ON THE ALGEBRAIC INDEPENDENCE OF CERTAIN POWER SERIES OF ALGEBRAIC NUMBERS

Letf_v(z)=∑a_(v,,k)z~(λ_(v,k))(v=1,…,s)be s power series with algebraic coefficients a_(v,k),convergence radii R_v>0 and sufficientlyrapidly increasing integers λ_(v,k).It is shown that under certain conditions depending only ona_(v,k) and λ_(v,k),(i)f_1(θ_1),…,f_s(θ_s)are algebraically independent for arbitrary algebraicnumbers θ_1,…,θ_s with θ<丨θ_v丨

ON THE ALGEBRAIC INDEPENDENCE OF CERTAIN POWER SERIES OF ALGEBRAIC NUMBERS

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\]$.