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

Hilbert showed that for most $(n,m)$ there exist positive semidefinite forms $p(x_1,\dots,x_n)$ 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 $p(x_1,\dots,x_n)$ of degree $m$.