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Josnei Novacoski

Publications and source records attributed to Josnei Novacoski.

At least 19 recordsLinked to original sources
arXiv2026

Ramification ideals for products of pure subgroups

Let $\mathcal E=(L/K,v)$ be a finite Galois extension of henselian valued fields. We study the ramification ideals $I_H$, for subgroups $H\leq {\rm Gal}(L/K)$, when the Galois group is a product of subgroups $H_i$ such that $L/K_{H_i}$ is pure (depth one). We first recall an explicit formula for ramification ideals of pure extensions and use it to obtain a lower bound for the ideals attached to arbitrary subgroups of a product of pure subgroups. A natural question is whether every $I_H$ coincides with one of the ideals coming from the pure factors. We show that this is not true, already for a defectless extension with Galois group $C_p\times C_p$. The counterexample is a compositum of two Artin--Schreier extensions with different ramification breaks; the failure comes from choosing a decomposition which is not compatible with the ramification filtration. Motivated by this example, we introduce ramification-adapted decompositions and prove a filtration-theoretic substitute for the conjectural statement in elementary abelian $p$-extensions. Since every flag of $\mathbb F_p$-vector spaces admits an adapted basis, every elementary abelian $p$-extension admits such a decomposition, and every subgroup ideal is represented by one adapted cyclic factor. If the adapted factors are pure, this representation can be written in the distance-set form occurring in the original conjecture. We also prove that, for an adapted decomposition $\mathcal G=H_1\times\cdots\times H_r$, all ramification ideals are principal if and only if every degree-$p$ extension $L/K_{H_i}$ is defectless. Finally, we discuss the degree-$p^2$ example constructed by Kuhlmann in \cite[Section 3.5]{Topics}. Consequently every basis is ramification-adapted, while principality of all ramification ideals still does not characterize defectlessness.

math.AC↗
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↗
arXiv2026

A topological approach to key polynomials

In this paper we present characterizations of the sets of key polynomials and abstract key polynomials for a valuation $μ$ of $K(x)$, in terms of (ultrametric) balls in the algebraic closure $\overline K$ of $K$ with respect to $v$, a fixed extension of $μ_{\mid K}$ to $\overline K$. In particular, we show that the ways of augmenting $μ$, in the sense of Mac Lane, are in one-to-one correspondence with the partition of a fixed closed ball $B(a,δ)$ associated to $μ$ into the disjoint union of open balls $B^\circ(a_i,δ)$, modulo the action of the decomposition group of $v$. We also present a similar characterization for the set of limit key polynomials for an increasing family of valuations of $K(x)$.

math.AC↗
arXiv2025

Purity and distances between conjugates of elements over henselian valued fields

For a henselian valued field $(K,v)$ and a separable-algebraic element $a\in\overline{K}\setminus K$, we consider the set $S_K(a):= \{ v(a-a^\prime) \mid a^\prime\neq a \text{ is a $K$-conjugate of $a$} \}$. The central aim of this paper is to provide a bound for the cardinality of the set $S_K(a)$, and to characterize the elements $a$ for which this set is a singleton. Connections of this set with the notion of \textit{depth} of $a$ has also been explored. We show that $S_K(a)$ is a singleton whenever $K(a)|K$ is a minimal extension. A stronger version of this result is obtained when $a$ has depth one over $K$. We also provide a host of examples illustrating that the bounds obtained are strict. Apart from being of independent interest, another primary motivation for considering this problem comes from the study of ramification ideals. In the depth one case, when $K(a)|K$ is a Galois extension, we obtain intimate connections between the cardinalities of $S_K(a)$ and the number of ramification ideals of the extension $(K(a)|K,v)$. In particular, we show that these cardinalities are same whenever the extension is defectless and non-tame, or whenever $(K,v)$ has rank one. In order to obtain these results, we provide comprehensive descriptions of the ramification ideals of $(K(a)|K,v)$ which extend the known results in this direction.

math.AC↗
arXiv2025

On the distances of an element to its conjugates

For a valued field $(K,v)$, with a fixed extension of $v$ to the algebraic closure $\overline K$ of $K$, and an element $θ\in\overline K$, we are interested in the possible values of $θ-θ'$ where $θ'$ runs through all the $K$-conjugates of $θ$. The study of these values is a classic problem in number theory and ramification theory. However, the classic results focus on tame, and in particular defectless, extensions. In this paper we focus on the study of defect extensions. We want to compare the number of such values to invariants of $θ$. The main invariant we have in mind is the depth of $θ$. We present various examples that show that, in the defect case, none of the equivalent of the classic results are true. We also discuss the relation between the number of such values and the number of ramification ideals of the extension $(K(θ)/K,v)$. In order to do so, we present some results about ramification ideals that have interest on their own.

math.AC↗
arXiv2025

Depth of Artin-Schreier defect towers

The depth of a simple algebraic extension $(L/K,v)$ of valued fields is the minimal length of the Mac Lane-Vaquié chains of the valuations on $K[x]$ determined by the choice of different generators of the extension. In a previous paper, we characterized the defectless unibranched extensions of depth one. In this paper, we analyze this problem for towers of Artin-Schreier defect extensions. Under certain conditions on $(K,v)$, we prove that the towers obtained as the compositum of linearly disjoint defect Artin-Schreier extensions of $K$ have depth one. We conjecture that these are the only depth one Artin-Schreier defect towers and we present some examples supporting this conjecture.

