An explicit Baker type lower bound of exponential values

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 $α_0=0$, $α_1,...,α_m$, 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.

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