Related papers: Universally defining $\mathbb{Z}$ in $\mathbb{Q}$ …

We prove that Z in definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Q-morphisms, whether there exists one that…

Julia Robinson has given a first-order definition of the rational integers $\mathbb Z$ in the rational numbers $\mathbb Q$ by a formula $(\forall \exists \forall \exists)(F=0)$ where the $\forall$-quantifiers run over a total of 8…

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…

It is shown that any finitely generated subring of a global field has a universal first-order definition in its fraction field. This covers Koenigsmann's result for the ring of integers and its subsequent extensions to rings of integers in…

We show that the complement of the ring of integers in a number field K is Diophantine. This means the set of ring of integers in K can be written as {t in K | for all x_1, ..., x_N in K, f(t,x_1, ..., x_N) is not 0}. We will use global…

We consider first-order definability and decidability questions over rings of integers of algebraic extensions of $\Q$, paying attention to the uniformity of definitions. The uniformity follows from the simplicity of our first-order…

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…

We study function fields of curves over a base field $K$ which is either a global field or a large field having a separable field extension of degree divisible by $4$. We show that, for any such function field, Hilbert's 10th Problem has a…

We establish that all rings of $S$-integers are universally definable in function fields in one variable over certain ground fields including global and non-archimedean local fields. That is, we show that the complement of such a ring of…

We show that the ring of integers of $\mathbb{Q}^{\text{tr}}$ is existentially definable in the ring of integers of $\mathbb{Q}^{\text{tr}}(i)$, where $\mathbb{Q}^{\text{tr}}$ denotes the field of all totally real numbers. This implies that…

We consider a finite universe U (more exactly - a family U of them) and second order quantifiers Q_K, where for each U this means quantifying over a family of n(K)-place relations closed under permuting U. We define some natural orders and…

Let $K$ be a large field such that $K[\sqrt{-1}]$ is not algebraically closed and $F/K$ a function field in one variable. Extending techniques and results from earlier work with Becher and Dittmann, we show that every valuation ring on $F$…

For a given number field $K$, we give a $\forall\exists\forall$-first order description of affine Darmon points over $\mathbb{P}^1_K$, and show that this can be improved to a $\forall\exists$-definition in a remarkable particular case.…

We offer a $\forall\exists$-definition for (affine) Campana points over $\mathbb{P}^1_K$ (where $K$ is a number field), which constitute a set-theoretical filtration between $K$ and $\mathcal{O}_{K,S}$ ($S$-integers), which are well-known…

We extend results of Denef, Zahidi, Demeyer and the second author to show the following. (1) Rational integers have a single-fold Diophantine definition over the ring of integral functions of any function field of characteristic 0. (2)…

We give model theoretic criteria for $\exists \forall$ and $\forall \exists$- formulas in the ring language to define uniformly the valuation rings $\mathcal{O}$ of models $(K, \mathcal{O})$ of an elementary theory $\Sigma$ of henselian…

We give a definition, in the ring language, of Z_p inside Q_p and of F_p[[t]] inside F_p((t)), which works uniformly for all $p$ and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula…

This paper gives a universal definition of $\mathbb{F}_q [t]$ in $\mathbb{F}_q (t)$ using 89 quantifiers, more direct than those that exist in the current literature. The language $\mathcal{L}_{\mbox{rings}, t}$ we consider here is the…

Given a totally real number field $F$, we show that there are only finitely many totally real extensions of $K$ of a fixed degree that admit a universal quadratic form defined over $F$. We further obtain several explicit classification…

Let $V$ be a valuation ring of a global field $K$. We show that for all positive integers $k$ and $1 < n_1 \leq \ldots \leq n_k$ there exists an integer-valued polynomial on $V$, that is, an element of $\text{Int}(V) = \{ f \in K[X] \mid…