Abstract
In 1902, P. Stäckel proved the existence of a transcendental function f(z), analytic in a neighbourhood of the origin, and with the property that both f(z) and its inverse function assume, in this neighbourhood, algebraic values at all algebraic points. Based on this result, in 1976, K. Mahler raised the question of the existence of such functions which are analytic in \(\mathbb {C}\). In this work, we provide a positive answer for this question by showing the existence of uncountable many of these functions.
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References
Mahler, K.: Lectures on Transcendental Numbers, Lecture Notes in Math., vol. 546. Springer, Berlin (1976)
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Communicated by A. Venkatesh.
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Marques, D., Moreira, C.G. A positive answer for a question proposed by K. Mahler. Math. Ann. 368, 1059–1062 (2017). https://doi.org/10.1007/s00208-016-1485-z
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DOI: https://doi.org/10.1007/s00208-016-1485-z