Estudos em Teoria de Valorizações: famílias crescentes, anéis graduados, polinômios chaves e o defeito de extensões valoradas de corpos

Abstract

In this thesis we study topics in Valuation Theory, with a particular focus on key polynomials, graded rings associated to valuations and the defect of a valued field extension. We first study, via direct limits, the graded ring associated to a valuation v that is an upper bound of an increasing family of valuations in the valuative tree. In particular, we describe the graded ring associated to a limit key polynomial via a direct limit of graded rings. We then parametrize subsets of the space of valuations using suprema of increasing families of valuations. These families are constructed from pseudo-Cauchy sequences and from approximation types. Finally, we study the defect of a valued field extension (L|K,v). We investigate the relationship between unibranched defectless valued field extensions and the complete sequences of key polynomials for the induced valuation on the polynomial ring K[x].