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Josnei Novacoski

Publications and source records attributed to Josnei Novacoski.

32 records · Page 2Linked to original sources
arXiv2021

Limit key polynomials as $p$-polynomials

The main goal of this paper is to characterize limit key polynomials for a valuation $ν$ on $K[x]$. We consider the set $Ψ_α$ of key polynomials for $ν$ of degree $α$. We set $p$ be the exponent characteristic of $ν$. Our first main result (Theorem 1.1) is that if $Q_α$ is a limit key polynomial for $Ψ_α$, then the degree of $Q_α$ is $p^rα$ for some $r\in\mathbb N$. Moreover, in Theorem 1.2, we show that there exist $Q\inΨ_α$ and $Q_α$ a limit key polynomial for $Ψ_α$, such that the $Q$-expansion of $Q_α$ only has terms which are powers of $p$.

math.AC↗
arXiv2020

On MacLane-Vaquié key polynomials

One of the main goals of this paper is to present the relation of limit key polynomials and limit MacLane-Vaquié key polynomials. This is a continuation of the work started in a paper by Decaup, Mahboub and Spivakovsky, where it is proved the equivalent result for key polynomials (which are not limit). Moreover, we present a result (Theorem 1.1) that generalizes various results in the literature.

math.AC↗
arXiv2019

Essentially finite generation of valuation rings in terms of classical invariants

The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field $(K,ν)$ and an extension $ω$ of $ν$ to a finite extension $L$ of $K$. Then we study when the valuation ring of $ω$ is essentially finitely generated over the valuation ring of $ν$. We present a necessary condition in terms of classic invariants of the extension by Hagen Knaf and show that in some particular cases, this condition is also sufficient. We also study when the corresponding extension of graded algebras is finitely generated. For this problem we present an equivalent condition (which is weaker than the one for the finite generation of the valuation rings).

math.AC↗
arXiv2019

Perron transforms and Hironaka's game

In this paper we present a matricial result that generalizes Hironaka's game and Perron transforms simultaneously. We also show how one can deduce the various forms in which the algorithm of Perron appears in proofs of local uniformization from our main result.

math.AC↗
arXiv2019

Extensions of a valuation from $K$ to $K[x]$

In this paper we give an introduction on how one can extend a valuation from a field $K$ to the polynomial ring $K[x]$ in one variable over $K$. This follows a similar line as the one presented by the author in his talk at ALaNT 5. We will discuss the objects that have been introduced to describe such extensions. We will focus on key polynomials, pseudo-convergent sequences and minimal pairs. Key polynomials have been introduced and used by various authors in different ways. We discuss these works and the relation between them. We also discuss a recent version of key polynomials developed by Spivakovsky. This version provides some advantages, that will be discussed in this paper. For instance, it allows us to relate key polynomials, in an explicit way, to pseudo-convergent sequences and minimal pairs. This paper also provides examples that ilustrate these objects and their properties. Our main goal when studying key polynomials is to obtain more accurate results on the problem of local uniformization. This problem, which is still open in positive characteristic, was the main topic of the paper of the author and Spivakovsky in the proceedings of ALaNT 3.

math.AC↗
arXiv2018

Key polynomials and minimal pairs

In this paper we establish the relation between key polynomials (as defined in \cite{SopivNova}) and minimal pairs of definition of a valuation. We also discuss truncations of valuations on a polynomial ring $K[x]$. We prove that a valuation $ν$ is equal to its truncation on some polynomial if and only if $ν$ is valuation-transcendental. Another important result of this paper is that if $μ$ is any extension of $ν$ to $\overline K[x]$ and $Λ$ is a complete sequence of key polynomials for $ν$, without last element, then for each $Q\in Λ$ there exists a suitable root $a_Q\in \overline K$ of $Q$ such that $\{a_Q\}_{Q\in Λ}$ is a pseudo-convergent sequence defining $μ$.

