Relative approximation degrees and the henselian rationality problem over perfect fields

Abstract

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.