Global and unglobal relations among values of \(E\)-functions (Q5960221)

Let \(f_1(z),\dots,f_s(z)\) be a family of KE-functions that is a solution to the system of linear differential equations \(\bar y' = A(z)\bar y+\overline B(z)\), where \(A(z)\) is an \((s\times s)\)-matrix and \(\overline B(z)\) is a vector-column consisting of rational functions. The author establishes an equivalence relation between the Shidlovskij hypothesis [see \textit{A. B. Shidlovskij}, Transcendental numbers, Nauka, Moscow (1987; Zbl 0629.10026)] and the relation \[ P(f_1(\alpha) ,\dots,f_s(\alpha)) = 0,\quad P(x_1,\dots,x_s) \in \mathbb{K}[x_1,\dots, x_s]. \tag{1} \] Besides, it is proved that the relation (1) remains valid for all conjugate fields iff \[ P(f_1(z),\dots,f_s(z)) = (z-\alpha) R(z,f_1(z),\dots,f_s(z)), \] where \(R\) is a polynomial with coefficients from the field~\(\mathbb{K}\).