  Classically, the Bezout matrix or simply Bezoutian of two polynomials is used
to locate the roots of the polynomial and, in particular, test for stability.
In this paper, we develop the theory of Bezoutians on real Riemann surfaces of
dividing type. The main result connects the signature of the Bezoutian of two
real meromorphic functions to the topological data of their quotient, which can
be seen as the generalization of the classical Cauchy index. As an application,
we propose a method to count the number of zeroes of a polynomial in a
quadrature domain using the inertia of the Bezoutian. We provide examples of
our method in the case of simply connected quadrature domains.
