Curve-rational functions

TL;DR

This paper introduces and studies classes of functions on real algebraic varieties that are rational on curves or arcs, establishing their equivalence under mild conditions and exploring their properties and applications.

Contribution

It defines curve-rational and arc-rational functions, proves their equivalence with continuous hereditarily rational functions, and extends results to complex algebraic varieties.

Findings

Curve-rational, arc-rational, and hereditarily rational functions coincide under mild assumptions.

Arc-rational functions on open sets are arc-analytic.

Characterization of regular functions via rationality on products of varieties.

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: ${\it curve-rational \, functions}$ (i.e., continuous rational on algebraic curves) and ${\it arc-rational\, functions}$ (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.