The second discriminant of a univariate polynomial

We define the second discriminantD(2)of a univariate polynomialfof degree greater than 2 as the product of the linear forms 2r(k)-r(i)-r(j)for all triples of rootsr(i),r(k),r(j)offwithi<jandj equivalent to k, k equivalent to i.D(2)vanishes if and only iffhas at least one root which is equal to the average of two other roots. We show thatD(2)can be expressed as the resultant offand a determinant formed with the derivatives off, establishing a new relation between the roots and the coefficients off. We prove several notable properties and present an application ofD(2).