Summary
This tutorial gives an introductory presentation of algebraic and geometric methods to solve a polynomial system ƒ1 = ⋯ = ƒm = 0. The algebraic methods are based on the study of the quotient algebra A of the polynomial ring modulo the ideal I = (ƒ1,..., ƒm). We show how to deduce the geometry of solutions from the structure of A and in particular, how solving polynomial equations reduces to eigenvalue and eigenvector computations of multiplication operators in A. We give two approaches for computing the normal form of elements in A, used to obtain a representation of multiplication operators. We also present the duality theory and its application to solving systems of algebraic equations. The geometric methods are based on projection operations which are closely related to resultant theory. We present different constructions of resultants and different methods for solving systems of polynomial equations based on these formulations. Finally, we illustrate these tools on problems coming from applications in computer-aided geometric design, computer vision, robotics, computational biology and signal processing.
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© 2005 Springer-Verlag Berlin Heidelberg
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Elkadi, M., Mourrain, B. (2005). Symbolic-numeric methods for solving polynomial equations and applications. In: Bronstein, M., et al. Solving Polynomial Equations. Algorithms and Computation in Mathematics, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-27357-3_3
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DOI: https://doi.org/10.1007/3-540-27357-3_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-24326-7
Online ISBN: 978-3-540-27357-8
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