A characterization of sums of 2nth powers of global meromorphic functions

Abstract

Hilbert's 17th Problem asked the following. Let f(x1,⋯,xn) be a real polynomial which for all real values α1,⋯,αn satisfies f(α1,⋯,αn)≥0. Is it true that f=∑(gi/hi)2 for polynomials gi,hi∈R[x1,⋯,xn]? (It was known that f=∑(gi)2, gi∈R[x1,⋯,xn], is not always true.) Artin gave a positive answer to this problem. In this paper the author proves an analogous theorem with f replaced by a real-analytic function on a compact analytic subvariety X of a real analytic manifold, the rational functions gi/hi replaced by meromorphic functions, the power 2 replaced by 2n and the condition "f(α1,⋯,αn)≥0 for all real α1,⋯,αn ''replaced by a suitable condition that reduces in the case n=1, X=Rn, to "f≥0''. This condition is as follows: For every analytic curve σ:(−ε,ε)→X for which (1) the germ of the image of σ at σ(0) is not contained in the germ of the singular set of X at σ(0) and (2) f∘σ(t)=atm+⋯(a≠0), one has a>0 and 2n∣m. The author studies the relationship between curves satisfying (1) and certain valuations on the field of meromorphic functions. On the other hand, Becker's theory of 2nth powers in formally real fields related the question of representing an element as the sum of 2nth powers to the study of valuations on the field. These two theories are combined to yield the 2nth power, analytic analogue of the original Hilbert problem.