An explicit Baker type lower bound of exponential values

Abstract

Let $\mathbb{I}$ denote an imaginary quadratic field or the field $\mathbb{Q}$ of rational numbers and ${\mathbb{Z}}_{\mathbb{I}}$ its ring of intergers. We shall prove an explicit Baker type lower bound for ${\mathbb{Z}}_{\mathbb{I}}$-linear form of the numbers \begin{equation}\label{1} 1,\ e^{α_1},...,\ e^{α_m},\quad m\ge 2, \end{equation} where ${\alpha }_0=0$, ${\alpha }_1,...,{\alpha }_m$, are $m+1$ different numbers from the field $\mathbb{I}$. Our work gives gives some improvements to the existing explicit versions of of Baker's work about exponential values at rational points. In particilar, dependences on $m$ are improved.