×

Valuations in fields of power series. (English) Zbl 1055.12004

Let \(k\) be an algebraically closed field and \(R=k[[x_{1},\dots,x_{n}]]\) be the formal power series ring in the variables \(x_{i}\) and \(K\) be its quotient field. This paper deals with discrete valuations of \(K\) which are trivial on \(k\) and whose center at \(R\) is the maximal ideal generated by \(x_{1},\dots,x_{n}.\) The authors give explicitly a description of all discrete rank one valuations of \(k((x_{1},x_{2}))/k\) and \(k((x_{1},x_{2},x_{3}))/k.\) Constructive examples of rank two valuations are also given. The description is different from the one already known in the function field case [cf. S. Khanduja and U. Garg, Mathematika 37, 97–105 (1990; Zbl 0689.12018)].

MSC:

12J10 Valued fields
13A18 Valuations and their generalizations for commutative rings

Citations:

Zbl 0689.12018
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML

References:

[1] Abhyankar, S. S.: On the valuations centered in a local domain. Amer. J. Math. 78 (1956), 321-348. · Zbl 0074.26301
[2] Briales, E. and Herrera, F. J.: Construcción explícita de las valo- raciones de un anillo de series formales en dos variables. In Actas X Jor- nadas Hispano-Lusas (Murcia, 1985), II, 1-10. Universidad de Murcia, Mur- cia, 1985.
[3] Briales, E.: Constructive theory of valuations. Comm. Algebra 17 (1989), no. 5, 1161-1177. · Zbl 0696.13004
[4] Vicente, J. L.: Aritmética en Zn. Revista de la Sociedad andaluza de profesores de matemáticas “THALES” (1986), no. 5, 8-32.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.