Related papers: Definable henselian valuation rings

We prove that the existential theory of any function field $K$ of characteristic $p> 0$ is undecidable in the language of rings provided that the constant field does not contain the algebraic closure of a finite field. We also extend the…

These notes form part of a joint research project on the logic of fields with many valuations, connected by a product formula. We define such structures and name them {\em globally valued fields} (GVFs). This text aims primarily at a proof…

We show how suitable extensions $(L|K,v)$ of prime degree of valued fields give rise to definable coarsenings of the valuation rings of $L$ and $K$. In the case of Artin-Schreier and Kummer extensions with wild ramification, we can also…

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…

We show that functors like algebraic $K$-theory (such as unitary or symplectic $K$-functors), as well as the higher Grothendieck--Witt groups, possess the local constancy condition for Henselian valuation rings. Namely, taken with finite…

We study the model theory of finitely ramified henselian valued fields of fixed initial ramification, obtaining versions of the Ax-Kochen-Ershov principle as follows. We identify the induced structure on the residue field and show that once…

We show that ${\mathbb Z}$ is definable in ${\mathbb Q}$ by a universal first-order formula in the language of rings. We also present an $\forall\exists$-formula for ${\mathbb Z}$ in ${\mathbb Q}$ with just one universal quantifier. We…

We give an 'arithmetic regularity lemma' for groups definable in finite fields, analogous to Tao's 'algebraic regularity lemma' for graphs definable in finite fields. More specifically, we show that, for any $M>0$, any finite field…

Building on work of Kuhlmann and Lisinski, we study the theory of the Hahn series field $\mathbb{F}_{q}(\!(\mathbb{Q})\!)$, over a finite field $\mathbb{F}_{q}$, equipped with the $t$-adic valuation, in a language of valued fields. We prove…

We give a definition of a class of Dedekind domains which includes the rings of integers of global fields and give a proof that all rings in this class have finite ideal class group. We also prove that this class coincides with the class of…

For a discrete valuation ring $R$ with quotient field $K$ and residue field $F$ both of characteristic not 2, we study low-dimensional quadratic forms with Witt class in the $n$-th power of the fundamental ideal of $F$ resp. $K$ and point…

Let $(K,\nu)$ be an arbitrary-rank valued field, $R_\nu$ its valuation ring, $K(\alpha)/K$ a separable finite field extension generated over $K$ by a root of a monic irreducible polynomial $f\in R_\nu[X]$. We give necessary and sufficient…

We prove a negative solution to the analogue of Hilbert's tenth problem for rings of one variable non-Archimedean entire functions in any characteristic. In the positive characteristic case we prove more: the ring of rational integers is…

We show pro-definability of spaces of definable types in various classical complete first order theories, including complete o-minimal theories, Presburger arithmetic, $p$-adically closed fields, real closed and algebraically closed valued…

We show that rings of $S$-integers of a global function field $K$ of odd characteristic are first-order universally definable in $K$. This extends work of Koenigsmann and Park who showed the same for $\mathbb{Z}$ in $\mathbb{Q}$ and the…

Let A and B be integral domains. Suppose A is Noetherian and B is a finitely generated A-algebra that contains A. Denote by A' the integral closure of A in B. We show that A' is determined by finitely many unique discrete valuation rings.…

Given an elliptic curve $E$ over a perfect defectless henselian valued field $(F,\mathrm{val})$ with perfect residue field $\textbf{k}_F$ and valuation ring $\mathcal{O}_F$, there exists an integral separated smooth group scheme…

Let O be a complete discrete valuation ring with finite residue field k of odd characteristic. Let G be a general or special linear group or a unitary group defined over O and let $\mathfrak{g}$ denote its Lie algebra. For every positive…

Consider a Henselian rank one valued field $K$ of equicharacteristic zero along with the language $\mathcal{L}^{P}$ of Denef--Pas. Let $f: A \to K$ be an $\mathcal{L}^{P}$-definable (with parameters) function on a subset $A$ of $K^{n}$. We…

Let $T$ be a complete, model complete o-minimal theory extending the theory of real closed ordered fields and assume that $T$ is power bounded. Let $K$ be a model of $T$ equipped with a $T$-convex valuation ring $\mathcal{O}$ and a…