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Explicit determinantal representations of up to quintic bivariate polynomials. (English) Zbl 1401.65035
Summary: For bivariate polynomials of degree \(n\leq 5\) we give fast numerical constructions of determinantal representations with \(n\times n\) matrices. Unlike some other existing constructions, our approach returns matrices of the smallest possible size \(n\times n\) for all (not just generic) polynomials of degree \(n\) and does not require any symbolic computation. We can apply these linearizations to numerically compute the roots of a system of two bivariate polynomials by using numerical methods for the two-parameter eigenvalue problems.

MSC:
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65H04 Numerical computation of roots of polynomial equations
65F50 Computational methods for sparse matrices
13P15 Solving polynomial systems; resultants
14M12 Determinantal varieties
14Q05 Computational aspects of algebraic curves
Software:
BiRoots; Matlab
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