The second discriminant of a univariate polynomial

We define the second discriminant D_(2)of a univariate polynomial f of degree greater than 2 as the product of the linear forms 2r_(k)-r_(i)-r_(j)for all triples of roots r_(i),r_(k),r_(j)of f with i<j and j≠k,k≠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).