On Hilbert 17th problem and real nullstellensatz for global analytic functions

Abstract

The author proves a Nullstellensatz for the ring of real analytic functions on a compact analytic manifold. The main results are the following. Theorem 1: Let X be a compact irreducible analytic set of a real analytic manifold M and f:X→R a nonnegative analytic function. Then f is a sum of squares of meromorphic functions. Theorem 2: Let I be a finitely generated ideal of O(M) with Z(I) compact. Then IZ(I)=I√R, where Z(I) is the zero set of I, IZ(I) the ideal of functions (in O(M)) vanishing on I, and I√R the real radical of I (i.e. the set of functions f in O(M) such that there exist g1,⋯,gk and an integer p with f2p + g2 1 + ⋯ +g2k ∈ I). Corollary: Let I be as in Theorem 2. Then IZ(I)=I if and only if I is real (i.e. I=I √ R). The proofs are based on results about extension of orders.