On global Nash functions

Abstract

Let M superset-of R be a compact Nash manifold, and N (M) [resp. O(M)] its ring of global Nash (resp. analytic) functions. A global Nash (resp. analytic) set is the zero set of finitely many global Nash (resp. analytic) functions, and we have the usual notion of irreducible set. Then we say that separation holds for M if every Nash irreducible set is analytically irreducible. The main result of this paper is that separation holds if and only if every semialgebraic subset of M described by s global analytic inequalities can also be described by s global Nash inequalities. In passing, we also prove that when separation holds, every Nash function on a Nash set extends to a global Nash function on M.