Algebraic versions of Hartogs’ theorem

Abstract

Let [Formula: see text] be an uncountable field of characteristic [Formula: see text]. For a given function [Formula: see text], with [Formula: see text], we prove that [Formula: see text] is regular if and only if the restriction [Formula: see text] is a regular function for every algebraic curve [Formula: see text] in [Formula: see text] which is either an affine line or is isomorphic to a plane curve in [Formula: see text] defined by the equation [Formula: see text], where [Formula: see text] are prime numbers. We also show that regularity of [Formula: see text] can be verified on other algebraic curves in [Formula: see text] with desired geometric properties. Furthermore, if the field [Formula: see text] is not algebraically closed, we construct a [Formula: see text]-valued function on [Formula: see text] that is not regular, but all its restrictions to nonsingular algebraic curves in [Formula: see text] are regular functions.