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

Fernando Galván, José Francisco (2008) On the positive extension property and Hilbert's 17th problem for real analytic sets. Journal für die reine und angewandte Mathematik, 618 . pp. 1-49. ISSN 0075-4102

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Abstract

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-0 of R-0(n) has the pe property if every positive semidefinite analytic function germ on X-0 has a positive semidefinite analytic extension to R-0(n); analogously one states the pe property for a global real analytic set X in an open set Q of R-0(n). These pe properties are natural variations of Hilbert's 17th problem. Here, we prove that: (1) A real analytic germ X-0 subset of R-0(3) has the pe 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 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.

Item Type:Article
Uncontrolled Keywords:Positive semidefinite analytic function; Positive Extension (PE) propert;Sum of squares; Hilbert’s 17th Problem;Singular points.
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:15124
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