Algebraic integers with small absolute size

Abstract

In this paper we show that for each k ∈ N there are innitely 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 → ∞. 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 Erdös and Turàn bound on the number of roots of a polynomial in a sector.

Keywords: Lattice, the shortest vector, algebraic number eld, absolute size, absolute normalised size