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On the regularity of a semialgebraic function

Volume 125 / 2020

Ali El-Siblani, Krzysztof Kurdyka Annales Polonici Mathematici 125 (2020), 25-46 MSC: Primary 14P10; Secondary 14P20, 32B20, 58A07. DOI: 10.4064/ap190719-19-3 Published online: 15 June 2020

Abstract

Let be a semialgebraic function of class C^1, defined on an open set {U \subset \mathbb R ^n}. Let P(x,y) \in \mathbb R [x_1,\dots , x_n, y] be a polynomial of degree d such that {P(x,f(x)) = 0}, x\in U. We prove that if f is of class C^K with K \gt \frac {1}{2}d^7, then f is analytic. If n=1, then it suffices that K \gt \frac {1}{2}d^2.

Authors

  • Ali El-SiblaniFaculté des sciences, section IV
    Université Libanaise
    Houche El-Oumaraa, Zahlé, Lebanon
    e-mail
  • Krzysztof KurdykaUniv. Savoie Mont Blanc
    CNRS UMR 5127 LAMA
    73000 Chambéry, France
    e-mail

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