Roots of bivariate polynomial systems via determinantal representations

Bor Plestenjak; Michiel E. Hochstenbach

doi:10.1137/140983847

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arXiv · 1506.02291

Roots of bivariate polynomial systems via determinantal representations

Bor Plestenjak·Michiel E. Hochstenbach

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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.

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Bor Plestenjak, Michiel E. Hochstenbach. 2015-06-07. Roots of bivariate polynomial systems via determinantal representations. https://doi.org/10.1137/140983847

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