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| ON THE ALGEBRAIC INDEPENDENCE OF CERTAIN POWER SERIES OF ALGEBRAIC NUMBERS |
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Citation： |
Zhu Yaochen.ON THE ALGEBRAIC INDEPENDENCE OF CERTAIN POWER SERIES OF ALGEBRAIC NUMBERS[J].Chinese Annals of Mathematics B,1984,5(1):109~118 |
| Page view： 776
Net amount： 687 |
Authors： |
Zhu Yaochen; |
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| Abstract： |
Let
$$\[{f_v}(s) = \sum\limits_{k = 1}^\infty {{a_{v,k}}{z^{{\lambda _{v,k}}}}} \begin{array}{*{20}{c}}
{(v = 1, \cdots ,s)}&{}
\end{array}\]$$
be s power series with algebraic coeffcients $\[{{a_{v,k}}}\]$ convergence radio $\[{R_v} > 0\]$ and sufficiently rapidly increasing integers $\[{{\lambda _{v,k}}}\]$. It is shown that under certain conditions depending only on
$\[{{a_{v,k}}}\]$ and $\[{{\lambda _{v,k}}}\]$ $\[(i){f_1}({\theta _1}), \cdots ,{f_s}({\theta _s})\]$are algebraically independent for arbitrary algebraic
numbers $\[{\theta _1} \cdots ,{\theta _s}\]$ with $\[0 < \left| {{\theta _v}} \right| < {R_v}(v = 1, \cdots ,v)\]$;
$\[(ii){f_v}({\theta _\mu })(v = 1, \cdots ,s;\mu = 1, \cdots ,t)\]$ are algebraically independent for t difierent algebraic numbers $\[{\theta _1}, \cdots ,{\theta _t}\]$ with $\[0 < \left| {{\theta _t}} \right| < \left| {{\theta _{t - 1}}} \right| < \cdots < \left| {{\theta _1}} \right| < \mathop {\min }\limits_{1 \le v \le s} R\]$. |
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