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An explicit Baker-type lower bound of exponential values. (English) Zbl 1387.11059
Summary: Let $$\mathbb I$$ denote an imaginary quadratic field or the field $$\mathbb Q$$ of rational numbers and let $$\mathbb Z_{\mathbb I}$$ denote its ring of integers. We shall prove a new explicit Baker-type lower bound for a $$\mathbb Z_{\mathbb I}$$-linear form in the numbers $$1, \mathrm{e}^{\alpha_1}, \ldots, \mathrm{e}^{\alpha_m}$$, $$m \geq 2$$, where $$\alpha_0 = 0$$, $$\alpha_1, \ldots, \alpha_m$$ are $$m + 1$$ different numbers from the field $$\mathbb I$$. Our work gives substantial improvements on the existing explicit versions of Baker’s work about exponential values at rational points. In particular, dependencies on $$m$$ are improved.

##### MSC:
 11J82 Measures of irrationality and of transcendence 11J72 Irrationality; linear independence over a field
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