Notes on symmetrization by Bezoutiant

Let p and q be a monic hyperbolic polynomials such that q separates p and let H be the Bezoutian (form) of p and q. Then H is nonnegative definite and symmetrizes the Sylvester matrix associated with p. When q = p' this fact is observed by E. Jannelli. We give a simple proof of this fact and at the same time prove that the family of Bezoutian of Nuij approximation of p and p' gives quasi-symmetrizers introduced by S. Spagnolo. A relation connecting H with the symmetrizer which was used by J. Leray for strictly hyperbolic polynomials is also given.