Related papers: Existentially defining valuations in function fiel…

We consider the two-variable fragment FO^2[<] of first-order logic over finite words. Numerous characterizations of this class are known. Th\'erien and Wilke have shown that it is decidable whether a given regular language is definable in…

We work with rational rank 1 valuations centered in regular local rings. Given an algebraic function field $K$ of transcendence degree 3 over $k$, a regular local ring $R$ with $QF(R)=K$ and a $k$-valuation $\nu$ of $K$, we provide an…

We use the differentiability of the arithmetic volume function and an arithmetic Bertini type theorem to classify when one can find a closed point on the generic fiber of an arithmetic variety, whose heights with respect to some finite…

Let $R$ be a valuation ring with fraction field $K$ and $2\in R^\times$. We give an elementary proof of the following known result: Two unimodular quadratic forms over $R$ are isometric over $K$ if and only if they are isometric over $R$.…

A study of the relation between a noetherian local domain with a given valuation and its associated graded ring with respect to the valuation, which in some cases is an esentially toric variety, possibly of infinite embedding dimension, but…

Let $K$ be a field, $\mathcal {O}_v$ a valuation ring of $K$ associated to a valuation $v$: $K\rightarrow\Gamma\cup\{\infty\}$, and ${\bf m}_v$ the unique maximal ideal of $\mathcal {O}_v$. Consider an ideal $\mathcal {I}$ of the free…

Let (K, v) be a henselian valued field of arbitrary rank. In this paper, we give an irreducibility criterion for multivariate polynomials over K using valuation theory.

We show that it is decidable whether or not a relation on the reals definable in the structure $\langle \mathbb{R}, +,<, \mathbb{Z} \rangle$ can be defined in the structure $\langle \mathbb{R}, +,<, 1 \rangle$. This result is achieved by…

We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general…

We prove the existence of definable retractions onto arbitrary closed subsets of $K^{n}$ definable over Henselian valued fields $K$. Hence directly follows non-Archimedian analogues of the Tietze--Urysohn and Dugundji theorems on extending…

We introduce a first-order theory of finite full binary trees and then identify decidable and undecidable fragments of this theory. We show that the analogue of Hilbert`s 10th Problem is undecidable by constructing a many-to-one reduction…

Let $F$ be a finitely generated regular field extension of transcendence degree $\geq 2$ over a perfect field $k$. We show that the multiplicative group $F^\times/k^\times$ endowed with the equivalence relation induced by algebraic…

Let $G$ be a finite group. Then there exists a first-order statement $S(G)$ in the language of rings without parameters and depending only on $G$ such that, for any field $K$, we have that $K\models S(G)$ if and only if $K$ has a Galois…

Let $\mathcal{F}=(F;+,\cdot,0,1,D)$ be a differentially closed field. We consider the question of definability of the derivation $D$ in reducts of $\mathcal{F}$ of the form $\mathcal{F}_{R}=(F;+,\cdot,0,1,P)_{P \in R}$ where $R$ is a…

Suppose that $k$ is an arbitrary field. Consider the field $k((x_1,...,x_n))$, which is the quotient field of the ring $k[[x_1,...,x_n]]$ of formal power series in the variables $x_1,...,x_n$, with coefficients in $k$. Suppose that $\sigma$…

Consider a simple algebraic valued field extension $(L/K,v)$ and denote by $\mathcal O_L$ and $\mathcal O_K$ the corresponding valuation rings. The main goal of this paper is to present, under certain assumptions, a description of $\mathcal…

In this paper natural necessary and sufficient conditions for quantifier elimination of matrix rings $M_n(K)$ in the language of rings expanded by two unary functions, naming the trace and transposition, are identified. This is used…

The Kochen-Specker theorem proves the inability to assign, simultaneously, noncontextual definite values to all (of a finite set of) quantum mechanical observables in a consistent manner. If one assumes that any definite values behave…

Suppose that $(K,v_0)$ is a valued field, $f(x)\in K[x]$ is a monic and irreducible polynomial and $(L,v)$ is an extension of valued fields, where $L=K[x]/(f(x))$. Let $A$ be a local domain with quotient field $K$ dominated by the valuation…

An expansion of a definably complete field either defines a discrete subring, or the image of a definable discrete set under a definable map is nowhere dense. As an application we show a definable version of Lebesgue's differentiation…