| Title |
The size of algebraic integers with many real conjugates |
| Authors |
Dubickas, Artūras |
| DOI |
10.26493/1855-3974.1348.d47 |
| Full Text |
|
| Is Part of |
ARS mathematica contemporanea.. Koper : Univerza na Primorskem FAMNIT. 2018, vol. 14, iss. 1, p. 165-176.. ISSN 1855-3966. eISSN 1855-3974 |
| Keywords [eng] |
Algebraic number field ; relative size ; relative normalised size ; Mahler measure ; Schur-Siegel-Smyth trace problem |
| Abstract [eng] |
In this paper we show that the relative normalised size with respect to a number field K of an algebraic integer alpha not equal -1, 0, 1 is greater than 1 provided that the number of real embeddings s of K satisfies s >= 0.828n, where n = [K : Q]. This can be compared with the previous much more restrictive estimate s >= n 0.192 root n/log n and shows that the minimum m(K) over the relative normalised size of nonzero algebraic integers alpha in such a field K is equal to 1 which is attained at alpha = +/- 1. Stronger than previous but apparently not optimal bound for m(K) is also obtained for the fields K satisfying 0.639 <= s/n < 0.827469 . . . . In the proof we use a lower bound for the Mahler measure of an algebraic number with many real conjugates. |
| Published |
Koper : Univerza na Primorskem FAMNIT |
| Type |
Journal article |
| Language |
English |
| Publication date |
2018 |
| CC license |
|
