preparation Theorem and Weierstrass polynomials. Continuity of roots, discriminant and generalized discriminant. Decomposition into irreducible factors. Puiseux’s theorem. References: [3], [6], [7], [8].

Lecture 3. Semi-analytic sets and distinguished stratifications. Real analytic sets, semi-analytic sets, Thom’s lemma, dimension. References: [1], [2], [4].

Lecture 4. Regular separation and Lojasiewicz’s inequality. Metric properties of seminalytic sets, curve selection lemma, Lojasiewicz’s inequality. References: [5], [1], [4].

References

[1] E. Bierstone and P. D. Milman, Semianalytic and Subanalytic sets, Publ. I.H.E.S.,

67 (1988), 5-42.

[2] M.Coste, Ensembles semi-alg´ebriques. Real algebraic geometry and quadratic forms

(Rennes, 1981), pp. 109–138, Lecture Notes in Math., 959, Springer, Berlin-New York,

1982

[3] R. Gunning, H. Rossi, Hugo Analytic functions of several complex variables. Prentice-

Hall, Inc., Englewood Cliffs, N.J. 1965 xiv+317 pp.

[4] K. Kurdyka, S. Lojasiewicz, M. Zurro Stratifications distingu´ees comme un util en

g´eom´etrie semi-analytique, Manuscripta Math. 86, 81–102, (1995).

[5] S. Lojasiewicz, Ensembles semi-analytiques,, preprint, I.H.E.S. (1965) available on

the web page ”http://perso.univ-rennes1.fr/michel.coste/Lojasiewicz.pdf”.

[6] S. Lojasiewicz, Introduction to complex analytic geometry, Birkh¨auser, Basel, 1991.

[7] R. Narasimhan, Introduction to the theory of analytic spaces. Lecture Notes in Math-

ematics, No. 25 Springer-Verlag, Berlin-New York 1966 iii+143 pp

[8] Ruiz, Jesus M. The basic theory of power series. Advanced Lectures in Mathematics.

Friedr. Vieweg & Sohn, Braunschweig, 1993. x+134 pp.

UNIVERSITE DE SAVOIE, Laboratoire de Mathematiques (LAMA), UMR

5127 CNRS, 73-376 Le Bourget-du-Lac cedex FRANCE