Minimal Mahler measures for generators of some fields
Artūras Dubickas
Vilnius University, Lithuania
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Abstract
We prove that for each odd integer there are infinitely many number fields of degree such that each generator of has Mahler measure greater than or equal to , where is the discriminant of the field . This, combined with an earlier result of Vaaler and Widmer for composite , answers negatively a question of Ruppert raised in 1998 about ‘small’ algebraic generators for every . We also show that for each and any , there exist infinitely many number fields of degree such that every algebraic integer generator of has Mahler measure greater than . On the other hand, every such field contains an algebraic integer generator with Mahler measure smaller that . This generalizes the corresponding bounds recently established by Eldredge and Petersen for .
Cite this article
Artūras Dubickas, Minimal Mahler measures for generators of some fields. Rev. Mat. Iberoam. 39 (2023), no. 1, pp. 269–282
DOI 10.4171/RMI/1331