Hartogs-type theorems in real algebraic geometry, II

Abstract

AbstractLet $$f:X\rightarrow \mathbb {R}$$ f : X → R be a function defined on a connected nonsingular real algebraic set X in $$\mathbb {R}^n$$ R n with $$\textrm{dim}X\ge 2.$$ dim X ≥ 2 . We prove that f is a regular function whenever the restriction $$f|_C$$ f | C is a regular function for every algebraic curve C in X that is an analytic submanifold homeomorphic to the unit circle and has at most one singular point. We also have a suitable version of this result for X not necessarily connected.