When should a polynomial’s root nearest to a real number be real itself?

Abstract:

The conditions are studied under which the root of an integer polynomial nearest to a given real number $y$ is real. It is proved that if a polynomial $P \in \mathbb {Z}[x]$ of degree $d \geq 2$ satisfies $|P(y)| \ll 1/M(P)^{2d-3}$ for some real number $y$, where the implied constant depends on $d$ only, then the root of $P$ nearest to $y$ must be real. It is also shown that the exponent $2d-3$ is best possible for $d=2,3$ and that it cannot be replaced by a number smaller than $(2d-3)d/(2d-2)$ for each $d \geq 4$.