Roots of bivariate polynomial systems via determinantal representations

TL;DR

This paper introduces two new determinantal representations for bivariate polynomials, enabling efficient root computation via eigenvalue problems, especially for polynomials up to degree 10 or with few terms.

Contribution

It presents novel determinantal representations suitable for different polynomial types, improving root-finding efficiency for bivariate systems.

Findings

Method is competitive with existing approaches for degree up to 10.

Effective for polynomials with few terms.

Two representations optimize for scalar and matrix coefficients.

Abstract

We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two-parameter eigenvalue problem. The first determinantal representation is suitable for polynomials with scalar or matrix coefficients, and consists of matrices with asymptotic order $n^2/4$, where $n$ is the degree of the polynomial. The second representation is useful for scalar polynomials and has asymptotic order $n^2/6$. The resulting method to compute the roots of a system of two bivariate polynomials is competitive with some existing methods for polynomials up to degree 10, as well as for polynomials with a small number of terms.