arXiv subjects

Pierre Touchard

Publications and source records attributed to Pierre Touchard.

10 recordsLinked to original sources

arXiv2026

Linear theories of global fields with absolute values

We study the theory of a global field k as a k-vector space with a predicate for one of the absolute values on k. For example, we prove that in this language a global field with an ultrametric or real archimedean absolute value has a decidable theory, while with a complex absolute value the theory is always undecidable. We also study the existential theories and axiomatize k together with predicates for all non-complex absolute values on k simultaneously.

math.LO↗

arXiv2025

Growing Spines: Ad Infinitum et Ad Infinitesimalia

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.

math.LO↗

arXiv2025

Growing Spines Ad Infinitum

We show that every non-trivial ordered abelian group $G$ is augmentable by infinite elements, i.e., we have $G\preccurlyeq H\oplus G$ for some non-trivial ordered abelian group $H$. As an application, we show that when $k$ is a field of characteristic 0, then $k$ is not $t$-henselian if and only if all henselian valuations with residue field $k$ are ($\emptyset$-)definable.

math.LO↗

arXiv2024

Mekler's Construction and Murphy's Law for 2-Nilpotent Groups

Mekler's construction is a powerful technique for building purely algebraic structures from combinatorial ones. Its power lies in the fact that it allows various model-theoretic tameness properties of the combinatorial structure to transfer to the algebraic one. In this paper, we push this ideology much further, describing a broad class of properties that transfer through Mekler's construction. This technique subsumes many well-known results and opens avenues for many more. As a straightforward application of our methods, we (1) obtain transfer principles for stably embedded pairs of Mekler groups and (2) construct strictly $\mathsf{NFOP}_k$ pure groups for all $k\in\mathbb{N}_{>2}$. We also answer a question of Chernikov and Hempel on transfer of burden.

math.LO↗

arXiv2023

Generalised Indiscernibles, Dividing Lines, and Products of Structures

Generalised indiscernibles highlight a strong link between model theory and structural Ramsey theory. In this paper, we use generalised indiscernibles as tools to prove results in both these areas. More precisely, we first show that a reduct of an ultrahomogenous $\aleph_0$-categorical structure which has higher arity than the original structure cannot be Ramsey. In particular, the only nontrivial Ramsey reduct of the generically ordered random $k$-hypergraph is the linear order. We then turn our attention to model-theoretic dividing lines that are characterised by collapsing generalised indiscernibles, and prove, for these dividing lines, several transfer principles in (full and lexicographic) products of structures. As an application, we construct new algorithmically tame classes of graphs.

math.LO↗

arXiv2023

Stably Embedded Pairs of Ordered Abelian Groups

We investigate when an ordered abelian group $G$ is stably embedded in a given elementary extension $H$. We focus on a large class of ordered groups which includes maximal ordered groups with interpretable archimedean valuation. We give a complete answer for groups in this class which takes the form of a transfer principle for valued groups. It follows in particular that all types in the lexicographic product $\prod_{i\in \omega} \mathbb{Z}$ are definable.

math.LO↗

arXiv2022

On the hyperfields associated to valued fields

One can associate to a valued field an inverse system of valued hyperfields $(\mathcal{H}_i)_{i \in I}$ in a natural way. We investigate when, conversely, such a system arise from a valued field. First, we extend a result of Krasner by showing that the inverse limit of certain systems are stringent valued hyperfields. Secondly, we describe a Hahn-like construction which yields a henselian valued field from a stringent valued hyperfield. In addition, we provide an axiomatisation of the theory of stringent valued hyperfields in a language consisting of two binary function symbols $\oplus$ and $\cdot$ and two constant symbols $\textbf{0}$ and $\textbf{1}$.

math.RA↗

arXiv2021

On Model Theory of Valued Vector Spaces

In analogy to valued fields, we study model-theoretic properties of valued vector spaces with variable base field by proving transfer principles down to the skeleton and down to the value set and base field. For instance, we give a formula which computes its burden in terms of the burden of its base field and its value set. To do this, we study these transfer principles in the context of lexicographic products of structures.

math.LO↗

arXiv2020

Stably embedded submodels of Henselian valued fields

We show a transfer principle for the property that all types realised in a given elementary extension are definable. It can be written as follows: a Henselian valued fields is stably embedded in an elementary extension if and only if its value group is stably embedded in its corresponding extension, its residue field is stably embedded in its corresponding extension, and the extension of valued fields satisfies a certain algebraic condition. We show for instance that all types over the Hahn field $\mathbb{R}((\mathbb{Z}))$ are definable. Similarly, all types over the quotient field of the Witt ring $W(\mathbb{F}_p^{\text{alg}})$ are definable. This extends a work of Cubides and Delon and of Cubides and Ye.

math.LO↗

arXiv2020

Burden in Henselian Valued Fields

In the spirit of the Ax-Kochen-Ershov principle, we show that in certain cases the burden of a Henselian valued field can be computed in terms of the burden of its residue field and that of its value group. To do so, we first see that the burden of such a field is equal to the burden of its RV-sort. These results are generalisations of a work of Chernikov and Simon.

math.LO↗