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Algebraic approximation of germs of real analytic sets


Authors: M. Ferrarotti, E. Fortuna and L. Wilson
Journal: Proc. Amer. Math. Soc. 138 (2010), 1537-1548
MSC (2000): Primary 14P15, 32B20, 32S05
DOI: https://doi.org/10.1090/S0002-9939-10-10283-4
Published electronically: January 19, 2010
MathSciNet review: 2587437
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Abstract: Two subanalytic subsets of $ \mathbb{R}^n$ are $ s$-equivalent at a common point, say $ O$, if the Hausdorff distance between their intersections with the sphere centered at $ O$ of radius $ r$ goes to zero faster than $ r^s$. In the present paper we investigate the existence of an algebraic representative in every $ s$-equivalence class of subanalytic sets. First we prove that such a result holds for the zero-set $ V(f)$ of an analytic map $ f$ when the regular points of $ f$ are dense in $ V(f)$. Moreover we present some results concerning the algebraic approximation of the image of a real analytic map $ f$ under the hypothesis that $ f^{-1}(O)=\{O\}$.


References [Enhancements On Off] (What's this?)

  • [Ab] S. ABHYANKAR: On the ramification of algebraic functions. Amer. J. Math., vol. 77 (1955), pp. 575-592. MR 0071851 (17:193c)
  • [Ar] F. AROCA: Puiseux parametric equations of analytic sets. Proc. Amer. Math. Soc., vol. 132, n. 10 (2004), pp. 3035-3045 (electronic). MR 2063125 (2005c:32032)
  • [BM] E. BIERSTONE, P. MILMAN: Semianalytic and subanalytic sets. Inst. Hautes Études Sci. Publ. Math., vol. 67 (1988), pp. 5-42. MR 972342 (89k:32011)
  • [B1] M. BILSKI: Approximation of analytic sets by Nash tangents of higher order. Math. Z., vol. 256, n. 4 (2007), pp. 705-716. MR 2308884 (2008h:32009)
  • [B2] M. BILSKI: Approximation of analytic sets with proper projection by algebraic sets. arXiv:0905.1881v1[math.CV] (2009).
  • [FFW] M. FERRAROTTI, E. FORTUNA AND L. WILSON: Local approximation of semialgebraic sets. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), vol. I, n. 1 (2002), pp. 1-11. MR 1994799 (2004f:14083)
  • [H] H. HIRONAKA: Stratification and flatness. In: Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 199-265. MR 0499286 (58:17187)
  • [KR] K. KURDYKA, G. RABY: Densité des ensembles sous-analytiques. Ann. Inst. Fourier (Grenoble), vol. 39 (1989), pp. 753-771. MR 1030848 (90k:32026)
  • [M] J. MILNOR: Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612 (39:969)
  • [TW] D. TROTMAN, L. WILSON: Stratifications and finite determinacy. Proc. London Math. Soc. (3), vol. 78, n. 2 (1999), pp. 334-368. MR 1665246 (2000h:58069)

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Additional Information

M. Ferrarotti
Affiliation: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
Email: ferrarotti@polito.it

E. Fortuna
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, I-56127 Pisa, Italy
Email: fortuna@dm.unipi.it

L. Wilson
Affiliation: Department of Mathematics, University of Hawaii, Manoa, Honolulu, Hawaii 96822
Email: les@math.hawaii.edu

DOI: https://doi.org/10.1090/S0002-9939-10-10283-4
Received by editor(s): January 9, 2009
Published electronically: January 19, 2010
Additional Notes: This research was partially supported by M.I.U.R. and by G.N.S.A.G.A
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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