## 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.

Original language | English (US) |
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Pages (from-to) | 39-69 |

Number of pages | 31 |

Journal | Mathematische Annalen |

Volume | 370 |

Issue number | 1-2 |

DOIs | |

State | Published - Feb 1 2018 |

## All Science Journal Classification (ASJC) codes

- General Mathematics

## Keywords

- 14P05
- 14P10
- 26C15