The Second Discriminant of a Univariate Polynomial

Abstract

We define the second discriminant $D_2$ of a univariate polynomial $f$ of degree greater than $2$ as the product of the linear forms $2\,r_k-r_i-r_j$ for all triples of roots $r_i, r_k, r_j$ of $f$ with $i<j$ and $j\neq k, k\neq i$. $D_2$ vanishes if and only if $f$ has at least one root which is equal to the average of two other roots. We show that $D_2$ can be expressed as the resultant of $f$ and a determinant formed with the derivatives of $f$, establishing a new relation between the roots and the coefficients of $f$. We prove several notable properties and present an application of $D_2$.