TY - JOUR

T1 - Algebraic independence over ℚp

AU - Bundschuh, Peter

AU - Nishioka, Kumiko

N1 - Funding Information:
This work was done during the second-named author's stay at the University of Cologne supported by the Alexander von Humboldt Foundation. Both authors are very grateful to the foundation for giving them the opportunity for collaboration.

PY - 2004

Y1 - 2004

N2 - 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)).

AB - 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)).

UR - http://www.scopus.com/inward/record.url?scp=85009962840&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85009962840&partnerID=8YFLogxK

U2 - 10.5802/jtnb.458

DO - 10.5802/jtnb.458

M3 - Article

AN - SCOPUS:85009962840

VL - 16

SP - 519

EP - 533

JO - Journal de Theorie des Nombres de Bordeaux

JF - Journal de Theorie des Nombres de Bordeaux

SN - 1246-7405

IS - 3

ER -