The Second Discriminant of a Univariate Polynomial

TL;DR

This paper introduces the second discriminant $D_2$ of a univariate polynomial, linking roots and coefficients through a new resultant expression, and explores its properties and applications.

Contribution

It defines the second discriminant $D_2$, relates it to resultants and derivatives, and demonstrates its properties and potential applications.

Findings

$D_2$ vanishes when a root equals the average of two others.

$D_2$ can be expressed as a resultant involving $f$ and its derivatives.

The paper establishes new relations between roots and coefficients.

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 $D_2$.