Abstract
The last chapter contains applications of linear algebra techniques to the problem of solving zero-dimensional polynomial systems over a field \(K\). If we are mainly interested in \(K\)-rational solutions, we can use an algorithm to find all 1-dimensional joint eigenspaces of the multiplication family: the eigenvalue method, or the eigenvector method. In general, if the base field \(K\) is finite, we can still obtain good results by using the techniques of cloning, univariate representations, and recursion. However, over the rational numbers, the technique of cloning leads to the hard problem of splitting polynomials over number fields, and it is here that the applications, as well as the book, come to their natural conclusion.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Auzinger, W., Stetter, H.J.: An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations. In: Agarwal, R.G., Chow, Y.M., Wilson, S.J. (eds.) Int. Conf. on Numerical Mathematics (Singapore 1988), vol. 86, pp. 11–30. Birkhäuser, Basel (1988)
Cox, D.A.: Solving equations via algebras. In: Dickenstein, A., Emiris, I. (eds.) Solving Polynomial Equations. Algorithms and Comp. in Math., vol. 14, pp. 63–124. Springer, Berlin (2005)
Ekedahl, T., Laksov, D.: Splitting algebras, symmetric functions, and Galois theory. J. Algebra Appl. 4, 59–75 (2005)
Jacobson, N.: Basic Algebra I, 2nd edn. Dover Publications, Mineola (2009)
Kreuzer, M., Robbiano, L.: Computational Commutative Algebra 1. Springer, Heidelberg (2000)
Kreuzer, M., Robbiano, L.: Computational Commutative Algebra 2. Springer, Heidelberg (2005)
Lazard, D.: Résolution des systèmes d’equations algébriques. Theor. Comput. Sci. 15, 77–110 (1981)
Matiyasevich, Y.: Hilbert’s Tenth Problem. MIT Press, Cambridge (1993)
Milne, J.S.: Fields and Galois theory. On-line lecture notes, Ver. 4.50 (2014). Available at http://www.jmilne.org/math/CourseNotes/FT.pdf
Möller, H.-M., Tenberg, R.: Multivariate polynomial system solving using intersections of eigenspaces. J. Symb. Comput. 32, 513–531 (2001)
Robbiano, L., Abbott, J. (eds.): Approximate Commutative Algebra. Springer, Vienna (2009)
Sommese, A., Wampler, C.W. II: The Numerical Solution of Systems of Polynomial Equations Arising in Engineering and Science. World Scientific Publ., Singapore (2005)
Stetter, H.: Numerical Polynomial Algebra. SIAM, Philadelphia (2004)
Author information
Authors and Affiliations
6.1 Electronic Supplementary Material
Below is the link to the electronic supplementary material.
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Kreuzer, M., Robbiano, L. (2016). Solving Zero-Dimensional Polynomial Systems. In: Computational Linear and Commutative Algebra. Springer, Cham. https://doi.org/10.1007/978-3-319-43601-2_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-43601-2_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-43599-2
Online ISBN: 978-3-319-43601-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)