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

Abstract

Hilbert showed that for most ( n , m ) (n,m) there exist positive semidefinite forms p ( x 1 , … , x n ) p(x_1,\dots ,x_n) of degree m 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 h so that h 2 p h^2p is a sum of squares of forms; that is, p p is a sum of squares of rational functions with denominator h h . We show that, for every such ( n , m ) (n,m) there does not exist a single form h h which serves in this way as a denominator for every positive semidefinite p ( x 1 , … , x n ) p(x_1,\dots ,x_n) of degree m m .