赋值到有理函数域的常见扩张
Common extensions of valuations to rational function fields
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中文总结 AI 辅助
研究赋值到有理函数域的常见扩张，刻画特定共轭a'使诱导赋值在K(X)上限制为w，还引入正则极限关键多项式等概念，证明正则性与根确定截断赋值的等价性，得出(K,v)稠密时w有正则关键多项式完全序列的结论。
中文摘要 AI 辅助
设(K(X)|K,w)为有理函数域的赋值超越扩张，取最小定义对(a,γ)。本文刻画了a的那些K共轭a'，使得由(a',γ)诱导的单项式赋值在K(X)上限制为w。特别地，我们证明a'满足此条件当且仅当a'和a在(K,v)的亨赛尔化上共轭。本文第二部分涉及抽象关键多项式。我们引入正则极限关键多项式的概念，更一般地，正则关键多项式完全序列的概念。我们证明正则性等同于每个关键多项式的每个根确定相应截断赋值的性质。这通过允许极限关键多项式和赋值代数扩张扩展了Mahboub、Mansour和Spivakovsky的早期工作。因此，当(K,v)在其亨赛尔化中稠密时，我们得到w总是允许一个正则关键多项式完全序列。
英文摘要
Let (K(X)|K,w) be a valuation transcendental extension of rational function fields and take a minimal pair of definition (a, gamma). In this paper, we characterize those K-conjugates a' of a such that the monomial valuation induced by the pair (a', gamma) restricts to w on K(X). In particular, we show that a' satisfies this if and only if a' and a are conjugates over the henselization of (K,v). The second part of the paper concerns abstract key polynomials. We introduce the notion of a regular limit key polynomial, and more generally, that of a regular complete sequence of key polynomials. We prove that regularity is equivalent to the property that every root of every key polynomial determines the corresponding truncated valuation. This extends earlier work of Mahboub, Mansour and Spivakovsky by allowing both limit key polynomials and valuation algebraic extensions. As a consequence, we obtain that w always admits a regular complete sequence of key polynomials whenever (K,v) is dense in its henselization.
