The Second Discriminant of a Univariate Polynomial

Dongming Wang; Jing Yang

paper

The Second Discriminant of a Univariate Polynomial

arXiv:1609.00840

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 $D_{2}$ .

24 pages, 1 table