On a global analytic Positivstellensatz

Abstract

We consider several modified versions of the Positivstellensatz for global analytic functions that involve infinite sums of squares and/or positive semidefinite analytic functions. We obtain a general local-global criterion which localizes the obstruction to have such a global result. This criterion allows us to get completely satisfactory results for analytic curves, normal analytic surfaces and real coherent analytic sets whose connected components are all compact.

Positive Semidefinite Analytic Functions on Real Analytic Surfaces

“…The fact that the set B(X) of non-normal points of X appears in the set of poles of a representation of a positive semidefinite analytic function on a C-analytic surface X as sums of squares of meromorphic functions makes it difficult, even in the coherent case, to obtain a weak Positivstellensatz in the sense of [ABF1].…”

The Nullstellensatz for real coherent analytic surfaces

“…Now unlike the algebraic case we can find positive semidefinite functions which can be represented as a sum of infinitely many squares of analytic functions and not as a finite sum (this phenomenon appears already in dimension 3 see [2,Example 5.16]). However, it is clear that if an infinite sum of squares ∞ m=1 f m 2 of global analytic functions vanishes on an analytic set, all the functions f m should vanish on it.…”