Algebraic independence over ℚ p

Peter Bundschuh[1]; Kumiko Nishioka[2]

  • [1] Mathematisches Institut Universität zu Köln Weyertal 86-90 50931 Köln, Germany
  • [2] Mathematics, Hiyoshi Campus Keio University 4-1-1 Hiyoshi, Kohoku-ku Yokohama 223-8521, Japan

Journal de Théorie des Nombres de Bordeaux (2004)

  • Volume: 16, Issue: 3, page 519-533
  • ISSN: 1246-7405

Abstract

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Let f ( x ) be a power series ∑ n ≥ 1 ζ ( n ) x e ( n ) , where ( e ( n ) ) is a strictly increasing linear recurrence sequence of non-negative integers, and ( ζ ( n ) ) a sequence of roots of unity in ℚ ¯ p satisfying an appropriate technical condition. Then we are mainly interested in characterizing the algebraic independence over ℚ p of the elements f ( α 1 ) , ... , f ( α t ) from ℂ p in terms of the distinct α 1 , ... , α t ∈ ℚ p satisfying 0 < | α τ | 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 ∈ ℚ p , 0 < | α | p < 1 } is algebraically independent over ℚ p if ( ζ ( n ) ) satisfies the “technical condition”. We close with a conjecture concerning more general sequences ( e ( n ) ) .

How to cite

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Bundschuh, Peter, and Nishioka, Kumiko. "Algebraic independence over $\mathbb{Q}_p$." Journal de Théorie des Nombres de Bordeaux 16.3 (2004): 519-533. <http://eudml.org/doc/249266>.

@article{Bundschuh2004,
abstract = {Let $f(x)$ be a power series $\sum _\{n\ge 1\}\zeta (n)x^\{e(n)\}$, where $(e(n))$ is a strictly increasing linear recurrence sequence of non-negative integers, and $(\zeta (n))$ a sequence of roots of unity in $\overline\{\mathbb\{Q\}\}_p$ satisfying an appropriate technical condition. Then we are mainly interested in characterizing the algebraic independence over $\mathbb\{Q\}_p$ of the elements $f(\alpha _1),\ldots ,$$f(\alpha _t)$ from $\mathbb\{C\}_p$ in terms of the distinct $\alpha _1,\ldots ,\alpha _t\in \mathbb\{Q\}_p$ satisfying $0 &lt; |\alpha _\tau |_p &lt; 1$ for $\tau = 1,\ldots ,t$. A striking application of our basic result says that, in the case $e(n) = n$, the set $\lbrace f(\alpha )|\, \alpha \in \mathbb\{Q\}_p, \, 0 &lt;|\alpha |_p &lt; 1\rbrace $ is algebraically independent over $\mathbb\{Q\}_p$ if $(\zeta (n))$ satisfies the “technical condition”. We close with a conjecture concerning more general sequences $(e(n))$.},
affiliation = {Mathematisches Institut Universität zu Köln Weyertal 86-90 50931 Köln, Germany; Mathematics, Hiyoshi Campus Keio University 4-1-1 Hiyoshi, Kohoku-ku Yokohama 223-8521, Japan},
author = {Bundschuh, Peter, Nishioka, Kumiko},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {3},
pages = {519-533},
publisher = {Université Bordeaux 1},
title = {Algebraic independence over $\mathbb\{Q\}_p$},
url = {http://eudml.org/doc/249266},
volume = {16},
year = {2004},
}

TY - JOUR
AU - Bundschuh, Peter
AU - Nishioka, Kumiko
TI - Algebraic independence over $\mathbb{Q}_p$
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2004
PB - Université Bordeaux 1
VL - 16
IS - 3
SP - 519
EP - 533
AB - Let $f(x)$ be a power series $\sum _{n\ge 1}\zeta (n)x^{e(n)}$, where $(e(n))$ is a strictly increasing linear recurrence sequence of non-negative integers, and $(\zeta (n))$ a sequence of roots of unity in $\overline{\mathbb{Q}}_p$ satisfying an appropriate technical condition. Then we are mainly interested in characterizing the algebraic independence over $\mathbb{Q}_p$ of the elements $f(\alpha _1),\ldots ,$$f(\alpha _t)$ from $\mathbb{C}_p$ in terms of the distinct $\alpha _1,\ldots ,\alpha _t\in \mathbb{Q}_p$ satisfying $0 &lt; |\alpha _\tau |_p &lt; 1$ for $\tau = 1,\ldots ,t$. A striking application of our basic result says that, in the case $e(n) = n$, the set $\lbrace f(\alpha )|\, \alpha \in \mathbb{Q}_p, \, 0 &lt;|\alpha |_p &lt; 1\rbrace $ is algebraically independent over $\mathbb{Q}_p$ if $(\zeta (n))$ satisfies the “technical condition”. We close with a conjecture concerning more general sequences $(e(n))$.
LA - eng
UR - http://eudml.org/doc/249266
ER -

References

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  9. K. Nishioka, p -adic transcendental numbers. Proc. Amer. Math. Soc. 108 (1990), 39–41. Zbl0687.10027MR994783
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