| Title |
Minimal Mahler measures for generators of some fields / |
| Authors |
Dubickas, Artūras |
| DOI |
10.4171/RMI/1331 |
| Full Text |
|
| Is Part of |
Revista matematica Iberoamericana.. Berlin : European Mathematical Society Publishing House. 2023, vol. 39, iss. 1, p. 269-282.. ISSN 0213-2230. eISSN 2235-0616 |
| Keywords [eng] |
discriminant ; generator ; Mahler measure ; monogenic field ; number field |
| Abstract [eng] |
We prove that for each odd integer d ≥ 3 there are infinitely many number fields K of degree d such that each generator α of K has Mahler measure greater d +1 than or equal to d-d |ΔK | d+1/d(2d -2, where ΔK is the discriminant of the field K. This, combined with an earlier result of Vaaler and Widmer for composite d, answers negatively a question of Ruppert raised in 1998 about 'small' algebraic generators for every d ≥ 3. We also show that for each d ≥ 2 and any ε > 0, there exist infinitely many number fields K of degree d such that every algebraic integer generator α of K has Mahler measure greater than (1 - ε)|ΔK |1/d. On the other hand, every such field K contains an algebraic integer generator α with Mahler measure smaller that |ΔK |1/d. This generalizes the corresponding bounds recently established by Eldredge and Petersen for d = 3. |
| Published |
Berlin : European Mathematical Society Publishing House |
| Type |
Journal article |
| Language |
English |
| Publication date |
2023 |
| CC license |
|
