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 Fh=K(Y)h for some Y∈ Fh. 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 Fh. We prove that Fh 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.
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BibTeX
@misc{dutta2026relative,
title = {{Relative approximation degrees and the henselian rationality problem over perfect fields}},
author = {Arpan Dutta and Rumi Ghosh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03451},
eprint = {2609.03451},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03451},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Relative approximation degrees and the henselian rationality problem over perfect fields
AU - Arpan Dutta
AU - Rumi Ghosh
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03451
UR - https://arxiv.org/abs/2609.03451
AB - 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∈ 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.
ER -