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Cited by 671Algebraic Geometry and Number TheoryAdvanced Topology and Set TheoryMathematical and Theoretical Analysis

Semianalytic and subanalytic sets

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Bierstone, Edward, Milman, Pierre D.·1988·Publications Mathématiques de l'IHÉS
DOI 10.1007/bf02699126
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Abstract

0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 I. The Tarski-Seidenberg theorem and Thorn's lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. Semianalytic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3. Subanalytic sets 16 4. Transforming an analytic function to normal crossings by blowings-up 21 5. Uniformization and rcctilincarization 30 6. I.ojasicwicz's inequality; metric properties of subanalytic sets 33 7. Smooth points of a subanalytic sct 37 Bibliography 42

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22 of 22 references verified0 citation errors · 0 not checkable

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  • Complements, finite intersections and finite unions of semianalytic sets are semianalytic again.

    “Complements, finite intersections and finite unions of semianalytic sets are semianalytic again [10].”

    From Representations of the Weyl Algebra in Quantum Geometry (2009) · cites this paper as [10]

  • Every open covering of the manifold admits a compatible semianalytic partition of unity.

    “Technically, the locality is expressed by the fact that every open covering of Σ admits a compatible semianalytic partition of unity.”

    From Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras (2006) · cites this paper as [3]

  • In the semianalytic category, the intersection of every two connected submanifolds is locally a finite union of connected submanifolds.

    “The special property of the semianalytic category so relevant for us in this paper, is that the intersection between every two connected submanifolds, locally, is a finite union of connected submanifolds.”

    From Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras (2006) · cites this paper as [3]

  • For every semianalytic partition P(U,h) of an open U⊂R^n, every point x∈U has a neighborhood that admits an analytic partition finer than P(Ũ,h).

    “For every semianalytic partition P ( U , h ) of an open U ⊂ R n , every point x ∈ U has a neighborhood ˜ U which admits an analytic partition finer than P ( ˜ U , h ) .”

    From Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras (2006) · cites this paper as [3]

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Authors
Bierstone, E., & Milman, P. D.
Affiliations
University of Toronto (CA)
Venue
Publications Mathématiques de l'IHÉS
Publisher
MathDoc/Centre Mersenne
Published
1988-01-07
Volume
67
Pages
5-42
ISSN
  • 0073-8301 (ISSN-L)
  • 1618-1913
DOI
10.1007/bf02699126
OpenAlex
W2080803951
Cited by
671
References
22
Citations by year
2026: 8 · 2025: 14 · 2024: 22 · 2023: 30 · 2022: 18 · 2021: 34 · 2020: 29 · 2019: 33 · 2018: 26 · 2017: 20 · 2016: 20 · 2015: 32 · 2014: 32 · 2013: 16 · 2012: 32
FWCI
7.64
Type
journal article
Language
en
Open access
Licence
www.springer.com/tdm (tdm)
Locations
  • Publications mathématiques de l IHÉS (publishedVersion)
  • French digital mathematics library (Numdam) (submittedVersion)
Topics
  • Physical Sciences › Mathematics › Geometry and Topology › Algebraic Geometry and Number Theory
  • Physical Sciences › Mathematics › Geometry and Topology › Advanced Topology and Set Theory
  • Physical Sciences › Mathematics › Mathematical Physics › Mathematical and Theoretical Analysis
Keywords
semianalytic sets, subanalytic sets, normal crossings, uniformization, Łojasiewicz inequality, smooth points, o-minimality, real analytic geometry

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