Fernando Galván, José Francisco2023-06-202023-06-2020080075-410210.1515/CRELLE.2008.032https://hdl.handle.net/20.500.14352/49896In 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.engjournal articlehttp://www.maths.manchester.ac.uk/raag/preprints/0189.pdfhttp://www.degruyter.com/restricted access512.7Positive semidefinite analytic functionPositive Extension (PE) propertSum of squaresHilbert’s 17th ProblemSingular points.Geometria algebraica1201.01 Geometría Algebraica