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Local algebraic approximation of semianalytic sets


Authors: M. Ferrarotti, E. Fortuna and L. Wilson
Journal: Proc. Amer. Math. Soc. 143 (2015), 13-23
MSC (2010): Primary 14P15, 32B20, 32S05
DOI: https://doi.org/10.1090/S0002-9939-2014-12212-X
Published electronically: September 3, 2014
MathSciNet review: 3272727
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Abstract: Two subanalytic subsets of $ \mathbb{R}^n$ are called $ s$-equivalent at a common point $ P$ if the Hausdorff distance between their intersections with the sphere centered at $ P$ of radius $ r$ vanishes to order $ >s$ when $ r$ tends to 0. In this paper we prove that every $ s$-equivalence class of a closed semianalytic set contains a semialgebraic representative of the same dimension. In other words any semianalytic set can be locally approximated to any order $ s$ by means of a semialgebraic set and hence, by previous results, also by means of an algebraic one.


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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/S0002-9939-2014-12212-X
Received by editor(s): July 10, 2012
Received by editor(s) in revised form: January 16, 2013, January 18, 2013, and February 25, 2013
Published electronically: September 3, 2014
Additional Notes: The first and second authors’ research was partially supported by M.I.U.R. and by G.N.S.A.G.A
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2014 American Mathematical Society