Hilbert's 17th Problem and the Champagne Problem

Abstract

1 +t2 B( t) = 2 + t2 E (0( t) is a sum of 2n-th powers of elements in '0(t) for all n. To prove this surprising fact, Becker used his newly developed theory of higher level orderings on fields; the proof was not constructive. He then proposed the following problem: Find an explicit formula giving a representation of B(t) as a sum of 2n-th powers (in '0(t)) for all n. Becker promised a bottle of champagne to the first person to solve this; as a result, the problem became known as the Champagne Problem. The problem still remains unsolved in the form stated by Becker; however, recent work of B. Reznick gives an explicit formula for B(t) as a sum of 2n-th powers of elements in R(t). The theory of higher level orderings on fields, and hence the Champagne Problem, has its genesis in Hilbert's 17th Problem and E. Artin's solution to it. In this paper, we trace the history of these roots of the Champagne Problem and briefly describe Reznick's solution.