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

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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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Official URL: http://www.maths.manchester.ac.uk/raag/preprints/0189.pdf


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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
Deposited On:09 May 2012 11:05
Last Modified:06 Sep 2018 15:42

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