## Abstract

In this work we study the Positive Extension property and Hilbert's 17th problem for real analytic germs and sets. A real analytic germ *X*
_{0} of has the property if every positive semidefinite analytic function germ on *X*
_{0} has a positive semidefinite analytic extension to ; analogously one states the property for a global real analytic set *X* in an open set Ω of ℝ* ^{n}*. These properties are natural variations of Hilbert's 17th problem. Here, we prove that: (1) A real analytic germ has the property if and only if every positive semidefinite

*analytic*function germ on

*X*

_{0}is a sum of squares of

*analytic*function germs on

*X*

_{0}; and (2) a global real analytic set

*X*of dimension ≦ 2 and local embedding dimension ≦ 3 has the property if and only if it is coherent and all its germs have the property. If that is the case, every positive semidefinite analytic function on

*X*is a sum of squares of analytic functions on

*X*. Moreover, we classify the singularities with the property.

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