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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Quantitative transcendence results for numbers associated with Liouville numbers


Author: Deanna M. Caveny
Journal: Proc. Amer. Math. Soc. 120 (1994), 349-357
MSC: Primary 11J82; Secondary 11J85
MathSciNet review: 1165048
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Abstract: In 1937, Franklin and Schneider generalized the Gelfond-Schneider result on the transcendence of $ {\alpha ^\beta }$. They proved the following theorem: If $ \beta $ is an algebraic, irrational number and $ \alpha $ is "suitably well-approximated by algebraic numbers of bounded degree", then $ {\alpha ^\beta }$ is transcendental. In 1964, Feldman established the algebraic independence of $ \alpha $ and $ {\alpha ^\beta }$ under similar conditions. We use results concerning linear forms in logarithms to give quantitative versions of the Franklin-Schneider and Feldman results.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1165048-8
PII: S 0002-9939(1994)1165048-8
Keywords: Quantitative, transcendence, Liouville numbers
Article copyright: © Copyright 1994 American Mathematical Society