Fundamental Problems of Algorithmic Algebra [Hardcover]

Popular computer algebra systems such as Maple, Macsyma, Mathematica, and REDUCE are now basic tools on most computers. Efficient algorithms for various algebraic operations underlie all these systems. Computer algebra, or algorithmic algebra, studies these algorithms and their properties and represents a rich intersection of theoretical computer science with classical mathematics.

Fundamental Problems of Algorithmic Algebraprovides a systematic and focused treatment of a collection of core problemsthe computational equivalents of the classical Fundamental Problem of Algebra and its derivatives. Topics covered include the GCD, subresultants, modular techniques, the fundamental theorem of algebra, roots of polynomials, Sturm theory, Gaussian lattice reduction, lattices and polynomial factorization, linear systems, elimination theory, Grobner bases, and more.
Features
?? Presents algorithmic ideas in pseudo-code based on mathematical concepts and can be used with any computer mathematics system
?? Emphasizes the algorithmic aspects of problems without sacrificing mathematical rigor
?? Aims to be self-contained in its mathematical development
?? Ideal for a first course in algorithmic or computer algebra for advanced undergraduates or beginning graduate students

O INTRODUCTION
1. Fundamental Problem of Algebra
2. Fundamental Problem of Classical Algebraic Geometry
3. Fundamental Problem of Ideal Theory
4. Representation and Size
5. Computational Models
6. Asymptotic Notations
7. Complexity of Multiplication
8. On Bit versus Algebraic Complexity
9. Miscellany
10. Computer Algebra Systems
I ARITHMETIC
1. The Discrete Fourier Transform
2. Polynomial Multiplication
3. Modular FFT
4. Fast Integer Multiplication
5. Matrix Multiplication
II THE GCD
1. Unique Factorization Domain
2. Euclid's Algorithm
3. Euclideanlƒ<