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arXiv · 0810.2479

On Izumi's theorem on comparison of valuations

Mohammad Moghaddam
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Abstract

We prove that the sequence of MacLane key polynomials constructed in \cite{Mac1} and \cite{Sp2} 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 $(R,\frak{m})\subset K[x]$. 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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Mohammad Moghaddam. 2010-03-18. On Izumi's theorem on comparison of valuations. https://arxiv.org/abs/0810.2479

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