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Selecta Mathematica

, Volume 23, Issue 4, pp 2523–2552 | Cite as

Solutions of quasianalytic equations

  • André Belotto da Silva
  • Iwo Biborski
  • Edward BierstoneEmail author
Article

Abstract

The article develops techniques for solving equations \(G(x,y)=0\), where \(G(x,y)=G(x_1,\ldots ,x_n,y)\) is a function in a given quasianalytic class (for example, a quasianalytic Denjoy–Carleman class, or the class of \({\mathcal C}^\infty \) functions definable in a polynomially-bounded o-minimal structure). We show that, if \(G(x,y)=0\) has a formal power series solution \(y=H(x)\) at some point a, then H is the Taylor expansion at a of a quasianalytic solution \(y=h(x)\), where h(x) is allowed to have a certain controlled loss of regularity, depending on G. Several important questions on quasianalytic functions, concerning division, factorization, Weierstrass preparation, etc., fall into the framework of this problem (or are closely related), and are also discussed.

Keywords

Quasianalytic Denjoy–Carleman class Blowing up Power substitution Resolution of singularities Analytic continuation Weierstrass preparation 

Mathematics Subject Classification

Primary 03C64 26E10 32S45 Secondary 30D60 32B20 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • André Belotto da Silva
    • 1
    • 2
  • Iwo Biborski
    • 1
  • Edward Bierstone
    • 1
    Email author
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Université Paul Sabatier, Institut de Mathématiques de ToulouseToulouse Cedex 9France

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