相关论文: Basis discrepancies for extensions of valued field…
A new criterion on normal bases of finite field extension $\mathbb{F}_{q^n} / \mathbb{F}_{q}$ is presented and explicit criterions for several particular finite field extensions are derived from this new criterion.
We study completeness in partial differential varieties. We generalize many results from ordinary differential fields to the partial differential setting. In particular, we establish a valuative criterion for differential completeness and…
Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued…
If $L/K$ is a finite Galois extension of local fields, we say that the valuation criterion $VC(L/K)$ holds if there is an integer $d$ such that every element $x \in L$ with valuation $d$ generates a normal basis for $L/K$. Answering a…
In Formal Concept Analysis, a base for a finite structure is a set of implications that characterizes all valid implications of the structure. This notion can be adapted to the context of Description Logic, where the base consists of a set…
Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on 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 $K$.…
A classical theorem of P. McMullen describes all valuations on polytopes that are invariant under translations and weakly continuous, i.e., continuous with respect to parallel displacements of the facets of a polytope. While it is typically…
Recently, it has been proved that if we have a field extension, then it has linear matching property if and only if L is purely transcendental or is an extension of prime degree. In this note we provide a counterexample for this result.
We extend the characterization of extremal valued fields given in \cite{[AKP]} to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that…
Argumentation frameworks, consisting of arguments and an attack relation representing conflicts, are fundamental for formally studying reasoning under conflicting information. We use methods from mathematical logic, specifically…
We investigate valued fields which admit a valuation basis. Given a countable ordered abelian group G and a real closed, or algebraically closed field F, we give a sufficient condition for a valued subfield of the field of generalized power…
The ruled residue theorem characterises residue field extensions for valuations on a rational function field. Under the assumption that the characteristic of the residue field is different from $2$ this theorem is extended here to function…
For a valued field $(K,v)$, with a fixed extension of $v$ to the algebraic closure $\overline K$ of $K$, and an element $\theta\in\overline K$, we are interested in the possible values of $\theta-\theta'$ where $\theta'$ runs through all…
Let $(K,\nu)$ be an arbitrary-rank valued field, $R_\nu$ its valuation ring, $K(\alpha)/K$ a separable finite field extension generated over $K$ by a root of a monic irreducible polynomial $f\in R_\nu[X]$. We give necessary and sufficient…
In this Note we show that the notion of a basis of a finite-dimensional vector space could be introduced by an argument much weaker than Gauss' reduction method. Our aim is to give a short proof of a simply formulated lemma, which in fact…
For important cases of algebraic extensions of valued fields, we develop presentations of the associated K\"ahler differentials of the extensions of their valuation rings. We compute their annihilators as well as the associated Dedekind…
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…
Suppose $F$ is a field with a nontrivial valuation $v$ and valuation ring $O_{v}$, $E$ is a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We study the topology induced by $w$. We prove that the quasi-valuation…
We consider deformations of finite or infinite dimensional Lie algebras over a field of characteristic 0. There is substantial confusion in the literature if one tries to describe all the non-equivalent deformations of a given Lie algebra.…