### Abstract

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 _{2}p 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 language | English (US) |
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Pages (from-to) | 2829-2834 |

Number of pages | 6 |

Journal | Proceedings of the American Mathematical Society |

Volume | 133 |

Issue number | 10 |

DOIs | |

State | Published - Oct 1 2005 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics