Roots of bivariate polynomial systems via determinantal representations

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.

22 pages, 9 figures

Cited by in corpus (5)

• Solving singular generalized eigenvalue problems by a rank-completing perturbation
• On the singular two-parameter eigenvalue problem II
• Minimal determinantal representations of bivariate polynomials
• Simple determinantal representations of up to quintic bivariate polynomials
• Randomized methods for computing joint eigenvalues, with applications to multiparameter eigenvalue problems and root finding