On associated discriminants for polynomials in one variable

In this note we introduce a family Σi, i = 0, . . . , n− 2 of discriminants in the space Pn of polynomials of degree n in one variable and study some of their algebraic and topological properties following [Ar]-[Va] and [GKZ]. The discriminant Σi consists of all polynomials p such that some nontrivial linear combination α0p+α1p′+ · · ·+αip (i) has a zero of multiplicity greater or equal i+2. In particular, using the inversion of differential operators with constant coefficients (which induces the nonlinear involution on Pn) we obtain the algebraic isomorphism of Σi and Σn−2−i for all i.