Schanuel’s Conjecture and the Transcendence of Power Towers

Eva Trojovská and Pavel Trojovský
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Eva Trojovská: Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic
Pavel Trojovský: Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic

Mathematics, 2021, vol. 9, issue 7, 1-6

Abstract: We give three consequences of Schanuel’s Conjecture. The first is that P ( e ) Q ( e ) and P ( ? ) Q ( ? ) are transcendental, for any non-constant polynomials P ( x ) , Q ( x ) ? Q ¯ [ x ] . The second is that ? ? ? ? , for any algebraic numbers ? and ? . The third is the case of the Gelfond’s conjecture (about the transcendence of a finite algebraic power tower) in which all elements are equal.

Keywords: Schanuel’s Conjecture; Gelfond–Schneider Theorem; Hermite–Lindemann Theorem; algebraic independence; transcendence degree; power tower (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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