Henselian rationality over perfect fields without tameness

Establish henselian rationality for immediate valued function-field extensions of transcendence degree one over perfect fields of positive characteristic after removing the tameness hypothesis, namely determine whether every such extension becomes rational after henselization.

Background

The paper studies immediate extensions of function fields of transcendence degree one and asks when they become rational after henselization. Henselian rationality is known over tame fields, but tameness in positive characteristic requires both defectlessness and perfectness. Since henselian rationality can fail over defectless nonperfect fields, the authors identify the corresponding question over perfect fields as the unresolved general problem.

The paper develops relative approximation degrees, Artin–Schreier reduction, and implicit constant-field methods toward this problem. It proves several sufficient results, including rationality after passage to the implicit constant field under an absolute-ramification hypothesis and rationality for Type II extensions, but does not settle the general perfect-field case.

References

The problem remains open once we remove the tameness hypothesis. Over base fields of positive characteristic, tame fields are precisely those which are defectless and perfect.

— Relative approximation degrees and the henselian rationality problem over perfect fields  (2609.03451 - Dutta et al., 3 Sep 2026) in Section 1, Introduction

However, we do not presently know how to produce such a perturbation in general.

— Relative approximation degrees and the henselian rationality problem over perfect fields  (2609.03451 - Dutta et al., 3 Sep 2026) in Section 1, Introduction, subsection “Artin-Schreier reduction over perfect fields”

Since every such extension is either of Type I or Type II, the preceding corollary leaves precisely the following case open: Let (K,v) be a perfect valued field of positive characteristic satisfying K=Kr, and let (F|K,w) be an immediate extension of valued fields of transcendence degree one. Set L= IC(F|K,w). Assume that (K(X)|K,w) is of Type I for every separating transcendental element X\in F\setminus K. Is (F|K,w) henselian rational? Equivalently, given that Fh = L(Y)h for some element Y, does there necessarily exist Z\in Fh, transcendental over K, such that Fh = K(Z)h?

— Relative approximation degrees and the henselian rationality problem over perfect fields  (2609.03451 - Dutta et al., 3 Sep 2026) in Problem 1, Section “Concluding remarks and open problems”