On Izumi's theorem on comparison of valuations

TL;DR

This paper proves the finiteness of MacLane key polynomials for divisorial valuation extensions and uses this to establish Izumi's theorem, providing explicit bounds for valuation comparison constants.

Contribution

It demonstrates the finiteness of MacLane key polynomials for certain valuation extensions and derives Izumi's theorem with explicit bounds, linking key polynomials to valuation comparison.

Findings

Finiteness of MacLane key polynomials for divisorial valuations.

Existence of Izumi constants is equivalent to polynomial sequence finiteness.

Explicit bounds for Izumi constants in terms of key polynomials.

Abstract

We prove that the sequence of MacLane key polynomials constructed in \cite{Mac1} and \cite{Sp2} for a valuation extension $(K,\nu)\subset (K(x),\mu)$ is finite, provided that both $\nu$ and $\mu$ are divisorial and $\mu$ is centered over an analytically irreducible local domain $(R,\frak{m})\subset K[x]$. As a corollary, we prove Izumi's theorem on comparison of divisorial valuations. %We show that the existence of Izumi constants is equivalent to the finiteness of the sequence of the MacLane key-polynomials. 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.