Simple determinantal representations of up to quintic bivariate polynomials

TL;DR

This paper presents a fast, symbolic-free method for constructing minimal-size determinantal representations of bivariate polynomials up to degree five, enabling efficient numerical root finding.

Contribution

It introduces a new numerical approach for minimal determinantal representations of bivariate polynomials of degree up to five, avoiding symbolic computation.

Findings

Constructs $n\times n$ matrices for polynomials of degree $n\le 5$

Enables numerical root computation via two-parameter eigenvalue problems

Provides faster, symbolic-free linearizations for bivariate polynomials

Abstract

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.