Mahler, Kurt

Reprint: Zur Approximation der Exponentialfunktion und des Logarithmus. I, II (1931/32)

Doc. Math. Extra Vol. Mahler Selecta, 281-314 (2019)
DOI: 10.25537/dm.2019.SB-281-314

Summary

In Part I, Mahler introduces his classification of complex numbers and the following two results are proved. Let \(\vartheta_1,\vartheta_2,\ldots,\vartheta_N\) be \(N\) algebraic numbers that are linearly independent over the rationals and let \(\lambda\) be a Liouville number. Then, the numbers \(e^{\vartheta_1},e^{\vartheta_2},\ldots,e^{\vartheta_N},\lambda\) are algebraically independent over the field of algebraic numbers. Let \(z\) be the real logarithm of a positive rational number not equal to one and let \(\lambda\) be a Liouville number. Then, \(z\) and \(\lambda\) are algebraically independent over the field of algebraic numbers. \par Part II continues the study of the same title by giving various bounds on polynomials evaluated at logarithms and exponentials. A new proof of the transcendence of \(\pi\) is given as an application. \par Reprint of the author's papers [J. Reine Angew. Math. 166, 118--136 (1931; Zbl 0003.15101; JFM 57.0242.03); ibid. 166, 137--150 (1932; Zbl 0003.38805; JFM 58.0207.01)].

Mathematics Subject Classification

11-03, 11J85

Keywords/Phrases

approximation, exponential function, logarithm, algebraic independence

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