Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 1

Rational representation of real functions

Pages: 249 – 268

DOI: https://dx.doi.org/10.4310/PAMQ.2021.v17.n1.a7


Wojciech Kucharz (Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland; and Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Krakow, Poland)

Krzysztof Kurdyka (Laboratoire de Mathématiques, UMR 5127 de CNRS, Université Savoie Mont Blanc, Le Bourget-du-Lac, France)


Let $X$ be an irreducible smooth real algebraic variety of dimension at least $2$ and let $f : U \to \mathbb{R}$ be a function defined on a connected open subset $U \subset X(\mathbb{R})$. Assume that for every irreducible smooth real algebraic curve $C \subset X$, for which $C(\mathbb{R})$ is the boundary of a disc embedded in $U$, the restriction ${f \vert}_{C(\mathbb{R})}$ is continuous and has a rational representation. Then $f$ has a rational representation. This is a significant refinement of a recent result of J. Kollár and the authors. The novelty is that existence of rational representation is tested on a much smaller and more rigid class of curves. We also consider the case where $U$ is not necessarily connected and test rationality on subvarieties of dimension greater than $1$. For semialgebraic functions our results hold under slightly weaker assumptions.


real algebraic variety, rational function, rational representation, semialgebraic function, Nash function

2010 Mathematics Subject Classification

Primary 14P05, 14P10. Secondary 26C15.

W. Kucharz was partially supported by the National Science Center (Poland) under grant number 2014/15/ST1/00046.

K. Kurdyka was partially supported by the ANR project LISA (France).

Received 16 June 2020

Accepted 21 July 2020

Published 11 April 2021