Related papers: Algebraic series and valuation rings over nonclose…

As in Zariski's Uniformization Theorem we show that a valuation ring $V$ of characteristic $p>0$ of dimension one is a filtered direct limit of smooth ${\bf F}_p$-algebras under some conditions of transcendence degree. Under mild…

We establish necessary and sufficient conditions for a quadratic polynomial to be irreducible in the ring $Z[[x]]$ of formal power series with integer coefficients. For $n,m\ge 1$ and $p$ prime, we show that $p^n+p^m\beta x+\alpha x^2$ is…

We give an example of a valued field $(K,A)$ such that the valuation ring $A$ is definable by an $L_{\text{ring}}$-formula without parameters, but there is no $\exists\forall\exists$ or $\forall\exists\forall$-formula in $L_{\text{ring}}$…

The main goal of this paper is to characterize the module of K\"ahler differentials for an extension of valuation rings. More precisely, we consider a simple algebraic valued field extension $(L/K,v)$ and the corresponding valuation rings…

We prove the following result. Let k be an infinite perfect field of positive characteristic and assume that strong resolution of singularities holds over k. Let R be a localization of a commutative d-dimensional k-algebra of finite type…

A definable set in a pair (K, k) of algebraically closed fields is co-analyzable relative to the subfield k of the pair if and only if it is almost internal to k. To prove this and some related results for tame pairs of real closed fields…

Let $(R,\mathfrak{m}_R,k)$ be a one-dimensional complete local reduced $k$-algebra over a field of characteristic zero. R. Berger conjectured that $R$ is regular if and only if the universally finite module of differentials $\Omega_R$ is…

Let $f(x)=x^n+ax^2+bx+c \in \Z[x]$ be an irreducible polynomial with $b^2=4ac$ and let $K=\Q(\theta)$ be an algebraic number field defined by a complex root $\theta$ of $f(x)$. Let $\Z_K$ deonote the ring of algebraic integers of $K$. The…

The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field $(K,\nu)$ and an extension $\omega$ of $\nu$ to a finite extension $L$ of $K$.…

These notes form part of a joint research project on the logic of fields with many valuations, connected by a product formula. We define such structures and name them {\em globally valued fields} (GVFs). This text aims primarily at a proof…

We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power…

In this paper we study the Kummer extensions of the power series field $K=k((X_1,...,X_n)$, where $k$ is an algebraically closed field of arbitrary characteristic.

We present a sufficient condition for irreducibility of forcing algebras and study the (non)-reducedness phenomenon. Furthermore, we prove a criterion for normality for forcing algebras over a polynomial base ring with coefficients in a…

The set of formal power series with coefficients in an associative but noncommutative algebra becomes a loop with the substitution product. We initiate the study of this loop by describing certain Lie and Sabinin algebras related to it.…

In this paper we provide necessary and sufficient conditions for $ R=A\propto E $ to be a valuation ring where $E$ is a non-torsion or finitely generated $A-$module. Also, we investigate the $ (n,d) $ property of the valuation ring.

Let $A$ be a right noetherian algebra over a field $k$. If the base field extension $A \otimes_k K$ remains right noetherian for all extension fields $K$ of $k$, then $A$ is called stably right noetherian over $k$. We develop an inductive…

We prove that every non-trivial valuation on an infinite superrosy field of positive characteristic has divisible value group and algebraically closed residue field. In fact, we prove the following more general result. Let $K$ be a field…

We investigate the computability-theoretic properties of valued fields, and in particular algebraically closed valued fields and $p$-adically closed valued fields. We give an effectiveness condition, related to Hensel's lemma, on a valued…

We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility…

Let $R$ be a commutative $k-$algebra over a field $k$. Assume $R$ is a noetherian, infinite, integral domain. The group of $k-$automorphisms of $R$,i.e.$Aut_k(R)$ acts in a natural way on $(R-k)$.In the first part of this article, we study…