Notes on symmetrization by Bezoutiant

Abstract

Let $p$ be a monic hyperbolic polynomial and let $H$ be the Bezoutian matrix of $p$ and $p^{\prime }$. Then $H$ symmetrizes the Sylvester matrix associated with $p$. This fact is observed by E.Jannelli. We give a simple proof of this fact and at the same time show that the family of Bezoutian matrices of Nuij approximation of $p$ gives quasi-symmetrizers introduced by S.Spagnolo. A relation connecting $H$with the symmetrizer which was used by J.Leray for strictly hyperbolic polynomial is given.