Minimal determinantal representations of bivariate polynomials

TL;DR

This paper presents a simple, fast numerical algorithm to construct minimal size matrix representations of square-free bivariate polynomials, enabling efficient solutions to polynomial systems via eigenvalue methods.

Contribution

It introduces the first straightforward numerical method for minimal determinantal representations of bivariate polynomials of any degree.

Findings

Provides a numerical algorithm for minimal matrix representations

Enables efficient solving of bivariate polynomial systems

Speeds up computations in polynomial eigenvalue problems

Abstract

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.