math.AC↗
arXiv2025

Valuation rings in simple algebraic extensions of valued fields

Consider a simple algebraic valued field extension $(L/K,v)$ and denote by $\mathcal O_L$ and $\mathcal O_K$ the corresponding valuation rings. The main goal of this paper is to present, under certain assumptions, a description of $\mathcal O_L$ in terms of generators and relations over $\mathcal O_K$. The main tool used here are complete sequences of key polynomials. It is known that if the ramification index of $(L/K,v)$ is one, then every complete set gives rise to a set of generators of $\mathcal O_L$ over $\mathcal O_K$. We show that we can find a sequence of key polynomials for $(L/K,v)$ which satisfies good properties (called neat). Then we present explicit ``neat" relations that generate all the relations between the corresponding generators of $\mathcal O_L$ over $\mathcal O_K$.

math.AC↗
arXiv2025

Depth of extensions of valuations

In this paper we develop the theory of the depth of a simple algebraic extension of valued fields $(L/K,v)$. This is defined as the minimal number of augmentations appearing in some Mac Lane-Vaquié chain for the valuation on $K[x]$ determined by the choice of some generator of the extension. In the defectless and unibranched case, this concept leads to a generalization of a classical result of Ore about the existence of $p$-regular generators for number fields. Also, we find what valuation-theoretic conditions characterize the extensions having depth one.

math.AC↗
arXiv2023

Generators for extensions of valuation rings

For a finite valued field extension $(L/K,v)$ we describe the problem of find sets of generators for the corresponding extension $\mathcal O_L/\mathcal O_K$ of valuation rings. The main tool to obtain such sets are complete sets of (key) polynomials. We show that when the initial index coincide with the ramification index, sequences of key polynomials naturally give rise to sets of generators. We use this to prove Knaf's conjecture for pure extensions.

math.AC↗
arXiv2023

Kähler differentials, pure extensions and minimal key polynomials

The main object of study in this paper is the module $Ω$ of Kähler differentials of an extension of valuation rings. We show that in the case of pure extensions $Ω$ has a very good description. Namely, it is isomorphic to the quotient of two modules defined by final segments in the value group of the valuation. This allows us to describe the annihilator of $Ω$ and to give criteria for $Ω$ to be finitely generated and to be finitely presented. We also discuss how to relate the module of Kähler differentials of pure extensions to minimal key polynomials.

math.AC↗
arXiv2023

Minimal limit key polynomials

In this paper, we extend the theory of minimal limit key polynomials of valuations on the polynomial ring $\kx$. We use the theory of cuts on ordered abelian groups to show that the previous results on bounded sets of key polynomials of rank-one valuations, extend to vertically bounded sets of key polynomials of valuations of an arbitrary rank. We discuss as well properties of minimal limit key polynomials in the vertically unbounded case.

math.AC↗
arXiv2023

The module of Kähler differentials for extensions of valuation rings

The main goal of this paper is to characterize the module of Kähler differentials for an extension of valuation rings. More precisely, we consider a simple algebraic valued field extension $(L/K,v)$ and the corresponding valuation rings $\VR_L$ and $\VR_K$. In the case when $e(L/K,v)=1$ we present a characterization for $Ω_{\VR_L/\VR_K}$ in terms of a given sequence of key polynomials for the extension. Moreover, we use our main result to present a characterization for when $Ω_{\VR_L/\VR_K}=\{0\}$.

math.AC↗
arXiv2022

A characterization for the defect of rank one valued field extensions

In this paper we present a characterization for the defect of a simple algebraic extension of rank one valued fields using the key polynomials that define the valuation. As a particular example, this gives the classification of defect extensions of degree $p$ as dependent or independent presented by Kuhlmann.

math.AC↗
arXiv2022

The defect formula

In this paper we present a characterization for the defect of a simple algebraic extensions of valued fields. This characterization generalizes the known result for the henselian case, namely that the defect is the product of the relative degrees of limit augmentations. The main tool used here is the graded algebra associated to a valuation on a polynomial ring. Let $\kh$ be a henselization of a valued field $K$. Another relevant result proved in this paper is that for every valuation $\muh$ on $\khx$, with restriction $μ$ on $\kx$, the corresponding map $\mathcal G_μ\hk\mathcal G_{\muh}$ of graded algebras is an isomorphism.

math.AC↗
arXiv2021

Valuations on $K[x]$ approaching a fixed irreducible polynomial

For a fixed irreducible polynomial $F$ we study the set $\mathcal V_F$ of all valuations on $K[x]$ bounded by valuations whose support is $(F)$. The first main result presents a characterization for valuations in $\mathcal V_F$ in terms of their graded rings. We also present a result which gives, for a fixed $ν\in \mathcal V_F$ and a key polynomial $Q\in{\rm KP}(ν)$, the maximum value that augmented valuations in $\mathcal V_F$ can assume on $Q$. This value is presented explicitly in terms of the slopes of the Newton polygon of $F$ with respect to $Q$. Finally, we present some results about Artin-Schreier extensions that illustrate the applications that we have in mind for the results in this paper.

math.AC↗
arXiv2021

Generating sequences and key polynomials

The main goal of this paper is to study the different definitions of generating sequences appearing in the literature. We present these definitions and show that under certain situations they are equivalent. We also present an example that shows that they are not, in general, equivalent. We also present the relation of generating sequences and key polynomials.

math.AC↗
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↗