@article{f5ae8ea8c89245388fa4768d64605b4f,

title = "Curve-rational functions",

abstract = "Let W be a subset of the set of real points of a real algebraic variety X. We investigate which functions f: W→ 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.",

keywords = "14P05, 14P10, 26C15",

author = "J{\'a}nos Koll{\'a}r and Wojciech Kucharz and Krzysztof Kurdyka",

note = "Funding Information: 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 f{\"u}r Mathematik in Bonn. Partial support for KK was provided by the ANR project STAAVF (France). Funding Information: 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 f?r Mathematik in Bonn. Partial support for KK was provided by the ANR project STAAVF (France). Publisher Copyright: {\textcopyright} 2017, Springer-Verlag Berlin Heidelberg.",

year = "2018",

month = feb,

day = "1",

doi = "10.1007/s00208-016-1513-z",

language = "English (US)",

volume = "370",

pages = "39--69",

journal = "Mathematische Annalen",

issn = "0025-5831",

publisher = "Springer New York",

number = "1-2",

}