Real Algebraic and Analytic Geometry
Submission: 2006, January 18.
In this work we study the Positive Extension (PE) property and Hilbert's 17th Problem for real analytic germs and sets. A real analytic germ X_o of R^n_o has the PE property if every positive semidefinite analytic function germ on X_o has a positive semidefinite analytic extension to R^n_o; analogously one states the PE property for a global real analytic set X in an open set of R^n. These PE properties are natural variations of Hilbert's 17th Problem. Here, we prove that: (1) A real analytic germ X_o of R^3_o has the PE property if and only if every positive semidefinite \em analytic \em function germ on X_o is a sum of squares of analytic function germs on X_o; and (2) a global real analytic set X of dimension 1 or 2 and local embedding dimension less than or equal to 3 has the PE property if and only if it is coherent and all its germs have the PE 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 PE property .
Mathematics Subject Classification (2000): 14P99, 11E25, 32B10, 32S05.
Keywords and Phrases: positive semidefinite analytic function, Positive Extension (PE) property, sum of squares, Hilbert's 17th Problem, singular points.
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