Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


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

  • [BFGO] L. Birbrair, A. Fernandes, V. Grandjean, and D. O'Shea, Choking horns in Lipschitz geometry of complex algebraic varieties. arXiv:1206.3105
  • [BK] J. W. Bruce and N. P. Kirk, Generic projections of stable mappings, Bull. London Math. Soc. 32 (2000), no. 6, 718-728. MR 1781584 (2001g:58066), https://doi.org/10.1112/S0024609300007530
  • [FFW1] Massimo Ferrarotti, Elisabetta Fortuna, and Les Wilson, Local approximation of semialgebraic sets, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), no. 1, 1-11. MR 1994799 (2004f:14083)
  • [FFW2] M. Ferrarotti, E. Fortuna, and L. Wilson, Approximation of subanalytic sets by normal cones, Bull. Lond. Math. Soc. 39 (2007), no. 2, 247-254. MR 2323456 (2008j:32007), https://doi.org/10.1112/blms/bdl034
  • [FFW3] M. Ferrarotti, E. Fortuna, and L. Wilson, Algebraic approximation of germs of real analytic sets, Proc. Amer. Math. Soc. 138 (2010), no. 5, 1537-1548. MR 2587437 (2011b:14129), https://doi.org/10.1090/S0002-9939-10-10283-4
  • [KO] Krzysztof Kurdyka and Patrice Orro, Distance géodésique sur un sous-analytique, Real algebraic and analytic geometry (Segovia, 1995), Rev. Mat. Univ. Complut. Madrid 10 (French, with French summary). (1997), Special Issue, suppl., 173-182. MR 1485298 (98m:32008)
  • [Ł] S. Łojasiewicz, Sur la séparation régulière, Geometry seminars, 1985 (Italian) (Bologna, 1985) Univ. Stud. Bologna, Bologna, 1986, pp. 119-121 (French). MR 877540 (88d:32017)
  • [OW] Donal B. O'Shea and Leslie C. Wilson, Limits of tangent spaces to real surfaces, Amer. J. Math. 126 (2004), no. 5, 951-980. MR 2089078 (2005f:14110)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14P15, 32B20, 32S05

Retrieve articles in all journals with MSC (2010): 14P15, 32B20, 32S05


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

American Mathematical Society