On the positive extension property and Hilbert's 17th problem for real analytic sets

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 X0 of has the property if every positive semidefinite analytic function germ on X0 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 ℝn. 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 X0 is a sum of squares of analytic function germs on X0; 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.