Root separation for reducible monic quartics
For a polynomial \(P\) with integer coefficients one defines its height \(H(P)\), which is the maiximal modulus of its coefficients, and \(\text{sep}(P)\), the minimal distance between two distinct roots of it, and one puts \(\text{sep}(P)=H(P)^{-e(P)}\). There are still many open questions about the behaviour of \(e(P)\). Here the Authors prove that \(\limsup e(P)=2\) when \(P\) runs over all reducible monic quartic integer polynomials. This is the first case where an exact limit value of \(e(P)\) is proved for quartic polynomials. The proof is constructive.
scientific article; zbMATH DE number 6005070
scientific article