关于Izumi的估值比较定理
On Izumi's theorem on comparison of valuations
摘要：
我们证明了在\cite{Mac1}和\cite{Sp2}中构造的MacLane关键多项式序列对于一个赋值扩张$(K,ν)\subset (K(x),μ)$是有限的，只要$ν$和$μ$都是整除的，并且$μ$集中在一个解析不可约的局部域$(R,\frak{m})\subset K[x]$上.作为推论，我们证明了Izumi关于除数赋值比较的定理。我们证明了Izumi常数的存在性等价于MacLane键多项式序列的有限性。我们给出了明确的界限的泉常数的关键多项式的估值。我们证明了在某些情况下可以达到这个界限。
英文摘要：
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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