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Subanaliticity and the second part of Hilbert's 16th problem

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Real Algebraic Geometry

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1524))

Abstract

The aim of this paper is to clarify the role of subanalytic sets in the proofs concerning the second part of Hilbert's 16th problem.

After a brief introduction (§ 1), we give (§ 2) a somehow modified version of the subanalytic proofs contained, among other results, in the work of J.-P. Françoise and C.C. Pugh ([FP], 1986).

Finite cyclicity of elliptic points is proved in § 2 both by the methods of [FP] and as a consequence of the subanalyticity of the Poincaré map (this is our main modification).

In § 3, which is based mostly on the remarks of R. Roussarie, we give the reasons for which singular points other than elliptic do not admit the same subanalytic treatment.

The author of this paper had worked with subanalytic sets for over 10 years, so naturally the paper of J.-P. Françoise and C.C. Pugh ([FP]), published in 1986, immediately drew her attention, as it contains, among other results, some very ingenious applications of the subanalyticity to Hilbert's 16th problem have their limitations and why.

The author gathered together both the subanalytic results (§ 2) and the calculus that explains why direct generalization was impossible and what obstacles appear for singular points other than elliptic one (§ 3).

The author is very grateful to P. Biler (Wroclaw, visiting Orsay) for numerous suggestions concerning the proofs in § 3 as well as for his invaluable help in finding the references.

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Authors
  1. Zofia Denkowska
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Michel Coste Louis Mahé Marie-Françoise Roy

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© 1992 Springer-Verlag

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Denkowska, Z. (1992). Subanaliticity and the second part of Hilbert's 16th problem. In: Coste, M., Mahé, L., Roy, MF. (eds) Real Algebraic Geometry. Lecture Notes in Mathematics, vol 1524. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084622

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  • DOI: https://doi.org/10.1007/BFb0084622

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55992-4

  • Online ISBN: 978-3-540-47337-4

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