Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 601-606

SMALL DISCS CONTAINING CONJUGATE ALGEBRAIC INTEGERS

Artūras Dubickas

Vilnius University, Faculty of Mathematics and Informatics, Institute of Mathematics
Naugarduko 24, LT-03225 Vilnius, Lithuania; arturas.dubickas 'at' mif.vu.lt

Abstract. In this note we show that, for any ξ ∈ R, there is an infinite set of positive integers S such that, for each d ∈ S, the open disc with center at ξ and radius 1 + (log log d)²/(2 log d) contains a full set of conjugates of an algebraic integer of degree d. A slightly better bound on the radius is established when ξ ∈ Q \ Z.

2010 Mathematics Subject Classification: Primary 11R04, 11R09.

Key words: Conjugate algebraic integers, disc with real center, diameter of algebraic integer, Rouché's theorem.

Reference to this article: A. Dubickas: Small discs containing conjugate algebraic integers. Ann. Acad. Sci. Fenn. Math. 45 (2020), 601-606.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4524