zbMATH — the first resource for mathematics

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.

11J82 Measures of irrationality and of transcendence
11J72 Irrationality; linear independence over a field
Full Text: DOI arXiv
[1] Number theory IV. Transcendental numbers. 44 (1998)
[2] On G-functions. 2 pp 1– (1981)
[3] DOI: 10.4153/CJM-1965-061-8 · Zbl 0147.30901 · doi:10.4153/CJM-1965-061-8
[4] Handbook of mathematical functions with formulas, graphs, and mathematical tables. 55 (1964) · Zbl 0171.38503
[5] DOI: 10.1070/SM1996v187n12ABEH000178 · Zbl 0878.11030 · doi:10.1070/SM1996v187n12ABEH000178
[6] DOI: 10.4153/CMB-2005-013-5 · Zbl 1064.11054 · doi:10.4153/CMB-2005-013-5
[7] Illinois J. Math. 6 pp 64– (1962)
[8] Acta Arith. 27 pp 61– (1975)
[9] DOI: 10.1006/jnth.1995.1103 · Zbl 0839.11027 · doi:10.1006/jnth.1995.1103
[10] DOI: 10.1016/j.jnt.2012.02.018 · Zbl 1276.11124 · doi:10.1016/j.jnt.2012.02.018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.