Solving Polynomial Equation Systems I
The Kronecker-Duval Philosophy

Computational algebra; computational number theory; commutative algebra; handbook; reference; algorithmic; modern.

Teo Mora (Author)

9780521811545, Cambridge University Press

Hardback, published 27 March 2003

438 pages
24.2 x 16.2 x 2.9 cm, 0.747 kg

'The book [is] a thorough success …'. Zentralblatt MATH

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.

Subject Areas: Number theory [PBH], Algebra [PBF]