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.
Keywords: System of bivariate polynomial equations, determinantal representation, two-parameter eigenvalue problem, polynomial multiparameter eigenvalue problem

Original language | English |
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Publisher | s.n. |
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Number of pages | 22 |
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Publication status | Published - 2015 |
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Name | arXiv |
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Volume | 1506.02291 [math.NA] |
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