  We give two determinantal representations for a bivariate polynomial. They
may be used to compute the zeros of a system of two of these polynomials via
the eigenvalues of a two-parameter eigenvalue problem. The first determinantal
representation is suitable for polynomials with scalar or matrix coefficients,
and consists of matrices with asymptotic order $n^2/4$, where $n$ is the degree
of the polynomial. The second representation is useful for scalar polynomials
and has asymptotic order $n^2/6$. The resulting method to compute the roots of
a system of two bivariate polynomials is competitive with some existing methods
for polynomials up to degree 10, as well as for polynomials with a small number
of terms.
