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On the arithmetic nature of hypertranscendental functions at complex points. (English) Zbl 1239.11079
Author’s summary: “Most well-known transcendental functions usually take transcendental values at algebraic points belonging to their domains, the algebraic exceptions forming the so-called exceptional set. For instance, the exceptional set of the function \(e^{z- \sqrt 2}\) is the set \(\{\sqrt 2\}\), as follows from the Hermite-Lindemann theorem. In this paper, we use interpolation formulae to prove that any subset of \(\overline {\mathbb Q}\) is the exceptional set of uncountably many hypertranscendental entire functions with order of growth as small as we wish. Moreover these functions are algebraically independent over \(\mathbb C\).”
Reviewer’s remark: The paper essentially gives an interesting overview of old and recent results on the characterisation of entire functions which assume special values at certain algebraic sets; for example look at the given generalization of theorems of P. Stäckel [Math. Ann. 46, 513–520 (1895; JFM 26.0426.01)] and G. Faber and their proofs. For more information consult the references of the paper.

11J81 Transcendence (general theory)
Full Text: DOI
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