Algebraic versions of Hartogs' theorem.

Let be an uncountable field of characteristic 0. For a given function f : n → , with n ≥ 2 , we prove that f is regular if and only if the restriction f | C is a regular function for every algebraic curve C in n which is either an affine line or is isomorphic to a plane curve in 2 defined by the equation X p − Y q = 0 , where p < q are prime numbers. We also show that regularity of f can be verified on other algebraic curves in n with desired geometric properties. Furthermore, if the field is not algebraically closed, we construct a -valued function on n that is not regular, but all its restrictions to nonsingular algebraic curves in n are regular functions. [ABSTRACT FROM AUTHOR]