Conjugate algebraic numbers close to a symmetric set

Abstract

A new proof is presented for the Motzkin theorem saying that if a set consists of d− 1 complex points and is symmetric relative to the real axis, then there exists a monic, irreducible, and integral polynomial of degree d whose roots are as close to each of these d−1 points as we wish. Unlike the earlier proofs, the new proof is efficient, i.e., it gives both an explicit construction of the polynomial in question and the location of its dth root. §