Related papers: Ordered henselian valued fields: definability and …
We discuss definability of henselian valuation rings in the Macintyre language $\mathcal{L}_{\rm Mac}$, the language of rings expanded by n-th power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian…
We prove relative quantifier elimination for Pal's multiplicative valued difference fields with an added lifting map of the residue field. Furthermore, we generalize a $\mathrm{NIP}$ transfer result for valued fields by Jahnke and Simon to…
Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued…
We show quantifier elimination theorems for real closed valued fields with separated analytic structure and overconvergent analytic structure in their natural one-sorted languages and deduce that such structures are weakly o-minimal. We…
We study directed sets definable in o-minimal structures, showing that in expansions of ordered fields these admit cofinal definable curves, as well as a suitable analogue in expansions of ordered groups, and furthermore that no analogue…
Henselian elements are roots of polynomials which satisfy the conditions of Hensel's Lemma. In this paper we prove that for a finite field extension $(F|L,v)$, if $F$ is contained in the absolute inertia field of $L$, then the valuation…
In his unpublished preprint "Definable Valuations" Koenigsmann shows that every field that admits a t-henselian topology is either real closed or separably closed or admits a definable valuation inducing the t-henselian topology. To show…
A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V=HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are…
We show that arithmetic definability of henselian valuations is preserved by the tilting correspondence. Moreover, we show that if a perfectoid valuation is arithmetically definable, then no parameters are needed. We also investigate…
This paper aims at developing model-theoretic tools to study interpretable fields and definably amenable groups, mainly in $\mathrm{NIP}$ or $\mathrm{NTP_2}$ settings. An abstract theorem constructing definable group homomorphisms from…
Twenty years have passed since Kechris' seminal survey paper [A. S. Kechris, New directions in descriptive set theory. Bull. Symbolic Logic, 5(2):161-174, 1999]. As a follow-up of that work, we review some ot the (anti-)classification…
Consider an expansion $\mathcal R=(R,<,+,\ldots)$ of an ordered divisible Abelian group of finite burden defining no nonempty subset $X$ of $R$ which is dense and codense in a definable open subset $U$ of $R$ with $X \subseteq U$. We…
We are concerned with topology of Hensel minimal structures on non-trivially valued fields $K$, whose axiomatic theory was introduced in a recent paper by Cluckers-Halupczok-Rideau. We additionally require that every definable subset in the…
This paper proves a representation theorem regarding sequences of random elements that take values in a Borel space and are measurable with respect to the sigma algebra generated by an arbitrary union of sigma algebras. This, together with…
Suppose $E \subseteq \mathbb{R}$ is nowhere dense. If $(\mathbb{R},<,+,(x \mapsto \lambda x)_{\lambda \in \mathbb{R} }, E)$ does not define every bounded Borel subset of every $\mathbb{R}^n$ then for every $s > 0$ we have $$ | \{ k \in…
We study the behaviour of forking in valued fields, and we give several sufficient conditions for parameter sets in a Henselian valued field of residue characteristic zero to be an extension base. Notably, we consider arbitrary (potentially…
Here it is shown that standard set theory can be interpreted in a theory about order. The ordering here is about non-extensional flat classes, i.e. classes that are not elements of classes. So, stipulating a nearly well order over all those…
We show that any theory of tame henselian valued fields is NIP if and only if the theory of its residue field and the theory of its value group are NIP. Moreover, we show that if $(K,v)$ is a henselian valued field of residue characteristic…
A celebrated 1969 theorem of Michael Rabin is that the MSO theory of the real order where the monadic quantifier is allowed only to range over the sets of rational numbers, is decidable. In 1975 Saharon Shelah proved that if the monadic…
The notion of newtonianity is central to the study of the ordered differential field of logarithmic-exponential transseries done by Aschenbrenner, van den Dries, and van der Hoeven; see Chapter 14 of arxiv:1509.02588. We remove the…