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Mathematische Annalen

, Volume 370, Issue 1–2, pp 39–69 | Cite as

Curve-rational functions

  • János Kollár
  • Wojciech Kucharz
  • Krzysztof KurdykaEmail author
Article

Abstract

Let W be a subset of the set of real points of a real algebraic variety X. We investigate which functions \(f: W \rightarrow \mathbb {R}\) are the restrictions of rational functions on X. We introduce two new notions: curve-rational functions (i.e., continuous rational on algebraic curves) and arc-rational functions (i.e., continuous rational on arcs of algebraic curves). We prove that under mild assumptions the following classes of functions coincide: continuous hereditarily rational (introduced recently by the first named author), curve-rational and arc-rational. In particular, if W is semialgebraic and f is arc-rational, then f is continuous and semialgebraic. We also show that an arc-rational function defined on an open set is arc-analytic (i.e., analytic on analytic arcs). Furthermore, we study rational functions on products of varieties. As an application we obtain a characterization of regular functions. Finally, we get analogous results in the framework of complex algebraic varieties.

Mathematics Subject Classification

14P05 14P10 26C15 

Notes

Acknowledgements

We thank J. Bochnak, C. Fefferman and J. Siciak for useful comments, and S. Yakovenko for making us aware of the relevance of [10] for our project. Partial financial support to JK was provided by the NSF under Grant No. DMS-1362960. For WK, research was partially supported by the National Science Centre (Poland) under Grant No. 2014/15/B/ST1/00046. Furthermore, WK acknowledges with gratitude support and hospitality of the Max-Planck-Institut für Mathematik in Bonn. Partial support for KK was provided by the ANR project STAAVF (France).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • János Kollár
    • 1
  • Wojciech Kucharz
    • 2
  • Krzysztof Kurdyka
    • 3
    Email author
  1. 1.Department of Mathematics, Fine HallPrinceton UniversityPrincetonUSA
  2. 2.Faculty of Mathematics and Computer Science, Institute of MathematicsJagiellonian UniversityKrakówPoland
  3. 3.Laboratoire de Mathématiques (LAMA), UMR 5127, CNRSUniversité Savoie Mont BlancLe Bourget-du-Lac CedexFrance

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