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Some remarks about the Dedekind-Mertens lemma

Jakub Byszewski (2016)

Banach Center Publications

The Dedekind-Mertens lemma relates the contents of two polynomials and the content of their product. Recently, Epstein and Shapiro extended this lemma to the case of power series. We review the problem with a special emphasis on the case of power series, give an answer to a question posed by Epstein-Shapiro and investigate extensions of some related results. This note is of expository character and discusses the history of the problem, some examples and announces some new results.

Sur la composition de séries formelles à croissance contrôlée

Augustin Mouze (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let F be a holomorphic map from ℂ s to ℂ s defined in a neighborhood of zero such that F ( 0 ) = 0 . If the jacobian determinant of F is not identically zero, P. M. Eakin and G. A. Harris proved the following result: any formal power series 𝒜 such that 𝒜 ∘ F is analytic is itself analytic. If the jacobian determinant of F is identically zero, they proved that the previous conclusion is no more true. J. Chaumat and A.-M. Chollet extended this result in the case of formal power series satisfying growth conditions, of...

Sur la constante d’Eisenstein

Rachid Mechik (2008)

Annales mathématiques Blaise Pascal

On cherche à donner une méthode effective de calcul de la constante d’Eisenstein [3] d’une fonction algébrique. On commence en précisant les liens entre cette constante et les rayons de convergence p -adiques de la fonction pour les différents nombres premiers p . Puis on donne une démonstration entièrement effective du résultat bien connu liant fonctions algébriques et diagonales de fractions rationnelles. Enfin on explique comment en déduire une méthode de calcul générale. On illustre la méthode...

Sur la formule d’inversion de Lagrange

Charles Delorme (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

On se propose de démontrer que la formule d’inversion de Lagrange est encore valide sur un anneau commutatif, même pour une série ayant quelques termes à coefficients nilpotents avant le terme de degré 1 (dont le coefficient est inversible). On n’use que de techniques algébriques.

The Abhyankar-Jung theorem for excellent henselian subrings of formal power series

Krzysztof Jan Nowak (2010)

Annales Polonici Mathematici

Given an algebraically closed field K of characteristic zero, we prove the Abhyankar-Jung theorem for any excellent henselian ring whose completion is a formal power series ring K[[z]]. In particular, examples include the local rings which form a Weierstrass system over the field K.

The ring of arithmetical functions with unitary convolution: Divisorial and topological properties

Jan Snellman (2004)

Archivum Mathematicum

We study ( 𝒜 , + , ⊕ ) , the ring of arithmetical functions with unitary convolution, giving an isomorphism between ( 𝒜 , + , ⊕ ) and a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell-Everett [NumThe] between the ring ( 𝒜 , + , · ) of arithmetical functions with Dirichlet convolution and the power series ring [ [ x 1 , x 2 , x 3 , ⋯ ] ] on countably many variables. We topologize it with respect to a natural norm, and show that all ideals are quasi-finite. Some elementary results on factorization into atoms...

The valuated ring of the arithmetical functions as a power series ring

Emil Daniel Schwab, Gheorghe Silberberg (2001)

Archivum Mathematicum

The paper examines the ring A of arithmetical functions, identifying it to the domain of formal power series over 𝐂 in a countable set of indeterminates. It is proven that A is a complete ultrametric space and all its continuous endomorphisms are described. It is also proven that A is a quasi-noetherian ring.

Théorèmes de préparation Gevrey et étude de certaines applications formelles

Augustin Mouze (2003)

Annales Polonici Mathematici

We consider subrings A of the ring of formal power series. They are defined by growth conditions on coefficients such as, for instance, Gevrey conditions. We prove preparation theorems of Malgrange type in these rings. As a consequence we study maps F from ℂ s to ℂ p without constant term such that the rank of the Jacobian matrix of F is equal to 1. Let be a formal power series. If F is a holomorphic map, the following result is well known: ∘ F is analytic implies there exists a convergent power series...

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