Related papers: Notes on valuation theory for Krasner hyperfields
We previously obtained a generalization and refinement of results about the ramification theory of Artin-Schreier extensions of discretely valued fields in characteristic $p$ with perfect residue fields to the case of fields with more…
In this paper, we give a valuation formula for rational top differential forms of function fields in characteristic zero for arbitrary Abhyankar places generalizing the classical valuation at prime divisors. This enables us to define log…
Given a valued field $(K,v)$ and its completion $(\widehat{K},v)$, we study the set of all possible extensions of $v$ to $\widehat{K}(X)$. We show that any such extension is closely connected with the underlying subextension $(K(X)|K,v)$.…
We develop a first-order theory of ordered transexponential fields in the language $\{+,\cdot,0,1,<,e,T\}$, where $e$ and $T$ stand for unary function symbols. While the archimedean models of this theory are readily described, the study of…
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.
The continuous logic of globally valued fields -- A globally valued field is a field endowed with a family of absolute values that satisfy a product formula. Number fields and function fields in one variable give classical and fundamental…
We study the behaviour of forking in valued fields, and we give several sufficient conditions for parameter sets in a Henselian valued field of residue characteristic zero to be an extension base. Notably, we consider arbitrary (potentially…
Admitting a non-trivial $p$-henselian valuation is a weaker assumption on a field than admitting a non-trivial henselian valuation. Unlike henselianity, $p$-henselianity is an elementary property in the language of rings. We are interested…
In this essay we explore the notion of essential dimension using the theory of valuations of fields. Given a field extension K/k and a valuation on K that is trivial on k, we prove that the rank of the valuation cannot exceed the…
We build on previous work on multirings (\cite{roberto2021quadratic}) that provides generalizations of the available abstract quadratic forms theories (special groups and real semigroups) to the context of multirings…
A quasi-order is a binary, reflexive and transitive relation. In the Journal of Pure and Applied Algebra 45 (1987), S.M. Fakhruddin introduced the notion of (totally) quasi-ordered fields and showed that each such field is either an ordered…
We extend Greenberg's strong approximation theorem to schemes of finite presentation over valuation rings with arbitrary value group, using the ultraproduct method of Becker, Denef, Lipshitz and van den Dries. As an application, we prove a…
Based on the work of Hahn, Baer, Ostrowski, Krull, Kaplansky and the Artin-Schreier theory, and stimulated by a paper of S. Lang in 1953, the theory of real places and convex valuations has witnessed a remarkable development and has become…
We improve results of Belair, Macintyre, and Scanlon on valued fields with a valuation preserving automorphism by weakening their assumptions on the residue difference field. In the equicharacteristic zero case we also determine the induced…
In this study, we examine some properties of $r$-hyperideals in the commutative Krasner hyperrings. Some properties of $pr$-hyperideals are also studied. The relation between prime hyperideals and $r$-hyperideals is investigated. We show…
The paper establishes a relationship between finite separable extensions and norm groups of strictly quasilocal fields with Henselian discrete valuations, which yields a generally nonabelian one-dimensional local class field theory.
The notion of key polynomials was first introduced in 1936 by S. Maclane in the case of discrete rank 1 valuations. . Let K -> L be a field extension and {\nu} a valuation of K. The original motivation for introducing key polynomials was…
We develop some tools for analyzing dp-finite fields, including a notion of an ``inflator'' which generalizes the notion of a valuation/specialization on a field. For any field $K$, let $\operatorname{Sub}_K(K^n)$ denote the lattice of…
We give a first-order definition of key polynomials, we show the links with previous definitions, that it is relevant to study key degrees, and to use a kind of valuations that we call partially multiplicative. We also prove or reprove…
We discuss definability in the language of rings without parameters of the unique canonical henselian valuation of a field. We show that in most cases where the canonical henselian valuation is definable, it is already definable by a…