Related papers: Axiomatizing the existential theory of Fq((t))
We study the existential (and parts of the universal-existential) theory of equicharacteristic henselian valued fields. We prove, among other things, an existential Ax-Kochen-Ershov principle, which roughly says that the existential theory…
We study fragments of the existential theory of henselian valued fields with parameters. This includes the $\exists_n$-fragment in the equicharacteristic or unramified mixed characteristic case, the $\exists_n\exists_1$-fragment in the…
We study existential theories of henselian valued fields of positive characteristic with parameters from a trivially valued subfield. Compared to previous work, we relax perfectness and separability assumptions, and instead work with the…
We set up general machinery to study interpretations of fragments of theories. We then apply this to existential fragments of theories of fields, and especially of henselian valued fields. As an application we prove many-one reductions…
Assuming a certain form of resolution of singularities, we prove a general existential Ax-Kochen/Ershov principle for tamely ramified fields in all characteristics. This specializes to well-known results in residue characteristic $0$ and…
We resolve the strong Elementary Equivalence versus Isomorphism Problem for finitely generated fields. That is, we show that for every field in this class there is a first-order sentence which characterizes this field within the class up to…
A field is existentially t-henselian if it is has the same existential theory in the first-order language of rings as a field that admits a nontrivial henselian valuation. This property turns out to be equivalent to $\mathbb{Z}$-largeness,…
We study various universal-existential fragments of first-order theories of fields, in particular of function fields and of equicharacteristic henselian valued fields. For example we discuss to what extent the theory of a field k determines…
We provide axiomatization and relative quantifier elimination for valued fields equipped with an automorphism, in residue characteristic zero. Similar results are known under strong assumptions on the interaction between the automorphism…
Recently, Anscombe and Koenigsmann gave an existential 0-definition of the ring of formal power series F[[t]] in its quotient field in the case where F is finite. We extend their method in several directions to give general definability…
We construct an existentially undecidable complete discretely valued field of mixed characteristic with existentially decidable residue field and decidable algebraic part, answering a question by Anscombe-Fehm in a strong way. Along the…
In this paper we study solution attempts for a problem posed by Ali Enayat: can there be a finitely axiomatized consistent sequential theory that interprets itself plus the (sentential or non-uniform) Tarski biconditionals? We provide a…
We prove two completeness results, one for the extension of dependence logic by a monotone generalized quantifier Q with weak interpretation, weak in the meaning that the interpretation of Q varies with the structures. The second result…
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 investigate the theory of finite observables, i.e., resolutions of the finite-dimensional identity by means of positive operators, that have a physical interpretation in terms of measurement schemes. We focus on extremal and rank-one…
We study valued fields equipped with an automorphism. We prove that all of them have an extension admitting an equivariant cross-section of the valuation. In residual characteristic zero, and in the presence of such a cross-section, we show…
A theory of simultaneous resolution of singularities for families of embedded varieties (over a field of characteristic zero) parametrized by the spectrum of a suitable artinian ring, and compatible with a given algorithm of resolution, is…
In concurrency theory, weak bisimilarity is often used to relate processes exhibiting the same observable behaviour. The probabilistic environment gives rise to several generalisations; we study the infinitary semantics, which abstracts…
So far there exist just a few results about the uniqueness of maximal immediate valued differential field extensions and about the relationship between differential-algebraic maximality and differential-henselianity; see arXiv:1509.02588,…
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…