Related papers: Existentially defining valuations in function fiel…

We show a universal algebraic local characterisation of the expressive power of finite-valued languages with domains of arbitrary cardinality and containing arbitrary many cost functions.

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…

Let $R$ be a Dedekind ring, $K$ its quotient field, and $L=K(\alpha)$ a finite field extension of $K$ defined by a monic irreducible polynomial $f(x)\in R[x]$. We give an easy version of Dedekind's criterion which computationally improves…

We present a logical separability analysis for a functional quantum computation language. This logic is inspired by previous works on logical analysis of aliasing for imperative functional programs. Both analyses share similarities notably…

Let $K$ be a characteristic zero algebraic function field with a valuation $\nu$. Let $L$ be a finite extension of $K$ and $\omega$ be an extension of $\nu$ to $L$. We establish that the valuation ring $V_{\omega}$ of $\omega$ is…

We consider the fragment F of first order arithmetic in which quantification is restricted to ''for all but finitely many.'' We show that the integers form an F-elementary substructure of the real numbers. Consequently, the F-theory of…

Existential rules are a positive fragment of first-order logic that generalizes function-free Horn rules by allowing existentially quantified variables in rule heads. This family of languages has recently attracted significant interest in…

In this short note we give some corollaries of the polynomial inverse function theorem for large fields. We prove inverse and implicit function theorems for Nash maps over large fields, characterize large fields as fields satisfying inverse…

Consider a complete discrete valuation ring $\mathcal{O}$ with quotient field $F$ and finite residue field. Then the inclusion map $\mathcal{O} \hookrightarrow F$ induces a map $\hat{\mathrm{K}}^\mathrm{M}_*\mathcal{O} \to…

We prove that a function definable with parameters in an o-minimal structure is bounded away from infinity as its argument goes to infinity by a function definable without parameters, and that this new function can be chosen independently…

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…

Let $F$ be a field, let $D$ be a subring of $F$ and let $Z$ be an irreducible subspace of the space of all valuation rings between $D$ and $F$ that have quotient field $F$. Then $Z$ is a locally ringed space whose ring of global sections is…

We consider the class of languages defined in the 2-variable fragment of the first-order logic of the linear order. Many interesting characterizations of this class are known, as well as the fact that restricting the number of quantifier…

Let $R$ be an o-minimal expansion of a group in a language in which $\textrm{Th}(R)$ eliminates quantifiers, and let $C$ be a predicate for a valuational cut in $R$. We identify a condition that implies quantifier elimination for…

In this paper we address the decision problem for a fragment of set theory with restricted quantification which extends the language studied in [4] with pair related quantifiers and constructs, in view of possible applications in the field…

We prove that an expansion of an algebraically closed field by $n$ arbitrary valuation rings is NTP${}_2$, and in fact has finite burden. It fails to be NIP, however, unless the valuation rings form a chain. Moreover, the incomplete theory…

We study structural limitations of purely algebraic reasoning in the analysis of arithmetic dynamical systems. Rather than addressing the truth of specific conjectures, we introduce a fragment - relative notion of algebraic refutability for…

We study groups definable in existentially closed geometric fields with commuting derivations. Our main result is that such a group can be definably embedded in a group interpretable in the underlying geometric field. Compared to earlier…

We associate to any given finite set of valuations on the polynomial ring in two variables over an algebraically closed field a numerical invariant whose positivity characterizes the case when the intersection of their valuation rings has…

We provide an irreducibility test in the ring K[[x]][y] whose complexity is quasi-linear with respect to the discriminant valuation, assuming the input polynomial F square-free and K a perfect field of characteristic zero or greater than…