Well-ordered Valuations on Characteristic Zero Rational Function Fields of Transcendence Degree Two

Given a valuation on the function field k(x, y), we examine the set of images of nonzero elements of the underlying polynomial ring k[x, y] under this valuation. For an arbitrary field k, a Noetherian power series is a map z : Q → k that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on k(x, y). Although the value groups corresponding to such valuations are well-understood, the restrictions of the valuations to underlying polynomial rings have yet to be characterized. Let Λn denote the images under the valuation v of all nonzero polynomials f ∈ k[x, y] of at most degree n in the variable y. We construct a bound for the growth of Λn with respect to n for arbitrary valuations, and then specialize to valuations that arise from Noetherian power series. We provide a sufficient condition for this bound to be tight.