On the dimension of discrete valuations of k ((X1, ..., Xn))

Let nu be a rank-one discrete valuation of the field K-n = k((X-1,...,X-n)). We know, after [1], that if n = 2 then the dimension of nu, is 1 and if nu is the usual order function over k((X-1,...,X-n)) its dimension is n - 1. In this paper we prove that, in the general case, the dimension of a rank-one discrete valuation can be any number between I and n - 1. (C) 2001 Academic des sciences/Editions scientifiques et medicales Elsevier SAS.