Contents Online

# 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

#### Authors

#### Abstract

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.

#### Keywords

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