Computational Algebra and Number Theory

Computers have stretched the limits of what is possible in mathematics. More: they have given rise to new fields of mathematical study; the analysis of new and traditional algorithms, the creation of new paradigms for implementing computational methods, the viewing of old techniques from a concrete algorithmic vantage point, to name but a few. Computational Algebra and Number Theory lies at the lively intersection of computer science and mathematics. It highlights the surprising width and depth of the field through examples drawn from current activity, ranging from category theory, graph theory and combinatorics, to more classical computational areas, such as group theory and number theory. Many of the papers in the book provide a survey of their topic, as well as a description of present research. Throughout the variety of mathematical and computational fields represented, the emphasis is placed on the common principles and the methods employed. Audience: Students, experts, and those performing current research in any of the topics mentioned above.

Inhalt

Preface. 1: Calculating Growth Functions for Groups Using Automata; M. Brazil. 2: The Minimal Faithful Degree of a Finite Commutative Inverse Semigroup; S. Byleveld, D. Easdown. 3: Generalizations of the Todd-Coxeter Algorithm; S. A. Linton. 4: Computing Left Kan Extensions Using the Todd-Coxeter Procedure; M. Leeming, R. F. C. Walters. 5: Computing Finite Soluble Quotients; A. C. Niemeyer. 6: Computing Automorphism Groups of p-Groups; E. A. O'Brien. 7: The Art and Science of Computing in Large Groups; S. A. Linton. 8: Does the Set of Points of an Elliptic Curve Determine the Group? H. W. Lenstra Jr., J. Pila. 9: An Implementation of the Elliptic Curve Integer Factorization Method; W. Bosma, A. K. Lenstra. 10: Continued Fractions of Algebraic Numbers; E. Bombieri, A. J. van der Poorten. 11: Bounds for Class Numbers of Quadratic Orders; S. Loubotin, R. A. Mollin. 12: Short Representation of Quadratic Integers; J. Buchmann, C. Thiel, H. Williams. 13: A Density Conjecture for the Negative Pell Equation; P. Stevenhagen. 14: Computing Aurifeuillian Factors; R. P. Brent. 15: Computation of Cyclotomic Polynomials with MAGMA; W. Bosma. 16: On Some Characteristics of Uniformity of Distribution and their Applications; I. E. Shparlinski. 17: Recent Progress on Consistency Testing for Polynomial Systems; S. McCallum. 18: A New Generalisation of the Kummer Congruence; H. S. Gunaratne. 19: Series Expansions ofAlgebraic Functions; D. Griffiths. 20: Generation of Cocyclic Hadamard Matrices; K. J. Horadam, W. de Launey. 21: Large Cayley Graphs and Digraphs with Small Degree and Diameter; P. R. Hafner. 22: Hyperbolic Pyritohedra constructed from the Coxeter Group [4,3,5]; P. Lorimer.