Searcharxiv⌕ Search

arXiv subjects

Enric Nart

Publications and source records attributed to Enric Nart.

At least 19 recordsLinked to original sources
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

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

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↗
arXiv2024

Okutsu sequences in Henselian fields

For $(K,v)$ a Henselian valued field, let $θ\in\overline{K}$ with minimal polynomial $F$ over $K$. Okutsu sequences of $θ$ have been defined only when the extension $K(θ)/K$ is defectless. In this paper, we extend this concept to arbitrary $θ\in\overline{K}$ and we show that these objects are essentially equivalent to Okutsu frames of $F$ and to Mac Lane-Vaquié chains of the natural valuation on $K[x]$ induced by $θ$.

math.NT↗
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

Okutsu frames of irreducible polynomials over henselian fields

For a henselian valued field $(K,v)$ we establish a complete parallelism between the arithmetic properties of irreducible polynomials $F\in K[x]$, encoded by their Okutsu frames, and the valuation-theoretic properties of their induced valuations $v_F$ on $K[x]$, encoded by their MacLane-Vaquié chains. This parallelism was only known for defectless irreducible polynomials.

math.AC↗
arXiv2022

Rigidity of valuative trees under henselization

Let $(K,v)$ be a valued field and let $(K^h,v^h)$ be the henselization determined by the choice of an extension of $v$ to an algebraic closure of $K$. Consider an embedding $v(K^*)\hookrightarrowΛ$ of the value group into a divisible ordered abelian group. Let $T(K,Λ)$, $T(K^h,Λ)$ be the trees formed by all $Λ$-valued extensions of $v$, $v^h$ to the polynomial rings $K[x]$, $K^h[x]$, respectively. We show that the natural restriction mapping $T(K^h,Λ)\to T(K,Λ)$ is an isomorphism of posets. As a consequence, the restriction mapping $T_v\to T_{v^h}$ is an isomorphism of posets too, where $T_v$, $T_{v^h}$ are the trees whose nodes are the equivalence classes of valuations on $K[x]$, $K^h[x]$ whose restriction to $K$, $K^h$ are equivalent to $v$, $v^h$, respectively.

math.AG↗
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↗
arXiv2022

Polynomial factorization over henselian fields

Given a valued field $(K,v)$ and an irreducible polynomial $g\in K[x]$, we survey the ideas of Ore, Maclane, Okutsu, Montes, Vaquié and Herrera-Olalla-Mahboub-Spivakovsky, leading (under certain conditions) to an algorithm to find the factorization of $g$ over a henselization of $(K,v)$.

math.AC↗
arXiv2022

Valuative trees over valued fields

For an arbitrary valued field $(K,v)$ and a given extension $v(K^*)\hookrightarrowΛ$ of ordered groups, we analyze the structure of the tree formed by all $Λ$-valued extensions of $v$ to the polynomial ring $K[x]$. As an application, we find a model for the tree of all equivalence classes of valuations on $K[x]$ (without fixing their value group), whose restriction to $K$ is equivalent to $v$. In the henselian case, we apply these results to show that there is a complete parallelism between the arithmetic properties of irreducible polynomials $F\in K[x]$, encoded by their Okutsu frames, and the valuation-theoretic properties of their induced valuations $v_F$ on $K[x]$, encoded by their MacLane-Vaquié chains. This parallelism was only known for defectless irreducible polynomials.

math.AG↗
arXiv2022

Valuations with infinite limit-depth

For a certain field $K$, we construct a valuation-algebraic valuation on the polynomial ring $K[x]$, whose Maclane--Vaquié chain consists of an infinite (countable) number of limit augmentations

math.AC↗
arXiv2022

Square-free OM computation of global integral bases

For a prime $p$, the OM algorithm finds the $p$-adic factorization of an irreducible polynomial $f\in\mathbb{Z}[x]$ in polynomial time. This may be applied to construct $p$-integral bases in the number field $K$ defined by $f$. In this paper, we adapt the OM techniques to work with a positive integer $N$ instead of $p$. As an application, we obtain an algorithm to compute global integral bases in $K$, which does not require a previous factorization of the discriminant of $f$.

math.NT↗
arXiv2021

Cuts and small extensions of abelian ordered groups

We classify cuts in (totally) ordered abelian groups $\g$ and compute the coinitiality and cofinality of all cuts in case $\g$ is divisible, in terms of data intrinsically associated to the invariance group of the cut. We relate cuts with small extensions of $\g$ in a natural way, which leads to an explicit construction of a totally ordered real vector space containing realizations of all cuts. This construction is applied to the problem of classifying all extensions of the valuation from a given valued field $K$ to the rational function field $K(x)$.

math.AC↗
arXiv2020

Small extensions of abelian ordered groups

Let $Γ$ be a totally ordered group. We use Hahn's embedding theorem to construct a totally ordered set $Γ\subset Γ_{\operatorname{sme}}$ which classifies small extensions of $Γ$. This small-extensions closure $Γ_{\operatorname{sme}}$ is complete and plays a crucial role in the description of equivalence classes of valuations on the polynomial ring $K[x]$ over a field $K$.

math.AC↗
arXiv2020

MacLane-Vaquié chains of valuations on a polynomial ring

Let $(K,v)$ be a valued field. We review some results of MacLane and Vaquié on extensions of $v$ to valuations on the polynomial ring $K[x]$. We introduce certain MacLane-Vaquié chains of residually transcendental valuations, and we prove that every valuation $μ$ on $K[x]$ is a limit of a finite or countably infinite MacLane-Vaquié chain. This chain underlying $μ$ is essentially unique and contains arithmetic data yielding an explicit description of the graded algebra of $μ$ as an algebra over the graded algebra of $v$.

math.AG↗
arXiv2020

Invariants of limit key polynomials

Let $ν$ be a valuation of arbitrary rank on the polynomial ring $K[x]$ with coefficients in a field $K$. We prove comparison theorems between MacLane-Vaquié key polynomials for valuations $μ\leν$ and abstract key polynomials for $ν$. Also, some results on invariants attached to limit key polynomials are obtained. In particular, if $\operatorname{char}(K)=0$ we show that all limit key polynomials of unbounded continuous MacLane chains have numerical character equal to one.

math.AG↗
arXiv2020

Computation of residual polynomial operators of inductive valuations

Let $(K,v)$ be a valued field, and $μ$ an inductive valuation on $K[x]$ extending $v$. Let $G_μ$ be the graded algebra of $μ$ over $K[x]$, and $κ$ the maximal subfield of the subring of $G_μ$ formed by the homogeneous elements of degree zero. In this paper, we find an algorithm to compute the field $κ$ and the residual polynomial operator $R_μ: K[x]\toκ[y]$, where $y$ is another indeterminate, without any need to perform computations in the graded algebra. This leads to an OM algorithm to compute the factorization of separable defectless polynomials over henselian fields.

math.AG↗