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Displaying 101 –
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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.
In this article, we compare different types of representations for series with coefficients in complete idempotent semirings. Each of these representations was introduced to solve a particular problem. We show how they are or are not included one in the other and we present a common generalization of them.
Let be a holomorphic map from to defined in a neighborhood of zero such that If the jacobian determinant of is not identically zero, P. M. Eakin and G. A. Harris proved the following result: any formal power series such that is analytic is itself analytic. If the jacobian determinant of 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...
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 -adiques de la fonction pour les différents nombres premiers . 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...
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.
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.
We study , the ring of arithmetical functions with unitary convolution, giving an isomorphism between and a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell-Everett [NumThe] between the ring of arithmetical functions with Dirichlet convolution and the power series ring 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 paper examines the ring of arithmetical functions, identifying it to the domain of formal power series over in a countable set of indeterminates. It is proven that is a complete ultrametric space and all its continuous endomorphisms are described. It is also proven that is a quasi-noetherian ring.
On montre que tout anneau local régulier complet muni d’une valuation de rang peut être plongé, en tant qu’anneau valué, dans un anneau de séries de Puiseux généralisées.
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 to 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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