Hartogs-type theorems in real algebraic geometry, I

Abstract Let f : X ? ? {f:X\rightarrow\mathbb{R}} be a function defined on connected nonsingular real algebraic set X in n {\mathbb{R}^{n}} . We prove that regularity of f can detected by controlling the restrictions to either curves or surfaces If dim ? ? 2 {\operatorname{dim}X\geq 2} and k is positive integer, then regular whenever restriction | C {f|_{C}} for every curve C ???? k {\mathcal{C}^{k}} submanifold homeomorphic unit circle has precisely one singularity. Moreover, latter case, singularity equivalent plane equation x p = y q {x^{p}=y^{q}} some primes < {p<q} 3 3} , S {f|_{S}} surface S 2-sphere. also have suitable versions these results not necessarily connected.