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Approximation of Holomorphic Solutions of a System of Real Analytic Equations

Published online by Cambridge University Press:  20 November 2018

M. Hickel  and
G. Rond
M. Hickel
Affiliation:
Université Bordeaux 1, I.M.B., Equipe d’Analyse et Géométrie, et I.U.T. Bordeaux 1 département Informatique, 33405 Talence Cedex, Francee-mail: hickel@math.u-bordeaux1.fr
G. Rond
Affiliation:
I.M.L. Faculté des Sciences de Luminy, Case 907, 163 av. de Luminy, 13288 Marseille Cedex 9, Francee-mail: rond@iml.univ-mrs.fr
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Abstract

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We prove the existence of an approximation function for holomorphic solutions of a system of real analytic equations. For this we use ultraproducts andWeierstrass systems introduced by J. Denef and L. Lipshitz. We also prove a version of the Płoski smoothing theorem in this case.

Keywords

13B4013L0514F12Artin approximationreal analytic equations
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 4 , 01 December 2012 , pp. 752 - 761
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Artin, M., On the solutions of analytic equations. Invent. Math. 5(1968), 277–291. http://dx.doi.org/10.1007/BF01389777 Google Scholar
[2] Artin, M., Algebraic approximation of structures over complete local rings. Inst. Hautes études Sci. Publ. Math. (1969), no. 36, 23–58,Google Scholar
[3] Becker, J., Denef, J., Lipshitz, L., and van den Dries, L.: Ultraproducts and approximation in local rings. I. Invent. Math. 51(1979), no. 2, 189–203. http://dx.doi.org/10.1007/BF01390228 Google Scholar
[4] Bloom, T. and Graham, I., On “type” conditions for generic real submanifolds of ℂ n . Invent. Math. 40(1977), no. 3, 217–243. http://dx.doi.org/10.1007/BF01425740 Google Scholar
[5] Bochnak, J., Coste, M., and Roy, M. F., Géométrie algébrique réelle. Ergebnisse der Mathematik und ihre Grenzgebiete 12. Springer-Verlag, Berlin, 1987.Google Scholar
[6] Chang, C. C. and Keisler, H. J., Model Theory. Third edition. Studies in Logic and the Foundations of Mathematics 73. North-Holland, Amsterdam, 1990.Google Scholar
[7] Chevalley, C., Some properties of ideals in rings of power series. Trans. Amer. Math. Soc. 55(1944), 68–84.Google Scholar
[8] Denef, J. and Lipshitz, L., Ultraproducts and approximation in local rings. II. Math. Ann. 253(1980), no. 1, 1–28. http://dx.doi.org/10.1007/BF01457817 Google Scholar
[9] Gabrièlov, A. M., The formal relations between analytic functions. Funkcional. Anal. i Prilozhen. 5(1971), no. 4, 64–65.Google Scholar
[10] Milman, P., Complex analytic and formal solutions of real analytic equations in ℂ n. Math. Ann. 233(1978), no. 1, 1–7. http://dx.doi.org/10.1007/BF01351492 Google Scholar
[11] Pfister, G. and Popescu, D., Die strenge Approximationseigenshaft lokaler Ringe. Invent. Math. 30(1975), no. 2, 145–174. http://dx.doi.org/10.1007/BF01425506 Google Scholar
[12] Płoski, A., Note on a theorem of M. Artin. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22(1974), 1107–1109.Google Scholar
[13] Teissier, B., Résultats récents sur l’approximation des morphismes en algèbre commutative. Séminaire Bourbaki 784(1993-94), 259–282.Google Scholar
[14] Wavrick, J. J., A theorem on solutions of analytic equations with applications to deformations of complex structures. Math. Ann. 216(1975), no. 2, 127–142. http://dx.doi.org/10.1007/BF01432540 Google Scholar