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Finite automata and algebraic extensions of function fields

Kiran S. Kedlaya (2006)

Journal de Théorie des Nombres de Bordeaux

We give an automata-theoretic description of the algebraic closure of the rational function field 𝔽 q ( t ) over a finite field 𝔽 q , generalizing a result of Christol. The description occurs within the Hahn-Mal’cev-Neumann field of “generalized power series” over 𝔽 q . In passing, we obtain a characterization of well-ordered sets of rational numbers whose base p expansions are generated by a finite automaton, and exhibit some techniques for computing in the algebraic closure; these include an adaptation to positive...

Fréchet algebras, formal power series, and automatic continuity

S. R. Patel (2008)

Studia Mathematica

We describe all those commutative Fréchet algebras which may be continuously embedded in the algebra ℂ[[X]] in such a way that they contain the polynomials. It is shown that these algebras (except ℂ[[X]] itself) always satisfy a certain equicontinuity condition due to Loy. Using this result, some applications to the theory of automatic continuity are given; in particular, the uniqueness of the Fréchet algebra topology for such algebras is established.

Further remarks on formal power series

Marcin Borkowski, Piotr Maćkowiak (2012)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we present a considerable simplification of the proof of a theorem by Gan and Knox, stating a sufficient and necessary condition for existence of a composition of two formal power series. Then, we consider the behavior of such series and their (formal) derivatives at the boundary of the convergence circle, obtaining in particular a theorem of Bugajewski and Gan concerning the structure of the set of points where a formal power series is convergent with all its derivatives.

Hardy fields in several variables

Leonardo Pasini (1985)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In questo lavoro si estende il concetto di campo di Hardy [Bou], al contesto dei germi di funzioni in più variabili che sono definite su insiemi semi-algebrici [Br.], [D.] e che risultano essere morfismi di categorie lisce [Pal.]. In tale contesto si dimostra che per ogni campo di Hardy di germi di una fissata categoria liscia 𝒞 , la sua chiusura algebrica relativa nell'anello G ⁢ 𝒞 , di tutti i germi nella stessa categoria liscia, è un campo di Hardy reale chiuso, che è l'unica chiusura reale del campo...

Invertible ideals and Gaussian semirings

Shaban Ghalandarzadeh, Peyman Nasehpour, Rafieh Razavi (2017)

Archivum Mathematicum

In the first section, we introduce the notions of fractional and invertible ideals of semirings and characterize invertible ideals of a semidomain. In section two, we define Prüfer semirings and characterize them in terms of valuation semirings. In this section, we also characterize Prüfer semirings in terms of some identities over its ideals such as ( I + J ) ( I ∩ J ) = I J for all ideals I , J of S . In the third section, we give a semiring version for the Gilmer-Tsang Theorem, which states that for a suitable family...

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.

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