Minimal determinantal representations of bivariate polynomials
For researchers in computational algebra and polynomial systems, this provides the first simple numerical construction of minimal determinantal representations, enabling efficient numerical solution of bivariate polynomial systems.
The paper presents a fast numerical algorithm to construct minimal n×n matrices A, B, C such that det(A+xB+yC)=p(x,y) for any square-free bivariate polynomial p of degree n, solving a long-standing problem. The representation speeds up solving systems of two bivariate polynomials via two-parameter eigenvalue problems.
For a square-free bivariate polynomial $p$ of degree $n$ we introduce a simple and fast numerical algorithm for the construction of $n\times n$ matrices $A$, $B$, and $C$ such that $\det(A+xB+yC)=p(x,y)$. This is the minimal size needed to represent a bivariate polynomial of degree $n$. Combined with a square-free factorization one can now compute $n \times n$ matrices for any bivariate polynomial of degree $n$. The existence of such symmetric matrices was established by Dixon in 1902, but, up to now, no simple numerical construction has been found, even if the matrices can be nonsymmetric. Such representations may be used to efficiently numerically solve a system of two bivariate polynomials of small degree via the eigenvalues of a two-parameter eigenvalue problem. The new representation speeds up the computation considerably.