Nesterenko, Yu. V. Measure of algebraic independence of the values of certain functions. (Russian) Zbl 0603.10033 Mat. Sb., N. Ser. 128(170), No. 4(12), 545-568 (1985). 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 Cited in 9 ReviewsCited in 7 Documents MSC: 11J81 Transcendence (general theory) 11J85 Algebraic independence; Gel’fond’s method Keywords:values of exponential function; functions satisfying functional equations; values at algebraic points; measures of algebraic independence Citations:Zbl 0567.10034 × Cite Format Result Cite Review PDF Full Text: EuDML