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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
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\}$.

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Additional Information

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

E. Fortuna
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, I-56127 Pisa, Italy

L. Wilson
Affiliation: Department of Mathematics, University of Hawaii, Manoa, Honolulu, Hawaii 96822

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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