On real and p-adic Bezoutians

We study the quadratic form induced by the Bezoutian of two polynomials p and q, considering four problems. First, over R, in the separable case we count the number of configurations of real roots of p and q for which the Bezoutian has a fixed signature. Second, over Qp we develop a formula for the Hasse invariant of the Bezoutian. Third, we formulate a conjecture for the behavior of the Bezoutian in the non separable case, and offer a proof over R. We wrote a Pari code to test it over Qp and Q and found no counterexamples. Fourth, we state and prove a theorem that we hope will help prove the conjecture in the near future.