Related papers: On the uniqueness of maximal immediate extensions …

Let $(K,v)$ be a henselian valued field. Let $\mathbb{P}^{dless}\subset K[x]$ be the set of monic, irreducible polynomials which are defectless and have degree greater than one. For a certain equivalence relation $\,\approx\,$ on…

The convergence of polyhomogeneous expansions of zero-rest-mass fields in asymptotically flat spacetimes is discussed. An existence proof for the asymptotic characteristic initial value problem for a zero-rest-mass field with…

In this paper, we characterize NIP henselian valued fields modulo the theory of their residue field, both in an algebraic and in a model-theoretic way. Assuming the conjecture that every infinite NIP field is either separably closed, real…

We continue the study of $n$-dependent groups, fields and related structures, largely motivated by the conjecture that every $n$-dependent field is dependent. We provide evidence towards this conjecture by showing that every infinite…

The u-invariant of a field is the largest dimension of an anisotropic quadratic torsion form over the field. In this article we obtain a bound on the u-invariant of function fields in one variable over a henselian valued field with…

Let $(K, \nu)$ be a valued field, the notions of \emph{augmented valuation}, of \emph{limit augmented valuation} and of \emph{admissible family} of valuations enable to give a description of any valuation $\mu$ of $K [x]$ extending $\nu$.…

A henselian valued field $K$ is called separably tame if its separable-algebraic closure $K^{\operatorname{sep}}$ is a tame extension, that is, the ramification field of the normal extension $K^{\operatorname{sep}}|K$ is…

We establish the first extension results for divergence-free (or solenoidal) elements of $\mathrm{L}^{1}$-based function spaces. Here, the key point is to preserve the solenoidality constraint while simultaneously keeping the underlying…

Following our first article, we continue to investigate ultrametic modules over a ring of twisted polynomials of the form $[K;\vfi]$, where $\vfi$ is a ring endomorphism of $K$. The main motivation comes from the the theory of valued…

Let v be a rank m discrete valuation of k[[X1,...,Xn]] with dimension n-m. We prove that there exists an inmediate extension L of K where the valuation is monomial. Therefore we compute explicitly the residue field of the valuation.

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 introduce a new method of constructing complete sequences of key polynomials for simple extensions of tame fields. In our approach the key polynomials are taken to be the minimal polynomials over the base field of suitably constructed…

We study asymptotic jumping numbers for graded sequences of ideals, and show that every such invariant is computed by a suitable real valuation of the function field. We conjecture that every valuation that computes an asymptotic jumping…

In this paper we present some basic uniqueness results for evolutive equations under density constraints. First, we develop a rigorous proof of a well-known result (among specialists) in the case where the spontaneous velocity field…

Given an extendible spacetime one may ask how much, if any, uniqueness can in general be expected of the extension. Locally, this question was considered and comprehensively answered in a recent paper of Sbierski, where he obtains local…

We investigate fields of characteristic 0 and dp-rank 2. While we do not obtain a classification, we prove that any unstable field of characteristic 0 and dp-rank 2 admits a unique definable V-topology. If this statement could be…

One can associate to a valued field an inverse system of valued hyperfields $(\mathcal{H}_i)_{i \in I}$ in a natural way. We investigate when, conversely, such a system arise from a valued field. First, we extend a result of Krasner by…

Gaussian random fields on Euclidean spaces whose variances reach their maximum values at unique points are considered. Exact asymptotic behaviors of probabilities of large absolute maximum of theirs trajectories have been evaluated using…

For an arbitrary valued field $(K,v)$ and a given extension $v(K^*)\hookrightarrow\Lambda$ of ordered groups, we analyze the structure of the tree formed by all $\Lambda$-valued extensions of $v$ to the polynomial ring $K[x]$. As an…

We develop a theory of extensions of hyperfields that generalizes the notion of field extensions. Since hyperfields have a multivalued addition, we must consider two kinds of extensions that we call weak hyperfield extensions and strong…