On Izumi's theorem on comparison of valuations

Details On Izumi's theorem on comparison of valuations On Izumi's theorem on comparison of valuations

2008-10-14

arxiv.org/abs/0810.2479v2

Mathematics

Commutative Algebra

Accepted for publication in "Kodai Mathematical Journal"

Scientific paper

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.

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