Rational functions without poles in a compact set

Abstract

Let X be an irreducible nonsingular complex algebraic set and let K be a compact subset of X. We study algebraic properties of the ring of rational functions on X without poles in K. We give simple necessary conditions for this ring to be a regular ring or a unique factorization domain. 1. Introduction and main results. Throughout this note X stands for an irreducible nonsingular algebraic set in C N , for some N. We write O for the sheaf of regular functions on X and regard O(X) and Ox, for any point x in X, as subrings of the field K(X) of rational functions on X. Thus O(X) = x2X Ox ⊆ K(X),