Rational representation of real functions
Résumé
Let X be an irreducible smooth real algebraic variety of dimension at least 2 and let f : U → R be a function defined on a connected open subset U ⊂ X(R). Assume that for every irreducible smooth real algebraic curve C ⊂ X, for which C(R) is the boundary of a disc embedded in U , the restriction f | C(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.
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