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

2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67752Hilbert 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 $m$.Submitted to Proceedings of the American Mathematical SocietyAlgebraic GeometryCommutative AlgebraNumber Theory11E10, 11E25, 11E76, 12D15, 14P99On the absence of uniform denominators in Hilbert's 17th problemtext