Root Separation for Reducible Monic Quartics

Abstract

We study root separation for reducible monic integer polynomials of degree four. If $\text{H}(P)$ is the height and $\mathrm{sep}(P)$ the minimal distance between two distinct roots of a separable integer polynomial $P(x)$, and $\mathrm{sep}(P)=\mathrm{H}(P{)}^{-e(P)}$, we show that $\mathrm{limsup}e(P)=2$, where limsup is taken over all reducible monic integer polynomials $P(x)$ of degree $4$.