Related papers: Bounded pregeometries and pairs of fields

We construct a topology on a given algebraically closed field with a distinguished subfield which is also algebraically closed. This topology is finer than Zariski topology and it captures the sets definable in the pair of algebraically…

Let $k$ be a complete non-archimedean non-trivial valued field. In this paper, we investigate whether every $k$-algebra homomorphism between $k$-affinoid algebras is automatically bounded. We show that this property holds if and only if…

Working over a field ${\mathbb{k}}$ of characteristic $\ne 2$, we study what we call bisector fields, which are arrangements of paired lines in the plane that have the property that each line in the arrangement crosses the paired lines in…

Every discrete definable subset of a closed asymptotic couple with ordered scalar field $\boldsymbol k$ is shown to be contained in a finite-dimensional $\boldsymbol k$-linear subspace of that couple. It follows that the differential-valued…

This paper is concerned with the model-theoretic study of pairs $(K,F)$ where $K$ is an algebraically closed field and $F$ is a distinguished subfield of $K$ allowing extra structure. We study the basic model-theoretic properties of those…

Let $G$ be a linear algebraic group over a field $k$ of characteristic 0. We show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over an algebraic closure of $k$ are actually conjugate over a finite field…

We investigate when a computable automorphism of a computable field can be effectively extended to a computable automorphism of its (computable) algebraic closure. We then apply our results and techniques to study effective embeddings of…

This paper proposes a new setup for studying pairs of structures. This new framework includes many of the previously studied classes of pairs, such as dense pairs of o-minimal structures, lovely pairs, fields with Mann groups, and…

We prove a triangulation theorem for semi-algebraic sets over a p-adically closed field, quite similar to its real counterpart. We derive from it several applications like the existence of flexible retractions and splitting for…

Marker and Steinhorn shown that given two models $M\prec N$ of an o-minimal theory, if all 1-types over $M$ realized in $N$ are definable, then all types over $M$ realized in $N$ are definable. In this article we characterize pairs of…

We introduce bounded cohomology for (pairs of) groupoids and develop homological algebra to deal with it. We generalise results of Ivanov, Frigerio and Pagliantini to this setting and show that (under topological conditions) the bounded…

We prove that two finite-dimensional commutative algebras over an algebraically closed field are isomorphic if and only if they give rise to isomorphic representations of the category of finite sets and surjective maps.

Given a pair of number fields with isomorphic rings of adeles, we construct bijections between objects associated to the pair. For instance we construct an isomorphism of Brauer groups that commutes with restriction. We additionally…

Let K be a field. For a given valuation on K[x], we determine the structure of its graded algebra and describe its set of key polynomials, in terms of any given key polynomial of minimal degree. We also characterize valuations not admitting…

We give an abstract framework to transfer generalized amalgamation from a simple theory to another, and we apply it to theories of lovely pairs and of bounded PAC structures. We show in particular that bounded pseudo-algebraically closed…

Let K be a number field. A finite group G is called K-admissible if there exists a G-crossed product K-division algebra. K-admissibility has a necessary condition called K-preadmissibility that is known to be sufficient in many cases. It is…

The concept of a skew root of a skew polynomial is used to introduce notions of algebraic closedness for $\sigma$-fields, that is, a field equipped with an endomorphism. It is shown that every $\sigma$-field can be embedded in algebraically…

Pseudo algebraically closed, pseudo real closed, and pseudo $p$-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately. In this text, we propose a unified framework for…

We develop general foundations of topological algebra over a linearly topologized ring k in a format applicable to both formal schemes and analytic adic spaces. We are especially interested in determining exact closed tensor categories of…

Let A be a finitely generated commutative algebra over a field K with a presentation A=K < X_{1}, ..., X_{n} | R >, where R is a set of monomial relations in the generators X_{1}, ..., X_{n}. So A = K[S], the semigroup algebra of the monoid…