# Algebraic independence over ${ℚ}_{p}$

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

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## Abstract

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Let $f\left(x\right)$ be a power series ${\sum }_{n\ge 1}\zeta \left(n\right){x}^{e\left(n\right)}$, where $\left(e\left(n\right)\right)$ is a strictly increasing linear recurrence sequence of non-negative integers, and $\left(\zeta \left(n\right)\right)$ a sequence of roots of unity in ${\overline{ℚ}}_{p}$ satisfying an appropriate technical condition. Then we are mainly interested in characterizing the algebraic independence over ${ℚ}_{p}$ of the elements $f\left({\alpha }_{1}\right),...,$$f\left({\alpha }_{t}\right)$ from ${ℂ}_{p}$ in terms of the distinct ${\alpha }_{1},...,{\alpha }_{t}\in {ℚ}_{p}$ satisfying $0<|{\alpha }_{\tau }{|}_{p}<1$ for $\tau =1,...,t$. A striking application of our basic result says that, in the case $e\left(n\right)=n$, the set $\left\{f\left(\alpha \right)|\phantom{\rule{0.166667em}{0ex}}\alpha \in {ℚ}_{p},\phantom{\rule{0.166667em}{0ex}}0<|\alpha {|}_{p}<1\right\}$ is algebraically independent over ${ℚ}_{p}$ if $\left(\zeta \left(n\right)\right)$ satisfies the “technical condition”. We close with a conjecture concerning more general sequences $\left(e\left(n\right)\right)$.

## 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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