On Izumi's theorem on comparison of valuations

Abstract

We prove that the sequence of MacLane key polynomials constructed in [7] and [3] for a valuation extension $(K, ν) \subset (K(x), μ)$ is finite, provided that both ν and μ are divisorial and μ is centered over an analytically irreducible local domain $(R, \mathfrak{m}) \subset K[x]$. As a corollary, we prove Izumi's theorem on comparison of divisorial valuations. We give explicit bounds for the Izumi constant in terms of the key polynomials of the valuations. We show that this bound can be attained in some cases.