Roots of bivariate polynomial systems via determinantal representations
Provides a new computational approach for solving bivariate polynomial systems, benefiting researchers in algebraic geometry and numerical analysis, though the improvement is incremental for general dense polynomials.
The paper introduces two determinantal representations for bivariate polynomials that enable root computation via two-parameter eigenvalue problems, achieving asymptotic matrix orders of n^2/4 and n^2/6. The method is competitive with existing approaches for polynomials up to degree 10 and for sparse polynomials.
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.