Rationality of semialgebraic functions

Let X be an algebraic subset of Rⁿ, and ƒ: X → R a semialgebraic function. We prove that if ƒ is continuous rational on each curve C ⊂ X then: 1) ƒ is arc-analytic, 2) ƒ is continuous rational on X. As a consequence we obtain a characterization of hereditarily rational functions recently studied by J. Kollár and K. Nowak

Some conjectures on continuous rational maps into spheres

Recently continuous rational maps between real algebraic varieties have attracted the attention of several researchers. In this paper we continue the investigation of approximation properties of continuous rational maps with values in spheres. We propose a conjecture concerning such maps and show that it follows from certain classical conjectures involving transformation of compact smooth submanifolds of nonsingular real algebraic varieties onto subvarieties. Furthermore, we prove our conjecture in a special case and obtain several related results.Comment: arXiv admin note: text overlap with arXiv:1403.512