On the absence of uniform denominators in Hilbert’s 17th problem

Hilbert showed that for most (n,m)(n,m) there exist positive semidefinite forms p(x1,…,xn)p(x_1,\dots ,x_n) of degree mm which cannot be written as a sum of squares of forms. His 17th problem asked whether, in this case, there exists a form hh so that h2ph^2p is a sum of squares of forms; that is, p...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2005-10, Vol.133 (10), p.2829-2834
Main Author: Reznick, Bruce
Format: Article
Language:English
Subjects:
Algebra
Algebraic geometry
Exact sciences and technology
Field theory and polynomials
Logical theorems
Mathematical inequalities
Mathematical theorems
Mathematics
Number theory
Polynomials
Research article
Sciences and techniques of general use
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Summary:Hilbert showed that for most (n,m)(n,m) there exist positive semidefinite forms p(x1,…,xn)p(x_1,\dots ,x_n) of degree mm which cannot be written as a sum of squares of forms. His 17th problem asked whether, in this case, there exists a form hh so that h2ph^2p is a sum of squares of forms; that is, pp is a sum of squares of rational functions with denominator hh. We show that, for every such (n,m)(n,m) there does not exist a single form hh which serves in this way as a denominator for every positive semidefinite p(x1,…,xn)p(x_1,\dots ,x_n) of degree mm.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-05-07879-2