A criterion for algebraic independence with some applications. (English) Zbl 0712.11041

For an algebraic number \(\beta\) let \(\partial \beta\) denote the degree of \(\beta\), H(\(\beta\)) the height of \(\beta\), and \(s(\beta)=\partial \beta +\log H(\beta)\) the size of \(\beta\). The author proves the following criterion for algebraic independence: Let \(\beta_ 1,...,\beta_ t\) be given complex numbers, and let g: \({\mathbb{N}}\to {\mathbb{R}}_+\) satisfy g(n)\(\to \infty\) as \(n\to \infty\). Suppose that for each \(\tau\in \{1,...,t\}\) there exist an infinite set \(N_{\tau}\subset {\mathbb{N}}\) and \(\tau\) sequences \((\beta_{1n})_{n\in {\mathbb{N}}_ 1},...,(\beta_{\tau n})_{n\in {\mathbb{N}}_{\tau}}\) of algebraic numbers such that for each \(n\in N_{\tau}\) the inequalities \[ g(n)\sum^{\tau -1}_{\sigma =1}| \beta_{\sigma}-\beta_{\sigma n}| <| \beta_{\tau}- \beta_{\tau n}| \leq \exp (- g(n)[{\mathbb{Q}}(\beta_{1n},...,\beta_{\tau n}):{\mathbb{Q}}]\sum^{\tau}_{\sigma =1}(s(\beta_{\sigma n})/\partial (\beta_{\sigma n}))) \] hold. Then \(\beta_ 1,...,\beta_ t\) are algebraically independent. The author succeeds in giving a short proof for this result by using a Liouville-type estimate of Fel’dman. The criterion turns out to be useful and strong in applications. It is proved in the paper that it implies many earlier results, e.g. by Amou, the author and Wylegala, Cijsouw, and Zhu, on algebraic independence of the values of gap power series with algebraic coefficients.
Reviewer: K.Väänänen


11J85 Algebraic independence; Gel’fond’s method
11J81 Transcendence (general theory)