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Measure of algebraic independence of the values of certain functions. (Russian) Zbl 0603.10033

In this important article the author gives the following two main results. (1) Let the set of complex numbers \(\{a_ 1,...,a_ p\}\) and \(\{b_ 1,...,b_ q\}\) be linearly independent over \({\mathbb{Q}}\). Then there are at least \(\frac{m_ i}{p+q}-1\) algebraically independent numbers among the (a) \(m_ 1=p+q\) numbers \(e^{a_ ib_ j}\); (b) \(m_ 2=pq+p\) numbers \(a_ i\), \(e^{a_ ib_ j}\), (c) \(m_ 3=pq+p+q\) numbers \(a_ i\), \(b_ j\), \(e^{a_ ib_ j}\) (1\(\leq i\leq p\), \(1\leq j\leq q)\). From (1) it follows that \[ tr \deg_{{\mathbb{Q}}}(\alpha^{\beta},...,\alpha^{\beta^{d-1}})\geq [\frac{d}{2}] \] where \(\alpha\neq 0,1\) is an algebraic number and \(\beta\) is an algebraic number of degree d. This corollary has been announced by P. Philippon [Prog. Math. 59, 219-233 (1985; Zbl 0567.10034)].
(2) Let \(f_ 1(z),..,f_ m(z)\) satisfy the functional equations \(f_ i(z^ d)=a_ i(z)f_ i(z)+b_ i(z)\), \(a_ i(z)\), \(b_ i(z)\in {\mathbb{K}}(z)\), \(i=1,...,m\). The author derives measures of algebraic independence for the values of the functions \(f_ 1(z),...,f_ n(z)\) at algebraic points.
Reviewer: Xu Guangshan

MSC:

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

Citations:

Zbl 0567.10034