Zeros of a polynomial, p(x)=0, are often determined by computing the eigenvalues of a companion matrix: a matrix A which satisfies det(A-xI)=p(x).
In this talk we consider polynomial systems, in particular in 2 variables: p(x,y)=0, q(x,y)=0.
We look for a determinantal representation for such a bivariate polynomial: matrices A, B, C such that det(A-xB-yC)=p(x,y).
This means that we can compute the zeros of the system by solving a 2-parameter eigenvalue problem.
This approach, which already goes back to a theorem by Dixon in 1902, leads to fast solution methods, as well as a multitude of interesting open research questions.
This is mainly joint work with Bor Plestenjak (Ljubljana), and additionally several colleagues in algebra, among which Ada Boralevi (Torino).