Recall that given two complex polynomials f and g, the Bezout matrix B(f,g)=(bij) of f and g is defined by f(t)g(s)−f(s)g(t)t−s=∑i,jbijtisj. It is a classical result of Hermite that given a polynomial p(z), the Bezout matrix of p(z) and pτ(z)=p(z¯) is skew self-adjoint. The number of common zeroes of p and pτ is the dimension of the kernel of −iB(p,pτ). Additionally, p has n+ roots in the upper half-plane and n− in the lower half-plane. Here n−+ and n− stand for the number of positive and negative eigenvalues of −iB(p,pτ), respectively. In this talk, I will describe how one can extend the notion of the Bezout matrix to a pair of meromorphic functions on a compact Riemann surface. If the surface is real and dividing the matrix −iB(f,fτ) is J-selfadjoint, for a certain signature matrix J. We the study the signature of the Bezoutian and obtain an extension of Hermite’s theorem to quadrature domains.