The growth of valuations on rational function fields in two variables

Mosteig, Edward; Sweedler, Moss E.

doi:10.1090/s0002-9939-04-07456-8

Cited by 3 publications

(3 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…• 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]. • The following papers only rely on the prior result [26, Theorem 1], and are thus unaffected: [6], [11], [34]. • The paper [36] references the transfinite Newton recurrence described in [27, §2], and is thus unaffected.…”

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

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

show abstract

“…For the entirety of this paper, we focus on the polynomial ring K[x, y] in two variables over a field K of arbitrary characteristic. Constructions of K-valuations v on K(x, y) with v(K[x, y] * ) reversely well-ordered are investigated in [MoSw1], [MoSw2], [Mo1] and [Mo2].…”

Section: Introductionmentioning

confidence: 99%

Reversely well-ordered valuations on rational function fields in two variables

Mosteig

2019

Communications in Algebra

Self Cite

View full text Add to dashboard Cite

We examine valuations on a rational function field K(x, y) and analyze their behavior when restricting to an underlying polynomial ring K[x, y]. Motivated to solve the ideal membership problem in polynomial rings using Moss Sweedler's framework of generalized Gröbner bases, we produce an infinite collection of valuations v : K(x, y) → Z ⊕ Z such that v(K[x, y] * ) is reversely well-ordered. In addition, we construct a surprising example where v(K[x, y] * ) is nonpositive, yet not reversely well ordered.

show abstract

“…Note that the triangle inequality was chosen to be opposite of the most common definition, which is so that our results most closely coincide with those concerning monomial orders. For more details, see [MoSw1], [MoSw2], and [M]. A valuation on F over k is a valuation on F such that its restriction to k * is the zero map.…”

Section: Introductionmentioning

confidence: 99%