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

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

2003-06-10

arxiv.org/abs/math/0306163v1

Mathematics

Algebraic Geometry

Submitted to Proceedings of the American Mathematical Society

Scientific paper

Hilbert showed that for most $(n,m)$ there exist psd 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 {\it every} psd $p(x_1,...,x_n)$ of degree $m$.

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