arXiv subjects

Caio Henrique Silva de Souza

Publications and source records attributed to Caio Henrique Silva de Souza.

6 recordsLinked to original sources

arXiv2026

On topologies on the space of valuations and the valuative tree

In this paper, we discuss topological aspects of the space of valuations $\mathbb{V}$ and the valuative tree $\mathcal{T}(v,Λ)$. We present a relation between the weak tree topology and the Scott topology in $\mathcal{T}(v,Λ)$ and describe the supremum of an increasing family of valuations in a special subtree. We also view the valuative tree as a subset of the product $(Λ_\infty)^{K[x]}$ and prove that it is closed if we consider the natural product topology.

math.AC↗

arXiv2025

On defect in finite extensions of valued fields

In recent decades, the defect of finite extensions of valued fields has emerged as the main obstacle in several fundamental problems in algebraic geometry such as the local uniformization problem. Hence, it is important to identify defectless fields and study properties related to defect. In this paper we study the relations between the following properties of valued fields: simply defectless, immediate-defectless and algebraically maximal. The main result of the paper is an example of an algebraically maximal field that admits a simple defect extension. For this, we introduce the notion of quasi-finite elements in the generalized power series field $k\left(\left(t^Γ\right)\right)$.

math.AC↗

arXiv2025

Tame fields, Graded Rings and Finite Complete Sequences of Key Polynomials

In this paper, we present a criterion for $(K,v)$ to be henselian and defectless in terms of finite complete sequences of key polynomials. For this, we use the theory of Mac Lane-Vaquié chains and abstract key polynomials. We then prove that a valued field $(K,v)$ is tame if and only if $vK$ is $p$-divisible, $Kv$ is perfect and every simple algebraic extension of $K$ admits a finite complete sequence of key polynomials. The properties $vK$ $p$-divisible and $Kv$ perfect are described by the Frobenius endomorphism on the associated graded ring. We also make considerations on simply defectless and algebraically maximal valued fields and purely inertial and purely ramified extensions.

math.AC↗

arXiv2023

Parametrizations of subsets of the space of valuations

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.

math.AC↗

arXiv2022

Graded rings associated to valuations and direct limits

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.

math.AC↗

arXiv2021

On truncations of 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.

math.AC↗