Explicit determinantal representations of up to quintic bivariate polynomials

Abstract

For bivariate polynomials of degree we give fast numerical constructions of determinantal representations with matrices. Unlike some other existing constructions, our approach returns matrices of the smallest possible size for all (not just generic) 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 the two-parameter eigenvalue problems.