ON THE POSITIVE EXTENSION PROPERTY AND HILBERT’S 17th PROBLEM FOR REAL ANALYTIC SETS
نویسنده
چکیده
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 Xo of Ro has the PE property if every positive semidefinite analytic function germ on Xo has a positive semidefinite analytic extension to Ro ; analogously one states the PE property for a global real analytic set X in an open set Ω of R. These PE properties are natural variations of Hilbert’s 17th Problem. Here, we prove that: (1) A real analytic germ Xo ( Ro has the PE property if and only if every positive semidefinite analytic function germ on Xo is a sum of squares of analytic function germs on Xo; and (2) a global real analytic set X of dimension ≤ 2 and local embedding dimension ≤ 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. AMS Subject Classification: Primary 14P99; secondary 11E25, 32B10, 32S05.
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