Stacked pseudo-convergent sequences and polynomial Dedekind domains

Peruginelli, Giulio

doi:10.2140/ant.2025.19.1947

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Stacked pseudo-convergent sequences and polynomial Dedekind domains

Giulio Peruginelli

Vol. 19 (2025), No. 10, 1947–1972

DOI: 10.2140/ant.2025.19.1947

Abstract

Let $p \in ℤ$ be a prime, $\bar{ℚ_{p}}$ a fixed algebraic closure of the field of $p$ -adic numbers and $\bar{ℤ_{p}}$ the absolute integral closure of the ring of $p$ -adic integers. Given a residually algebraic torsion extension $W$ of $ℤ_{(p)}$ to $ℚ (X)$ , by Kaplansky’s characterization of immediate extensions of valued fields, there exists a pseudo-convergent sequence of transcendental type $E = {s_{n}}_{n \in ℕ} \subset \bar{ℚ_{p}}$ such that

W = ℤ_{(p), E} = {ϕ \in ℚ (X) ∣ ϕ (s_{n}) \in \bar{ℤ_{p}} for all sufficiently large n \in ℕ} .

We show here that we may assume that $E$ is stacked, in the sense that, for each $n \in ℕ$ , the residue field (resp. the value group) of $\bar{ℤ_{p}} \cap ℚ_{p} (s_{n})$ is contained in the residue field (resp. the value group) of $\bar{ℤ_{p}} \cap ℚ_{p} (s_{n + 1})$ ; this property of $E$ allows us to describe the residue field and value group of $W$ . In particular, if $W$ is a DVR, then there exists $α$ in the completion $ℂ_{p}$ of $[t] \bar{ℚ_{p}}$ , $α$ transcendental over $ℚ$ , such that $W = ℤ_{(p), α} = {ϕ \in ℚ (X) ∣ ϕ (α) \in 𝕆_{p}}$ , where $𝕆_{p}$ is the unique local ring of $ℂ_{p}$ ; $α$ belongs to $[t] \bar{ℚ_{p}}$ if and only if the residue field extension $W ∕ M \supseteq ℤ ∕ p ℤ$ is finite. As an application, we provide a full characterization of the Dedekind domains between $ℤ [X]$ and $ℚ [X]$ .

To the memory of my mother

Keywords

pseudo-convergent sequence, residually algebraic extension, distinguished pair, minimal pair, Dedekind domain, integer-valued polynomials

Mathematical Subject Classification

Primary: 12J20, 13A18, 13F05, 13F20, 13F30

Milestones

Received: 21 March 2023

Revised: 10 June 2024

Accepted: 21 October 2024

Published: 5 September 2025

Authors

Giulio Peruginelli
Dipartimento di Matematica “Tullio Levi-Civita” Università di Padova Padova Italy

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Vol. 19, No. 10, 2025