The size of algebraic integers with many real conjugates

Artūras Dubickas

Abstract


In this paper we show that the relative normalised size with respect to a number field K of an algebraic integer α ≠ -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√(n/log n) and shows that the minimum m(K) over the relative normalised size of nonzero algebraic integers α in such a field K is equal to 1 which is attained at α = ±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.

Keywords


Algebraic number field, relative size, relative normalised size, Mahler measure, Schur-Siegel-Smyth trace problem

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ISSN: 1855-3974

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