Simple determinantal representations of up to quintic bivariate polynomials
This offers a practical numerical tool for solving bivariate polynomial systems, but the restriction to degree ≤5 and the incremental nature limit its impact.
The paper provides fast numerical constructions of determinantal representations for bivariate polynomials up to degree 5 using n×n matrices, enabling numerical root-finding for systems of two bivariate polynomials via two-parameter eigenvalue methods.
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.