TY - JOUR
T1 - On the absence of uniform denominators in Hilbert's 17th problem
AU - Reznick, Bruce
PY - 2005/10
Y1 - 2005/10
N2 - 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.
AB - 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.
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U2 - 10.1090/S0002-9939-05-07879-2
DO - 10.1090/S0002-9939-05-07879-2
M3 - Article
AN - SCOPUS:26444514585
SN - 0002-9939
VL - 133
SP - 2829
EP - 2834
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 10
ER -