Hartogs-type theorems in real algebraic geometry, I

TL;DR

This paper establishes criteria for the regularity of functions on real algebraic sets by examining their behavior on algebraic curves and surfaces, extending classical Hartogs-type theorems in real algebraic geometry.

Contribution

It introduces new conditions under which the regularity of a function can be inferred from its restrictions to algebraic curves and surfaces, including cases with singularities.

Findings

Regularity can be detected on algebraic curves in X.

Regularity can be detected on algebraic surfaces in X.

Results apply to both connected and disconnected sets.

Abstract

Let f:X-->R be a function defined on a connected nonsingular real algebraic set X in R^n. We prove that regularity of f can be detected on either algebraic curves or surfaces in X. If dimX>1 and k is a positive integer, then f is a regular function whenever the restriction f|C is a regular function for every algebraic curve C in X that is a C^k submanifold homeomorphic to the unit circle and is either nonsingular or has precisely one singularity. Moreover, in the latter case, the singularity of C is equivalent to the plane curve singularity defined by the equation x^p=y^q for some primes p<q. If dimX>2, then f is a regular function whenever the restriction f|S is a regular function for every nonsingular algebraic surface S in X that is homeomorphic to the unit 2-sphere. We also have suitable versions of these results for X not necessarily connected.