### Impacto

### Downloads

Downloads per month over past year

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

PDF
Restringido a Repository staff only 477kB |

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

## 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 |

### Origin of downloads

Repository Staff Only: item control page