Roots of bivariate polynomial systems via determinantal representations

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 twoparameter eigenvalue problem. The first determinantal representation is suitable for polynomials with scalar or matrix coecients and consists of matrices with asymptotic order n2=4, where n is the degree of the polynomial. The second representation is useful for scalar polynomials and has asymptotic order n2=6. The resulting method to compute the roots of a system of two bivariate polynomials is very competitive with some existing methods for polynomials up to degree 10, as well as for polynomials with a small number of terms.