Rank one discrete valuations of power series fields

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 an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in section 5. The constructions given in these sections are not effective in the general case, because we need either to use Zorn’s lemma or to know explicitly a section σ of the natural homomorphism Rv → ∆v between the ring and the residue field of the valuation v. However, as a consequence of this construction, in section 7, we prove that k((X1, . . . ,Xn)) can be embedded into a field L((Y1, . . . , Yn)), where L is an algebraic extension of k and the “extended valuation” is as close as possible to the usual order function. 1 Terminology and preliminaries Let k be a field of characteristic 0, Rn = k[[X1, . . . , Xn]] the formal power series ring in n variables, Mn = (X1, . . . , Xn) its maximal ideal and Kn = k((X1, . . . , Xn)) its quotient field. Let v be a rank-one discrete valuation of Kn|k, Rv the valuation ring, mv the maximal ideal and ∆v the residue field of v. The center of v in Rn is mv ∩Rn. Throughout this paper “discrete valuation of Kn|k” means “rank-one discrete valuation of Kn|k whose center in Rn is the maximal ideal Mn”. The dimension of v, dim(v), is the transcendence degree of ∆v over k. ∗Partially supported by MTM2004-07203-C02-01 and FEDER