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

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Hilbert showed that for most (n, m) there exist positive semidefinite forms p(x 1, ... , x n) of degree m which cannot be written as a sum of squares of forms. His 17th problem asked whether, in this case, there exists a form h so that h 2p is a sum of squares of forms; that is, p is a sum of squares of rational functions with denominator h. We show that, for every such (n, m) there does not exist a single form h which serves in this way as a denominator for every positive semidefinite p(x 1,..., x n) of degree m.

Original languageEnglish (US)
Pages (from-to)2829-2834
Number of pages6
JournalProceedings of the American Mathematical Society
Issue number10
StatePublished - Oct 1 2005


ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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