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.
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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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Communicated by Ngaiming Mok.
