Monomial discrete valuations in k[[X]]

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

Valuations in fields of power series

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 possible and give explicit constructions.Junta de AndalucíaMinisterio de Ciencia y Tecnologí

Ramificación de valoraciones de superficies algebroides

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 escribir un resumen del contenido de la presente memoria, debo dejar bien claro un hecho importante y es que éste es un trabajo abierto, en el sentido de que no resuelve un problema hasta sus últimas consecuencias, acabando una teoría hasta dejarla formalmente perfecta. Por supuesto que en esta memoria se resuelven problemas; pero este trabajo no es sino el comienzo de algo, cuyas bases se establecen, que deja ante sí un campo amplio para futuros desarrollos. Se resuelven unos problemas, pero se plantean otros, de forma explícita o implícita, mayores quizás a los aquí resueltos.

Key polynomials for simple extensions of valued fields

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 key polynomials and give explicit formulae for them. We give an explicit bound on the order type of the set of key polynomials

Key polynomials for simple extensions of valued fields

In this paper we present a refined version of MacLane's theory of key polynomials [16]-[17], similar to those considered by M. Vaquié [24]-[27], and reminiscent of related objects studied by Abhyankar and Moh (approximate roots [1], [2]) and T.C. Kuo [14], [15].Let (K, ν_0) be a valued field. Given a simple transcendental extension of valued fields ι : K → K(x) we associate to ι a countable well ordered set of polynomials of K[x] called key polynomials. We define limit key polynomials and give an explicit description of them. We show that the order type of the set of key polynomials is bounded by ω × ω. If char k_ν_0 = 0 and rk ν_0 = 1, the order type is bounded by ω + 1

Extending valuations to formal completions

International audienc