Constructive theory of valuations

Abstract

Let k be an algebraically closed field of characteristic p ≥ 0, v a discrete rank-one valuation of .We proved, by expliciting the valuation, the existence of discrete rank-one valuations with preassigned singular ideal.Let L be a linear form of Zn.Then the residue field of a discrete valuation v associated with Lisa purely transcendental extension of k, of trascendence degree n-1.In the case n=2, we describe an algorithm that construct the residue field of any discrete rank-one valuation of k2πk, whose center in R2is the maximal ideal, proving that the residue field is a purely transcendental extension of k, of trascendence degree 1. Throughout this paper, k is a fixed algebraically closed field of characteristic p ≥ 0.Let be a collection of indeterminates, and write .