Rationality of exponential functions at integer arguments
Let \(c, a_{\ell i}, \alpha_{\ell i}\in C\), \[ E_{\ell}(x)=a_{\ell 1}\alpha_{\ell 1}^x+\ldots +a_{\ell k_{\ell}}\alpha_{\ell k_{\ell}}^x \] and \[ E({\mathbf x})=E_1(x_1)+\ldots + E_n(x_n) +c \] in \({\mathbf x}=(x_1,\ldots,x_n)\in \mathbb Z^n\). Assuming \(k_1+\ldots +k_n\leq N\), the author considers the relation \[ E({\mathbf x})\in \mathbb Q. \] A solution \(\mathbf{x}\) of the above relation is called degenerate if in the sum there are vanishing subsums. Theorem 1 states, that if no \(\alpha_{\ell i}\) is a radical of an algebraic number of degree \(\leq N\), then the relation \(E({\mathbf x})\in \mathbb Q\) has \(\ll 1\) nondegenerate solutions. Assuming that no \(\alpha_{\ell i}\) is a root of unity Theorem 2 describes the nondegenerate solutions of the relation by using so called special transformations. The proof involves among others the application of the result of \textit{J. H. Evertse, H. P. Schlickewei} and \textit{W. M. Schmidt} [Ann. Math. (2) 155, 807--836 (2002; Zbl 1026.11038)]. The very interesting results of the paper will certainly have several applications.
scientific article; zbMATH DE number 2081163