Value groups, residue fields, and bad places of rational function fields

Kuhlmann, Franz-Viktor

doi:10.1090/s0002-9947-04-03463-4

Cited by 91 publications

(113 citation statements)

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“…, and [Ku5]). In Section 4.1 we prove the following fact which is a generalization of Theorem 1.2 of [Ku1]. It shows that even if the field seems to be simple it may admit defect extensions.…”

Section: Introductionmentioning

confidence: 71%

Infinite towers of Artin-Schreier defect extensions of rational function fields

Blaszczok

2014

Valuation Theory in Interaction

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Abstract. We consider Artin-Scheier defect extensions of rational function fields in two variables over fields of positive characteristic. We study the problem of constructing infinite towers of such extensions. We classify ArtinSchreier defect extensions into "dependent" and "independent" ones, according to whether they are connected with purely inseparable defect extensions, or not. To understand the meaning of the classification for the issue of local uniformization, we consider various valuations of the rational function field and investigate for which it admits an infinite tower of dependent or independent Artin-Schreier defect extensions. We give also a criterion for a valued field of positive characteristic p with p-divisible value group and perfect residue field to admit infnitely many parallel dependent Artin-Schreier defect extensions or an infinite tower of such extensions.

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Section: Introductionmentioning

confidence: 71%

Infinite towers of Artin-Schreier defect extensions of rational function fields

Blaszczok

2014

Valuation Theory in Interaction

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“…The theorem above can be seen as the version of Theorem 3.11 of [4] for key polynomials and truncations. In Section 3, we describe a complete sequence of key polynomials for ν.…”

Section: Of [6]) a Valuation ν On K[x] Is Valuation-transcendental Imentioning

confidence: 99%

Extensions of a valuation from $K$ to $K[x]$

Novacoski¹

2020

Banach Center Publ.

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In this paper we give an introduction on how one can extend a valuation from a field K to the polynomial ring K[x] in one variable over K. This follows a similar line as the one presented by the author in his talk at ALaNT 5. We will discuss the objects that have been introduced to describe such extensions. We will focus on key polynomials, pseudo-convergent sequences and minimal pairs. Key polynomials have been introduced and used by various authors in different ways. We discuss these works and the relation between them. We also discuss a recent version of key polynomials developed by Spivakovsky. This version provides some advantages that will be discussed in this paper. For instance, it allows us to relate key polynomials, in an explicit way, to pseudo-convergent sequences and minimal pairs. This paper also provides examples that illustrate these objects and their properties. Our main goal when studying key polynomials is to obtain more accurate results on the problem of local uniformization. This problem, which is still open in positive characteristic, was the main topic of the paper of the author and Spivakovsky in the proceedings of ALaNT 3.

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“…(4) In Theorem 4.1, we use a valuation basis for G to avoid the need for a condition like (iv). However, a real closed field of finite absolute transcendence degree need not admit a valuation transcendency basis; see [8] for a precise definition of valuation transcendency basis and counterexamples. §6.…”

Section: Case 1 (Immediatementioning

confidence: 99%