Title: Denominators in Hilbert’s 17th Problem
Abstract: A real form of degree d in n variables is “psd" if it only takes non negative values, and is “sos” if it can be written as a sum of squares of forms of degree d/2; sos implies psd. But Hilbert proved in 1888 that there are psd ternary sextic forms ((n,d) = (3,6)) which are not sos and in 1893 proved that every psd ternary form is a sum of squares of rational functions. (On taking the common denominator, if p is psd then there exists F so that F^2p is sos.) His 17th problem was to prove this for n > 3, which Artin did in the 1920s in a non-constructive way. The first explicit examples of psd-not-sos forms didn’t appear until the 1960s.
I’ll survey what is known about denominators. For example, if p is a psd form which is strictly definitive (p(u) > 0 for u in S^{n-1}), then for a computable large N, (\sum_j x_j^2)^N*p is sos. On the other hand, there exist psd forms p with the property that every odd power of p is not sos. The proofs will be elementary.