Growing Spines: Ad Infinitum et Ad Infinitesimalia

TL;DR

This work resolves automatic definability questions for henselian valuations by linking definability to augmentability in three domains: valued fields, ordered abelian groups, and coloured linear orders. It develops the spine framework of Schmitt and Delon–Lucas to reduce complex model-theoretic questions to convex-subgroup and chain-structure analyses, and then applies Beth’s definability and Ax–Kochen/Ershov principles to obtain precise characterizations. A central achievement is a complete characterization, in characteristic 0, of when an ordered abelian group yields automatic (with parameters) or automatic ∅-definability of valuations, described in terms of strong or weak augmentability by infinitesimals and related spine conditions; the results generalize and refine Krapp–Kuhlmann–Link’s questions. The paper also extends augmentability results to coloured linear orders, provides axiomatisations for spine theories, and demonstrates how the framework yields definability criteria for henselian valuations via the residue field and value group, with clear consequences for the definability landscape in valued fields and DOAGs.

Abstract

We prove that for every ordered abelian group $G$ there exists a non-trivial ordered abelian group $H$ such that $G\preccurlyeq H\oplus G$ with the lexicographic order, and give a first-order characterization of ordered abelian group $G$ such that $G\preccurlyeq G\oplus H$ for some non-trivial $H$. We apply this to characterize which ordered abelian groups (respectively fields) ensure that any henselian valuation with said value group (respectively residue field) is definable in the language of rings. This answers a question of Krapp, Kuhlmann, and Link.

Theorems & Definitions (107)

• Definition 1.1
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Corollary 2.2
• proof
• Example 2.3
• Corollary 2.4