VALUATIONS OF k((X1,…,Xn)) WITH PREASSIGNED GROUP OF VALUES

Abstract

Let Γ be a totally ordered abelian group of finite rational rank r. Consider s and n two non-negative integers with r+s≤n and denote by K a field. The main purpose of this paper is to construct a s-dimensional valuation v of the quotient field K((X1,…,Xn)) of the corresponding power series ring K[[X1,…,Xn]] such that v birrationally dominates K[[X1,…,Xn]] and has Γ as group of values. This is possible except for the case in which Γ=ℤ is the group of the integers, s=0, n≥2 and K does not admit an infinite algebraic extension. In this last case, we show that there does not exist such a valuation.