Solving Polynomial Equation Systems 1st Edition by Teo Mora 9780521811569 0521811562

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ISBN 10: 0521811562
ISBN 13: 9780521811569 
Author: Teo Mora

With the advent of computers, theoretical studies and solution methods for polynomial equations have 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, emphasizing 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.

Solving Polynomial Equation Systems 1st Table of contents:

Part one: The Kronecker – Duval Philosophy

1

Euclid

1.1 The Division Algorithm

1.2 Euclidean Algorithm

1.3 Bezout’s Identity and Extended Euclidean Algorithm

1.4 Roots of Polynomials

1.5 Factorization of Polynomials

1.6* Computing a ged

1.6.1* Coefficient explosion

1.6.2* Modular Algorithm

1.6.3* Hensel Lifting Algorithm

1.6.4* Heuristic ged

2

Intermezzo: Chinese Remainder Theorems

2.1 Chinese Remainder Theorems

2.2 Chinese Remainder Theorem for a Principal Ideal Domain

2.3 A Structure Theorem (1)

2.4 Nilpotents

2.5 Idempotents

2.6 A Structure Theorem (2)

2.7 Lagrange Formula

3

Cardano

3.1 A Tautology?

3.2 The Imaginary Number

3.3 An Impasse

3.4 A Tautology!

4 Intermezzo: Multiplicity of Roots

4.1 Characteristic of a Field

4.2 Finite Fields

4.3 Derivatives

4.4 Multiplicity

4.5 Separability

4.6 Perfect Fields

4.7 Squarefree Decomposition

Kronecker I: Kronecker’s Philosophy

5.1 Quotients of Polynomial Rings

5.2 The Invention of the Roots

5.3 Transcendental and Algebraic Field Extensions

5.4 Finite Algebraic Extensions

5.5 Splitting Fields

6 Intermezzo: Sylvester

6.1 Gauss Lemma

6.2 Symmetric Functions

6.3* Newton’s Theorem

6.4 The Method of Indeterminate Coefficients

6.5 Discriminant

6.6 Results

6.7 Resultants and Roots

Galois I: Finite Fields

7.1 Galois Fields

7.2 Roots of Polynomials over Finite Fields

7.3 Distinct Degree Factorization

7.4 Roots of Unity and Primitive Roots

7.5 Representation and Arithmetics of Finite Fields

7.6* Cyclotomic Polynomials

7.7* Cycles, Roots and Idempotents

7.8 Deterministic Polynomial-time Primality Test

Kronecker II: Kronecker’s Model

8.1 Kronecker’s Philosophy

8.2 Explicitly Given Fields

8.3 Representation and Arithmetics

8.3.1 Representation

8.3.2 Vector space arithmetics

8.3.3 Canonical representation

8.3.4 Multiplication

8.3.5 Inverse and division

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Tags: Teo Mora, Solving, Polynomial