Application of Hermite-Mahler polynomials to the approximation of the values ofp-adic exponential function

We shall give a further application of Hermite-Mahler polynomials to the consideration ofp-adic exponential function. An effective lower bound is obtained for max {| – |_p,P(e)|_p}, where is an algebraic number satisfying ||_p <p^–/(p–1), and 0 is ap-adic number with | |_p depending on the degree of the polynomialPZ[y]. The bound obtained implies the transcendence ofe if ap-adic number satisfying 0 < ||_p <p^–/(p–1) is algebraic or can be well approximated by algebraic numbers.This work was carried out while the author was a research fellow of the Alexander von Humboldt Foundation.