zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
An elementary and constructive solution to Hilbert’s 17th problem for matrices. (English) Zbl 1126.12001

A real matrix A is called positive semidefinite if it is symmetric with all nonnegative eigenvalues. The main aim of the paper is the following theorem: If a symmetric matrix A with entries in R[x 1 ,,x n ] is positive semidefinite for all sustitutions (x 1 ,,x n ) n , then A can be expressed as a sum of squares of symmetric matrices with entries in R(x 1 ,,x n )·

This generalizes the famous Artin’s theorem solving Hilbert’s 17th problem. The theorem proved by the authors was originally proved by C. Procesi and M. Schacher [Ann. Math. (2) 104, 395–406 (1976; Zbl 0347.16010)]. However, the proof in the paper under review is constructive modulo the fact that one can represent the entries of A as a sum of rational functions.

12D15Formally real fields
13J30Real algebra
15A21Canonical forms, reductions, classification
15A54Matrices over function rings