# Algebraic independence over ℚp

Peter Bundschuh, Kumiko Nishioka

## 抄録

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)).

本文言語 English 519-533 15 Journal de Theorie des Nombres de Bordeaux 16 3 https://doi.org/10.5802/jtnb.458 Published - 2004 はい

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