Solving Polynomial Equation Systems I: The Kronecker-Duval Philosophy: v. 1 (Encyclopedia of Mathematics and its Applications)
Cambridge University Press, 3/27/2003
EAN 9780521811545, ISBN10: 0521811546
Hardcover, 438 pages, 24.2 x 16.2 x 2.9 cm
Language: English
Polynomial equations have been long studied, both theoretically and with a view to solving them. Until recently, manual computation was the only solution method and the theory was developed to accommodate it. With the advent of computers, the situation changed dramatically. Many classical results can be more usefully recast within a different framework which in turn lends itself to further theoretical development tuned to computation. This first book in a trilogy is devoted to the new approach. It is a handbook covering the classical theory of finding roots of a univariate polynomial, emphasising computational aspects, especially the representation and manipulation of algebraic numbers, enlarged by more recent representations like the Duval Model and the Thom Codification. Mora aims to show that solving a polynomial equation really means finding algorithms that help one manipulate roots rather than simply computing them; to that end he also surveys algorithms for factorizing univariate polynomials.
Preface
Part I. The Kronecker-Duval Philosophy
1. Euclid
2. Intermezzo
Chinese remainder theorems
3. Cardano
4. Intermezzo
multiplicity of roots
5. Kronecker I
Kronecker's philosophy
6. Intermezzo
Sylvester
7. Galois I
finite fields
8. Kronecker II
Kronecker's model
9. Steinitz
10. Lagrange
11. Duval
12. Gauss
13. Sturm
14. Galois II
Part II. Factorization
15. Ouverture
16. Kronecker III
factorization
17. Berlekamp
18. Zassenhaus
19. Fermeture
Bibliography
Index.