Algebraic integers with small absolute size

Abstract

In this paper we show that for each k ∈ ℕ there are infinitely many algebraic integers with norm k and absolute normalized size smaller than 1. We also show that the lower bound (n + s log 2)/2 on the square of the absolute size ‖α‖ of an algebraic integer α of degree n with exactly s real conjugates over ℚ is best possible for each even s > 2. For this, for each pair s, k ∈ ℕ, where s is even, we construct algebraic integers α with exactly s real conjugates and norm of modulus k satisfying deg α = n and ‖α‖2 = (n + s log 2)/2 + log k + O(n−1) as n → ∞ to. Finally, using the third smallest Pisot number θ3, which is the root of the polynomial x5–x4–x3+x2–1, we construct algebraic integers α of degree n that have exactly one real conjugate and satisfy ‖α‖2 ≤ n/2 + 0.346981 … (which is quite close to the above lower bound (n + log 2)/2 = n/2 + 0.346573 … for s = 1). In the proofs we use some irreducibility theorems for lacunary polynomials and the Erdős and Turán bound on the number of roots of a polynomial in a sector.