Application of Hermite-Mahler polynomials to the approximation of the values of p-adic exponential function (Q749588)

The paper gives an application of Hermite-Mahler polynomials to the consideration of p-adic exponential function. An effective lower bound is obtained for \(\max \{| \alpha -\theta |_ p,\quad | P(e^{\alpha})|_ p\},\) where \(\theta\) is an algebraic number satisfying \(| \theta |_ p<p^{-1/(p-1)}\), and \(\alpha\neq 0\) is a p-adic number with \(| \alpha |_ p\) depending on the degree of the polynomial \(P\in {\mathbb{Z}}[y]\). The bound obtained implies the transcendence of \(e^{\alpha}\) if a p-adic number \(\alpha\) satisfying \(0<| \alpha |_ p<p^{-1/(p-1)}\) is algebraic or can be well approximated by algebraic numbers.