All valuations on $K(X)$

Alexandru, Victor; Popescu, Nicolae; Zaharescu, Alexandru

doi:10.1215/kjm/1250520072

Cited by 37 publications

(37 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…It has been observed by several authors that a valuation-algebraic extension of v from K to K(x) can be represented as a limit of an infinite sequence of residue-transcendental extensions. See, e.g., [APZ3], where the authors also derive the assertion of our Theorem 2.9 from this fact. A "higher form" of this approach is found in [S].…”

Section: Valuations On K(x)mentioning

confidence: 63%

“…Since then, an impressive number of papers have been written about the construction of such valuations and about their properties; the following list is by no means exhaustive: [AP], [APZ1]- [APZ3], [KH1]- [KH10], [KHG1]- [KHG6], [KHPR], [MO1], [MO2], [MOSW1], [O1]- [O3], [PP], [V]. From the paper [APZ3] the reader may get a good idea of how MacLane's original approach has been developed further. Since then, the notion of "minimal pairs" has been 4560 FRANZ-VIKTOR KUHLMANN adopted and studied by several authors (see, e.g., [KHPR]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Kuhlmann

2004

Trans. Amer. Math. Soc.

113

View full text Add to dashboard Cite

Abstract. We classify all possible extensions of a valuation from a ground field K to a rational function field in one or several variables over K. We determine which value groups and residue fields can appear, and we show how to construct extensions having these value groups and residue fields. In particular, we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated. In the case of a rational function field K(x) in one variable, we consider the relative algebraic closure of K in the henselization of K(x) with respect to the given extension, and we show that this can be any countably generated separablealgebraic extension of K. In the "tame case", we show how to determine this relative algebraic closure. Finally, we apply our methods to power series fields and the p-adics.

show abstract

Section: Valuations On K(x)mentioning

confidence: 63%

Section: Introductionmentioning

confidence: 99%

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

Kuhlmann

2004

Trans. Amer. Math. Soc.

113

View full text Add to dashboard Cite

show abstract

“…The first statement is easily verified. For the second part we consider P, Q ∈ K[X], where P = i≤d(P) a i P i and Q is given by (2). Then, by (4),…”

Section: Then Every Q ∈ K[x] Can Be Represented Uniquely In the Formmentioning

confidence: 99%

“…Some problems connected with the norms on p-adic vector spaces were solved by I. S. Cohen [5] and A. F. Monna [8], and then O. Goldmann and N. Iwahori were concerned in [6] with the intrinsic structure that is carried by the set of all norms on a given finite dimensional vector space over a locally compact field. When r = 1, the case of K-algebra non-Archimedean norms on K [X] which are multiplicative and extend | | has been treated in [1][2][3]. In Section 2 below we consider generalizations of the Gauss valuation.…”

Section: Introductionmentioning

confidence: 99%

ALL NON-ARCHIMEDEAN NORMS ON K[X₁, . . ., X_r]

Groza¹,

Popescu²,

Zaharescu³

2009

Glasgow Math. J.

View full text Add to dashboard Cite

Abstract. If K is a field with a non-trivial non-Archimedean absolute value (multiplicative norm) | |, we describe all non-Archimedean K-algebra norms on the polynomial algebra K[X 1 , . . . , X r ] which extend | |.2000 Mathematics Subject Classification. 11S75, 11C08. Introduction.Let K be a field with a non-trivial non-Archimedean absolute value (multiplicative norm) | |. In this paper, we study K-algebra non-Archimedean norms on K[X 1 , . . . , X r ] which extend | |. Some problems connected with the norms on p-adic vector spaces were solved by I. S. Cohen [5] and A. F. Monna [8], and then O. Goldmann and N. Iwahori were concerned in [6] with the intrinsic structure that is carried by the set of all norms on a given finite dimensional vector space over a locally compact field. When r = 1, the case of K-algebra non-Archimedean norms on K [X] which are multiplicative and extend | | has been treated in [1][2][3]. In Section 2 below we consider generalizations of the Gauss valuation. We investigate the case when a K-vector space norm is a K-algebra norm and we also address the question of when two norms are equivalent. In Section 3 we then discuss possible types of norms on K[X 1 , . . . , X r ] which extend a given non-trivial non-Archimedean absolute value on K. The completion of K[X 1 , . . . , X r ] with respect to a non-Archimedean Gauss norm is given in Section 4.There are many applications of non-Archimedean multiplicative norms on K[X 1 , . . . , X r ] in algebraic geometry where a basic tool is to describe all the absolute values on K(X 1 , . . . , X r ) which extend | |. In [7] F.-V. Kuhlmann determined which value groups and which residue fields can possibly occur in this case. In the case r = 1 the r.t. extensions | | L of | | to L = K(X) have been considered by M. Nagata [9], who

show abstract

“…In a subsequent paper we will study further the ultrafilter/constructible topology on the space Zar(K|A) providing some applications to the representations of integrally closed domains as intersections of valuation overrings [11] (see also [10]). This paper is dedicated to the memory of Nicolae Popescu who recently left us: his papers were an important source of inspiration (e.g., [1], [2] [24], [25], [26], and [19]). …”

Section: Introductionmentioning

confidence: 99%

Ultrafilter and Constructible Topologies on Spaces of Valuation Domains

Finocchiaro

Fontana

Loper

2013

Communications in Algebra

View full text Add to dashboard Cite

Abstract. Let K be a field and let A be a subring of K. We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on the space Zar(K|A) of all valuation domains having K as quotient field and containing A. We show that the ultrafilter topology coincides with the constructible topology on the abstract Riemann-Zariski surface Zar(K|A). We extend results regarding distinguished spectral topologies on spaces of valuation domains.

show abstract

All valuations on $K(X)$

Cited by 37 publications

References 0 publications

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

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

ALL NON-ARCHIMEDEAN NORMS ON K[X₁, . . ., X_r]

Ultrafilter and Constructible Topologies on Spaces of Valuation Domains

Contact Info

Product

Resources

About

All valuations on $K(X)$

Cited by 37 publications

References 0 publications

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

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

ALL NON-ARCHIMEDEAN NORMS ON K[X1, . . ., Xr]

Ultrafilter and Constructible Topologies on Spaces of Valuation Domains

Contact Info

Product

Resources

About

ALL NON-ARCHIMEDEAN NORMS ON K[X₁, . . ., X_r]