Symbolic-numeric methods for solving polynomial equations and applications View Full Text


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Chapter Info

DATE

2005-01-01

AUTHORS

Mohamed Elkadi , Bernard Mourrain

ABSTRACT

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. More... »

PAGES

125-168

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/3-540-27357-3_3

DOI

http://dx.doi.org/10.1007/3-540-27357-3_3

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1032090794


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