Algebraic independence over ℚp

Peter Bundschuh, Kumiko Nishioka

Research output: Contribution to journalArticle

Abstract

Let f(x) be a power series ∑n≥1 ζ(n)xe(n), where (e(n)) is a strictly increasing linear recurrence sequence of nonnegative integers, and (ζ(n)) a sequence of roots of unity in ℚp satisfying an appropriate technical condition. Then we are mainly interested in characterizing the algebraic independence over ℚp of the elements f(α1),…, f(αt) from ℂp in terms of the distinct α1,…, αt ∈ ℚp satisfying 0 < |ατ |p < 1 for τ = 1,…, t. A striking application of our basic result says that, in the case e(n) = n, the set {f(α)| α ∈ ℚp, 0 < |α|p < 1} is algebraically independent over ℚp if (ζ(n)) satisfies the “technical condition”. We close with a conjecture concerning more general sequences (e(n)).

Original languageEnglish
Pages (from-to)519-533
Number of pages15
JournalJournal de Theorie des Nombres de Bordeaux
Volume16
Issue number3
Publication statusPublished - 2004

ASJC Scopus subject areas

  • Algebra and Number Theory

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  • Cite this

    Bundschuh, P., & Nishioka, K. (2004). Algebraic independence over ℚp Journal de Theorie des Nombres de Bordeaux, 16(3), 519-533.