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Geometry of Subanalytic and Semialgebraic Sets

  • Masahiro Shiota

Part of the Progress in Mathematics book series (PM, volume 150)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Masahiro Shiota
    Pages 1-94
  3. Masahiro Shiota
    Pages 95-269
  4. Masahiro Shiota
    Pages 270-304
  5. Masahiro Shiota
    Pages 305-387
  6. Masahiro Shiota
    Pages 388-419
  7. Back Matter
    Pages 421-434

About this book

Introduction

Real analytic sets in Euclidean space (Le. , sets defined locally at each point of Euclidean space by the vanishing of an analytic function) were first investigated in the 1950's by H. Cartan [Car], H. Whitney [WI-3], F. Bruhat [W-B] and others. Their approach was to derive information about real analytic sets from properties of their complexifications. After some basic geometrical and topological facts were established, however, the study of real analytic sets stagnated. This contrasted the rapid develop­ ment of complex analytic geometry which followed the groundbreaking work of the early 1950's. Certain pathologies in the real case contributed to this failure to progress. For example, the closure of -or the connected components of-a constructible set (Le. , a locally finite union of differ­ ences of real analytic sets) need not be constructible (e. g. , R - {O} and 3 2 2 { (x, y, z) E R : x = zy2, x + y2 -=I- O}, respectively). Responding to this in the 1960's, R. Thorn [Thl], S. Lojasiewicz [LI,2] and others undertook the study of a larger class of sets, the semianalytic sets, which are the sets defined locally at each point of Euclidean space by a finite number of ana­ lytic function equalities and inequalities. They established that semianalytic sets admit Whitney stratifications and triangulations, and using these tools they clarified the local topological structure of these sets. For example, they showed that the closure and the connected components of a semianalytic set are semianalytic.

Keywords

Finite algebra analytic function analytic geometry class form function geometry information pdc proof set sets tool topology

Authors and affiliations

  • Masahiro Shiota
    • 1
  1. 1.Department of MathematicsNagoya UniversityFurocho, Chikusa, NagoyaJapan

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-2008-4
  • Copyright Information Birkhäuser Boston 1997
  • Publisher Name Birkhäuser, Boston, MA
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-7378-3
  • Online ISBN 978-1-4612-2008-4
  • Series Print ISSN 0743-1643
  • Series Online ISSN 2296-505X
  • Buy this book on publisher's site