arXiv subjects

Wael Mahboub

Publications and source records attributed to Wael Mahboub.

6 recordsLinked to original sources

arXiv2026

Common extensions of valuations to rational function fields

Let (K(X)|K,w) be a valuation transcendental extension of rational function fields and take a minimal pair of definition (a, gamma). In this paper, we characterize those K-conjugates a' of a such that the monomial valuation induced by the pair (a', gamma) restricts to w on K(X). In particular, we show that a' satisfies this if and only if a' and a are conjugates over the henselization of (K,v). The second part of the paper concerns abstract key polynomials. We introduce the notion of a regular limit key polynomial, and more generally, that of a regular complete sequence of key polynomials. We prove that regularity is equivalent to the property that every root of every key polynomial determines the corresponding truncated valuation. This extends earlier work of Mahboub, Mansour and Spivakovsky by allowing both limit key polynomials and valuation algebraic extensions. As a consequence, we obtain that w always admits a regular complete sequence of key polynomials whenever (K,v) is dense in its henselization.

math.AC↗

arXiv2026

Finite Generation in Polynomial Semirings

We study the semiring $\mathbb{N}_0[\alpha]$ as an additive monoid where $\alpha$ is a positive real algebraic number. In the atomic case, the atoms of $\mathbb{N}_0[\alpha]$ are precisely the powers $\alpha^n$ up to a certain nonnegative integer $n$, and finite generation is governed by divisibility of the minimal polynomial by a negative-tail polynomial. Our first main result gives a complete characterization when the minimal polynomial has the form $\mathfrak{m}_\alpha(X)=p_\alpha(X)-c$ with $c\in\mathbb{N}$. Our second main result shows that finite generation forces $\alpha$ to be a weak Perron number. As an application, we analyze cubic minimal polynomials and obtain a partial classification of rank-$3$ monoids $\mathbb{N}_0[\alpha]$ by generation and factorization type, including coefficient constraints, non--length-factoriality results for a large family, and examples with prescribed numbers of atoms.

math.AC↗

arXiv2020

On common extensions of valued fields

Given a valuation $v$ on a field $K$, an extension $\bar{v}$ to an algebraic closure and an extension $w$ to $K(X)$. We want to study the common extensions of $\bar{v}$ and $w$ to $\bar{K}(X)$. First we give a detailed link between the minimal pairs notion and the key polynomials notion. Then we prove that in the case when $w$ is a transcendental extension, then any sequence of key polynomials admits a maximal element, and in case this sequence does not contain a limit key polynomial, then any root of the last key polynomial, describe a common extension.

math.AC↗

arXiv2019

Key Polynomials in dimension 2

Let $R$ be a two-dimensional regular local ring. In this paper, we prove that there is a bijection between the set of all valuations of $Quot(R)$ centered at $R$ and valuations of $k(x,y)$ centered at $k[x,y]_{(x,y)}$, where $k$ is the residue field of $R$ and $x$ and $y$ are independent variables. Moreover, we give a new proof, for the fact that the set of all normalized real valuations centered at $R$ admits a structure of non metric tree.

math.AG↗

arXiv2016

Abstract key polynomials and comparison theorems with the key polynomials of Mac Lane -- Vaquie

Let $\iota:(K,\nu)\hookrightarrow(K(x),\mu)$ be a simple purely transcendental extension of valued fields. In order to study such an extension, M. Vaqui\'e, generalizing an earlier construction of S. Mac Lane, introduced the notion of Key polynomials. In this paper we define a related notion of \textbf{abstract key polynomials} associated to $\iota$ and study the relationship between them and key polynomials of Mac Lane -- Vaqui\'e. Associated to each abstract key polynomial $Q$, we define the truncation $\mu_{Q}$ of $\mu$ with respect to $Q$ and we study the properties of those truncations. Roughly speaking, $\mu_{Q}$ is an approximation to $\mu$ defined by the key polynomial $Q$. We also define the notion of an abstract key polynomial $Q'$ being an \textbf{immediate successor} of another abstract key polynomial $Q$ (in this situation we write $Q<Q'$). The main comparison results proved in this paper are as follows:(1): An abstract key polynomial for $\mu$ is a Mac Lane -- Vaqui\'e key polynomial for the truncated valuation $\mu_{Q}$.(2): If $Q<Q'$ are two abstract key polynomials for $\mu$ then $Q'$ is a Mac Lane -- Vaqui\'e key polynomial for $\mu_{Q}$. (3) which, for a monic polynomial $Q\in K[x]$ and a valuation $\mu'$ of $K(x)$, gives a sufficient condition for $Q$ to be an abstract key polynomial for $\mu'$. Combined with an earlier result of M. Vaqui\'e, this describes a class of pairs of valuations $(\mu,\mu')$ such that $Q$ is a Mac Lane -- Vaqui\'e key polynomial for $\mu$ and an abstract key polynomial for $\mu'$.

math.AC↗

arXiv2012

Key Polynomials

The notion of key polynomials was first introduced in 1936 by S. Maclane in the case of discrete rank 1 valuations. . Let K -> L be a field extension and ν a valuation of K. The original motivation for introducing key polynomials was the problem of describing all the extensions μ of ν to L. Take a valuation μ of L extending the valuation ν. In the case when ν is discrete of rank 1 and L is a simple algebraic extension of K Maclane introduced the notions of key polynomials for μ and augmented valuations and proved that μ is obtained as a limit of a family of augmented valuations on the polynomial ring K[x]. In a series of papers, M. Vaquié generalized MacLane's notion of key polynomials to the case of arbitrary valuations ν (that is, valuations which are not necessarily discrete of rank 1). In the paper Valuations in algebraic field extensions, published in the Journal of Algebra in 2007, F.J. Herrera Govantes, M.A. Olalla Acosta and M. Spivakovsky develop their own notion of key polynomials for extensions (K, ν) -> (L, μ) of valued fields, where ν is of archimedian rank 1 (not necessarily discrete) and give an explicit description of the limit key polynomials. Our purpose in this paper is to clarify the relationship between the two notions of key polynomials already developed by vaquié and by F.J. Herrera Govantes, M.A. Olalla Acosta and M. Spivakovsky.

math.AG↗