On a class of integer-valued functions (Q5918891)

Let \({\mathbb{Z}} v_1+{\mathbb{Z}} v_2\) be a lattice in \({\mathbb{C}}\). The authors study a class of entire functions \(f\) which, together with their first derivative, take values from a number field \(E\) at each point of the form \(a_1v_1+a_2v_2\) with \((a_1,a_2)\in{\mathbb{Z}}^2\), \(a_1\ge 0\), \(a_2\ge 0\). They assume that the functions do not grow faster than \(\exp(\gamma |z|^{6/5}(\log |z|)^{-1})\) for some \(\gamma>0\). There is a similar assumption on the heights of the values of the functions \(f\) and their first derivative at the given points. The conclusion is that \(f\) is either a polynomial or of the form \(e^{-m\alpha z}P(e^{\alpha z})\) for some \(m\in {\mathbb{Z}}\), \(m\ge 0\) and some \(\alpha\in E\). There is no arithmetic assumption on the complex numbers \(v_1\) and \(v_2\).