Algebraic independence of values of E-functions at algebraic points
Let \(f_ 1(z),...,f_ m(z)\) denote the set of Siegel E-functions satisfying a system of linear differential equations \(y'_ k=Q_{k,0}+\sum^{m}_{i=1}Q_{k,i}y_ i\), \(k=1,...,m\), where \(Q_{k,i}\in {\mathbb{C}}(z)\). In his well-known extension of Siegel's method the author [Izv. Akad. Nauk SSSR, Ser. Mat. 26, 877-910 (1962; Zbl 0116.039)] proved that if the transcendence degree of the set \(\{f_ 1(z),...,f_ m(z)\}\) over \({\mathbb{C}}(z)\) is \(\ell\), \(1\leq \ell \leq m\), and the functions \(f_ 1(z),...,f_{\ell}(z)\) are algebraically independent, then the numbers \(f_ 1(\alpha),...,f_{\ell}(\alpha)\) are algebraically independent over \({\mathbb{Q}}\) at all algebraic points \(\alpha\) \(\not\in A\), where A is a finite set of algebraic numbers. In the present work the author continues the investigation of the set A by generalizing his earlier results and certain results of V. G. Chirskij [Mat. Zametki 14, 83-94 (1973; Zbl 0279.10025)].
scientific article; zbMATH DE number 4130475