Algebraic series and valuation rings over nonclosed fields

Cutkosky, Steven Dale; Kashcheyeva, Olga

doi:10.1016/j.jpaa.2007.11.013

Cited by 6 publications

(4 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…which is 1-1 and such thatφ The behavior of such examples is particularly interesting in positive characteristic [15]. The following example shows an important distinction between the case when R is complete and when R is not.…”

Section: Regular Local Rings Of Dimensionmentioning

confidence: 94%

“…This subject has been explored in many papers, including [7,34]. The extension of valuations to the completion of a local ring, which becomes extremely difficult in higher dimension and rank, is studied in [9,10,13,13,15,28,30,42]. There is an extensive literature on the theory of complete ideals in local rings, beginning with Zariski's articles [46,49].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Valuation semigroups of two‐dimensional local rings

Cutkosky

Vinh

2013

Proceedings of the London Mathematical Society

Self Cite

View full text Add to dashboard Cite

We consider the question of when a semigroup is the semigroup of a valuation dominating a two-dimensional noetherian domain, giving some surprising examples. We give a necessary and sufficient condition for the pair of a semigroup S and a field extension L/k to be the semigroup and residue field of a valuation dominating a regular local ring R of dimension 2 with residue field k, generalizing the theorem of Spivakovsky for the case when there is no residue field extension.

show abstract

Section: Regular Local Rings Of Dimensionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Valuation semigroups of two‐dimensional local rings

Cutkosky

Vinh

2013

Proceedings of the London Mathematical Society

Self Cite

View full text Add to dashboard Cite

show abstract

“…Remark 1. 10 The case (3.vi) 'F = x 3 +axy 2 +by 3 irreducible' requires a special comment. Denote A := κ[[x, y, z]]/(z 2 − F), the real closure of κ with R (endowed with its unique ordering) and…”

Section: P(κ[[x Y]]) = P(κ[x][[y]]) = P(κ[[x]][y]) = P(κ((x Y))) = P(qf(κ[x][[y]]))mentioning

confidence: 99%

Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings

Fernando

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

View full text Add to dashboard Cite

A classical problem in real geometry concerns the representation of positive semidefinite elements of a ring A as sums of squares of elements of A. If A is an excellent ring of dimension $$\ge 3$$ ≥ 3 , it is already known that it contains positive semidefinite elements that cannot be represented as sums of squares in A. The one dimensional local case has been afforded by Scheiderer (mainly when its residue field is real closed). In this work we focus on the 2-dimensional case and determine (under some mild conditions) which local excellent henselian rings A of embedding dimension 3 have the property that every positive semidefinite element of A is a sum of squares of elements of A.

show abstract

“…• The following papers make only passing references to [26] and/or [27], with no reference to specific results, and are thus unaffected: [1], [2], [9], [10], [14], [15], [18], [19], [21], [22], [23], [24], [25], [30], [31], [35], [38], [39]. • The following papers only reference [26] (either explicitly or via [27]) in the case where K = F p , and are thus unaffected: [3], [28], [32], [35].…”

Section: Effect On the Literaturementioning

confidence: 99%

On the algebraicity of generalized power series

Kedlaya

2016

Beitr Algebra Geom

View full text Add to dashboard Cite

Abstract. Let K be an algebraically closed field of characteristic p. We exhibit a counterexample against a theorem asserted in one of our earlier papers, which claims to characterize the integral closure of K((t)) within the field of Hahn-Mal'cev-Neumann generalized power series. We then give a corrected characterization, generalizing our earlier description in terms of finite automata in the case where K is the algebraic closure of a finite field. We also characterize the integral closure of K(t), thus generalizing a well-known theorem of Christol and suggesting a possible framework for computing in this integral closure. We recover various corollaries on the structure of algebraic generalized power series; one of these is an extension of Derksen's theorem on the zero sets of linear recurrent sequences in characteristic p.

show abstract

Algebraic series and valuation rings over nonclosed fields

Cited by 6 publications

References 23 publications

Valuation semigroups of two‐dimensional local rings

Valuation semigroups of two‐dimensional local rings

Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings

On the algebraicity of generalized power series

Contact Info

Product

Resources

About