RT Journal Article
T1 On the positive extension property and Hilbert's 17th problem for real analytic sets.
A1 Fernando Galván, José Francisco
AB 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.
PB Walter de Gruyter
SN 0075-4102
YR 2008
FD 2008
LK https://hdl.handle.net/20.500.14352/49896
UL https://hdl.handle.net/20.500.14352/49896
LA eng
NO RAAG
NO GAAR
DS Docta Complutense
RD 21 abr 2025