  For bivariate polynomials of degree $n\le 5$ we give fast numerical
constructions of determinantal representations with $n\times n$ matrices.
Unlike some other available constructions, our approach returns matrices of the
smallest possible size $n\times n$ for all polynomials of degree $n$ and does
not require any symbolic computation. We can apply these linearizations to
numerically compute the roots of a system of two bivariate polynomials by using
numerical methods for two-parameter eigenvalue problems.
