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Journal für die reine und angewandte Mathematik

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

José F. Fernando1

1Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain. e-mail: josefer@mat.ucm.es

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2008, Issue 618, Pages 1–49, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2008.032, July 2008

Publication History

Received:
2006-07-12
Published Online:
2008-07-01

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 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 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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