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Lacunary formal power series and the Stern-Brocot sequence

Jean-Paul Allouche, Michel Mendès France (2013)

Acta Arithmetica

Let F ( X ) = ∑ n ≥ 0 ( - 1 ) ε ₙ X - λ ₙ be a real lacunary formal power series, where εₙ = 0,1 and λ n + 1 / λ ₙ > 2 . It is known that the denominators Qₙ(X) of the convergents of its continued fraction expansion are polynomials with coefficients 0, ±1, and that the number of nonzero terms in Qₙ(X) is the nth term of the Stern-Brocot sequence. We show that replacing the index n by any 2-adic integer ω makes sense. We prove that Q ω ( X ) is a polynomial if and only if ω ∈ ℤ. In all the other cases Q ω ( X ) is an infinite formal power series; we discuss its algebraic...

Langage de Łukasiewicz et diagonales de séries formelles

Isabelle Fagnot (1996)

Journal de théorie des nombres de Bordeaux

Dans un corps fini, toute série formelle algébrique en une indéterminée est la diagonale d'une fraction rationnelle en deux indéterminées (Furstenberg 67). Dans cet article, nous donnons une nouvelle preuve de ce résultat, par des méthodes purement combinatoires.

Large superdecomposable E(R)-algebras

Laszlo Fuchs, Rüdiger Göbel (2005)

Fundamenta Mathematicae

For many domains R (including all Dedekind domains of characteristic 0 that are not fields or complete discrete valuation domains) we construct arbitrarily large superdecomposable R-algebras A that are at the same time E(R)-algebras. Here "superdecomposable" means that A admits no (directly) indecomposable R-algebra summands ≠ 0 and "E(R)-algebra" refers to the property that every R-endomorphism of the R-module, A is multiplication by an element of, A.

Length 2 variables of A[x,y] and transfer

Eric Edo, Stéphane Vénéreau (2001)

Annales Polonici Mathematici

We construct and study length 2 variables of A[x,y] (A is a commutative ring). If A is an integral domain, we determine among these variables those which are tame. If A is a UFD, we prove that these variables are all stably tame. We apply this construction to show that some polynomials of A[x₁,...,xₙ] are variables using transfer.

Lifting the field of norms

Laurent Berger (2014)

Journal de l’École polytechnique — Mathématiques

Let K be a finite extension of Q p . The field of norms of a p -adic Lie extension K ∞ / K is a local field of characteristic p which comes equipped with an action of Gal ( K ∞ / K ) . When can we lift this action to characteristic 0 , along with a compatible Frobenius map? In this note, we formulate precisely this question, explain its relevance to the theory of ( ϕ , Γ ) -modules, and give a condition for the existence of certain types of lifts.

Linear gradings of polynomial algebras

Piotr Jędrzejewicz (2008)

Open Mathematics

Let k be a field, let G be a finite group. We describe linear G -gradings of the polynomial algebra k[x 1, ..., x m] such that the unit component is a polynomial k-algebra.

Local derivations in polynomial and power series rings

Janusz Zieliński (2002)

Colloquium Mathematicae

We give a description of all local derivations (in the Kadison sense) in the polynomial ring in one variable in characteristic two. Moreover, we describe all local derivations in the power series ring in one variable in any characteristic.

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