Um estudo de valorizações transcendentes e algébricas via polinômios-chaves e pares minimais

Abstract

The main goal of this work is to study transcendental valuations and algebraic valuations. To achieve this, we use some of the main objects in Valuation Theory, such as key polynomials, truncations and minimal pairs. These objects will lead us to the results which will build the central part of this text and will be seen as the specific goals of this work. We will begin studying valuations in a general way and then we focus on monomial valuations. We will explore the concept of key polynomials and truncations, proving many technical results. Then, we will present the idea of minimal pair of definition, relating it to key polynomials and truncations. After that, we will study transcendental valuations and complement results of Novacoski (2019). We will also study part of Bengus-Lasnier (2021) recent work on balls and diskoids. In the last chapter, we will study algebraic valuations. We will finish our work presenting a classification proposed by Alexandru, Popescu and Zaharescu (1988, 1990a, 1990b) for all valuations on K(x), the field of rational functions over a field K, and with this classification we will make a general overview of the results we presented before.