Mathematische Zeitschrift

, Volume 285, Issue 1–2, pp 287–323 | Cite as

Continuous functions on the plane regular after one blowing-up

  • Goulwen Fichou
  • Jean-Philippe Monnier
  • Ronan Quarez


We study rational functions admitting a continuous extension to the real affine space. First of all, we focus on the regularity of such functions exhibiting some nice properties of their partial derivatives. Afterwards, since these functions correspond to rational functions which become regular after some blowings-up, we work on the plane where it suffices to blow-up points and then we can count the number of stages of blowings-up necessary. In the latest parts of the paper, we investigate the ring of rational continuous functions on the plane regular after one stage of blowings-up. In particular, we prove a Positivstellensatz without denominator in this ring.


Regular function Regulous function Rational function Real algebraic variety Sum of squares 

Mathematics Subject Classification

14P99 11E25 26C15 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Goulwen Fichou
    • 1
  • Jean-Philippe Monnier
    • 2
  • Ronan Quarez
    • 1
  1. 1.IRMAR (UMR 6625)Université de Rennes 1Rennes CedexFrance
  2. 2.LUNAM Université, LAREMAUniversité d’AngersAngersFrance

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