Se construyen las valoraciones de un cuerpo de funciones heromorfas formales en dos variables y se estudia la ramificación de algunos tipos de ellas. Especial atención se dedica al estudio de las singularidades de superficies cuyo entorno completo está representado por una serie de Puiseux y singularidades casi-ordinarias. Antes de empezar a...
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April 16, 2015 (v1)PublicationUploaded on: March 27, 2023
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June 8, 2016 (v1)Publication
Let K → L be an algebraic field extension and ν a valuation of K. The purpose of this paper is to describe the totality of extensions {ν′} of ν to L using a refined version of MacLane's key polynomials. In the basic case when L is a finite separable extension and rk ν = 1, we give an explicit description of the limit key polynomials (which can...
Uploaded on: December 4, 2022 -
April 14, 2023 (v1)Publication
In this paper we present a refined version of MacLane's theory of key polynomials, similar to those considered by M. Vaqui\'e and reminiscent of approximate roots of Abhyankar and Moh. Given a simple transcendental extension of valued fields, we associate to it a countable well-ordered set of polynomials called key polynomials. We define limit...
Uploaded on: April 15, 2023 -
November 10, 2022 (v1)Publication
Let v be a rank m discrete valuation of k[[X1,...,Xn]] with dimension n-m. We prove that there exists an inmediate extension L of K where the valuation is monomial. Therefore we compute explicitly the residue field of the valuation.
Uploaded on: March 24, 2023 -
October 7, 2016 (v1)Publication
Let (R; m; k) be a local noetherian domain with field of fractions K and R_v a valuation ring, dominating R (not necessarily birationally). Let v|K be the restriction of v to K; by definition, v|K is centered at R. Let \hat{R} denote the m-adic completion of R. In the applications of valuation theory to commutative algebra and the study of...
Uploaded on: March 27, 2023 -
June 8, 2016 (v1)Publication
In this paper we study rank one discrete valuations of the field k((X1, . . . , Xn)) whose center in k[[X1, . . . , Xn]] is the maximal ideal. In sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote section 2 to finding...
Uploaded on: March 24, 2023 -
June 8, 2016 (v1)Publication
This paper deals with valuations of fields of formal meromorphic functions and their residue fields. We explicitly describe the residue fields of the monomial valuations. We also classify all the discrete rank one valuations of fields of power series in two and three variables, according to their residue fields. We prove that all our cases are...
Uploaded on: March 27, 2023