math.AC↗
arXiv2016

Key polynomials and pseudo-convergent sequences

In this paper we introduce a new concept of key polynomials for a given valuation $ν$ on $K[x]$. We prove that such polynomials have many of the expected properties of key polynomials as those defined by MacLane and Vaquié, for instance, that they are irreducible and that the truncation of $ν$ associated to each key polynomial is a valuation. Moreover, we prove that every valuation $ν$ on $K[x]$ admits a sequence of key polynomials that completely determines $ν$ (in the sense which we make precise in the paper). We also establish the relation between these key polynomials and pseudo-convergent sequences defined by Kaplansky.

math.AC↗
arXiv2015

On the local uniformization problem

In this paper we give a short introduction to the local uniformization problem. This follows a similar line as the one presented by the second author in his talk at ALANT 3. We also discuss our paper on the reduction of local uniformization to the rank one case. In that paper, we prove that in order to obtain local uniformization for valuations centered at objects of a subcategory of the category of noetherian integral domains, it is enough to prove it for rank one valuations centered at objects of the same category. We also announce an extension of this work which was partially developed during ALANT 3. This extension says that the reduction mentioned above also works for noetherian rings with zero divisors (including the case of non-reduced rings).

math.AC↗
arXiv2015

The Scott topology of a rooted non-metric tree

In this paper we prove that the Scott topology $\mathfrak S$ on a rooted non-metric tree $\mathcal T$ is strictly coarser than the weak tree topology. Moreover, for each $t\in \mathcal T$, we consider a natural order $\preceq_t$ on $\mathcal T$ under which $t$ is the root of $\mathcal T$. Then the weak tree topology is generated by the union of the Scott topologies $\mathfrak S_t$ associated to $\preceq_t$.

math.AC↗
arXiv2015

Topologies on spaces of valuations: a closeness criterion

This paper is part of a program to understand topologies on spaces of valuations. We fix an ordered abelian group $Γ$ and an integral domain $R$. We study the relation between a topology on $Γ_\infty$ and the induced topology on the set $\mathcal W_Γ$ of all valuations of $R$ taking values in $Γ_\infty$. For instance, we give a criterion for $ \mathcal W_Γ$ to be closed in $\left(Γ_\infty\right)^R$. We also discuss the effect of this criterion for natural topologies on $Γ_\infty$.

math.AC↗
arXiv2015

Reduction of local uniformization to the case of rank one valuations for rings with zero divisors

This is a continuation of a previous paper by the same authors. In the former paper, it was proved that in order to obtain local uniformization for valuations centered on local domains, it is enough to prove it for rank one valuations. In this paper, we extend this result to the case of valuations centered on rings which are not necessarily integral domains and may even contain nilpotents.

math.AC↗
arXiv2013

Henselian Elements

Henselian elements are roots of polynomials which satisfy the conditions of Hensel's Lemma. In this paper we prove that for a finite field extension $(F|L,v)$, if $F$ is contained in the absolute inertia field of $L$, then the valuation ring $\mathcal O_F$ of $(F,v)$ is generated as an $\mathcal O_L$-algebra by henselian elements. Moreover, we give a list of equivalent conditions under which $\mathcal O_F$ is generated over $\mathcal O_L$ by finitely many henselian elements. We prove that if the chain of prime ideals of $\mathcal O_L$ is well-ordered, then these conditions are satisfied. We give an example of a finite valued inertial extension $(F|L,v)$ for which $\mathcal O_F$ is not a finitely generated $\mathcal O_L$-algebra. We also present a theorem that relates the problem of local uniformization with the theory of henselian elements.

math.AC↗
arXiv2012

Reduction of Local Uniformization to the rank one case

The main result of this paper is that in order to prove the local uniformization theorem for local rings it is enough to prove it for rank one valuations. Our proof does not depend on the nature of the class of local rings for which we want to prove local uniformization. We prove also the reductions for different versions of the local uniformization theorem.

math.AC↗