发表机构
IIT Bhubaneswar（印度理工学院布巴内斯瓦尔分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究完美域上的亨塞尔理性问题，完善相对逼近次数理论，推广亨塞尔次数界，分析Artin–Schreier归约，证明特定条件下亨塞尔理性可下传至基域K。
AI 中文摘要
设$(F|K,w)$是特征为$p>0$的秩1完美赋值域$(K,v)$上的超越次数为1的即时赋值函数域，若存在$Y\in F^h$使得$F^h=K(Y)^h$，则称其为亨塞尔理性的。Kuhlmann已证明驯服域上的亨塞尔理性，本文研究其方法可推广至完美域的程度，相对逼近次数是Kuhlmann方法的核心要素。我们首先通过证明每个多项式的相对逼近次数及常数的存在性，完善亨塞尔域上的相关理论，包括代数型伪收敛序列的情形；利用相伴单项式赋值的$j$-不变量，我们通过泰勒展开直接描述这些不变量，并推广Kuhlmann与Vlahu的亨塞尔次数界。接下来研究亨塞尔理性论证基础的Artin–Schreier归约：在完美域上，每个多项式都Artin–Schreier等价于一个相对逼近次数属于$\{1,p\}$的多项式；我们构造了一个显式秩1例子，表明在模$K[X]$的Artin–Schreier像下，$p$不总能约化为1，但在该例子中，过渡到模$K(X)^h$的Artin–Schreier像等价后可约化为1，且所得Artin–Schreier函数域是亨塞尔理性的。最后，假设$K$等于其绝对分歧域，令$L=IC(F|K,w)$为$K$在$F^h$中的相对代数闭包，我们证明$F^h$在$L$上是亨塞尔理性的，且当$L|K$有限时亨塞尔理性可下传至$K$；当某个可分超越元诱导II型扩张时，该有限性条件成立，此时亨塞尔理性成立。
英文摘要
Let $(F|K,w)$ be an immediate valued function field of transcendence degree one over a rank-one perfect valued field $(K,v)$ of characteristic $p>0$. It is henselian rational if $F^h=K(Y)^h$ for some $Y\in F^h$. Kuhlmann proved henselian rationality over tame fields; we investigate how far his method extends to perfect fields. Relative approximation degrees are a central ingredient in Kuhlmann's approach. We first complete their theory over henselian fields by proving the existence of the relative approximation degree and constant of every polynomial, including for pseudo-convergent sequences of algebraic type. Using the $j$-invariants of associated monomial valuations, we describe these invariants directly through Taylor expansions and extend the henselian degree bound of Kuhlmann and Vlahu. We next study the Artin--Schreier reduction underlying the henselian rationality argument. Over perfect fields, every polynomial is Artin--Schreier equivalent to one whose relative approximation degree lies in ${1,p}$. We construct an explicit rank-one example showing that $p$ cannot always be reduced to one modulo the Artin--Schreier image of $K[X]$. Nevertheless, reduction to degree one becomes possible in this example after passing to equivalence modulo the Artin--Schreier image of $K(X)^h$, and the resulting Artin--Schreier function field is henselian rational. Finally, assume that $K$ equals its absolute ramification field, and let $L=IC(F|K,w)$ be the relative algebraic closure of $K$ in $F^h$. We prove that $F^h$ is henselian rational over $L$, and that henselian rationality descends to $K$ whenever $L|K$ is finite. This finiteness condition holds whenever some separating transcendental element induces an extension of Type II, yielding henselian rationality in this case.
