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 $f:W\to \mathbb{R}$ are the restrictions of rational functions on $X$. We introduce two new notions: $curve-rationalfunctions$ (i.e., continuous rational on algebraic curves) and $arc-rationalfunctions$ (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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Cited by in corpus (5)

• A proof of Kurdyka's conjecture on arc-analytic functions
• Integral Closures In Real Algebraic Geometry
• Extensions of arc-analytic functions
• On some regularity condition
• Arc spaces, motivic measure and Lipschitz geometry of real algebraic sets