arXiv subjects
Publications and source records attributed to Josnei Antonio Novacoski.
In this paper we present different ways to parametrize subsets of the space of valuations on $K[x]$ extending a given valuation on $K$. We discuss the methods using pseudo-Cauchy sequences and approximation types. The method presented here is slightly different than the ones in the literature and we believe that our approach is more accurate.
In this paper, we study the structure of the graded ring associated to a limit key polynomial $Q_n$ in terms of the key polynomials that define $Q_n$. In order to do that, we use direct limits. In general, we describe the direct limit of a family of graded rings associated to a totally ordered set of valuations. As an example, we describe the graded ring associated to a valuation-algebraic valuation as a direct limit of graded rings associated to residue-transcendental valuations.
In this paper we study the truncation $ν_q$ of a valuation $ν$ on a polynomial $q$. It is known that when $q$ is a key polynomial, then $ν_q$ is a valuation. It is also known that the converse does not hold. We show that when $q$ is a key polynomial, then $ν_q$ is the restriction of the truncation given by an optimizing root of $q$. We also discuss which conditions assure that $ν_q=ν$. Finally, we assume that $ν_q$ is a valuation and present some conditions, given in terms of the graded algebra, to assure that $q$ is a key polynomial.