In the years 1994, 1995, two EIDMA mini courses on Computer Algebra were given at the Eindhoven University of Technology by, apart from ourselves, various invited lecturers. (EIDMA is the Research School 'Euler Institute for Discrete Mathematics and its Applications'.) The idea of the courses was to acquaint young mathematicians with algorithms and software for mathemat ical research and to enable them to incorporate algorithms in their research. A collection of lecture notes was used at these courses. When discussing these courses in comparison with other kinds of courses one might give in a week's time, Joachim Neubüser referred to our courses as 'tapas'. This denomination underlined that the courses consisted of appe tizers for various parts of algorithmic algebra; indeed, we covered such spicy topics as the link between Gröbner bases and integer programming, and the detection of algebraic solutions to differential equations. As a collection, the not es turned out to have some appeal of their own, which is the main reason why the idea came up of transforming them into book form. We feIt however, that the book should be distinguishable from a standard text book on computer algebra in that it retains its appetizing flavour by presenting a variety of topics at an accessible level with a view to recent developments.
Arvustused
"Tapas are hors d'oeuvres or snacks in the Spanish cuisine. And really, the topics treated by various authors can be considered a great variety of appetizers, which reach from mild to hot ones. ... All in all a very recommendable book which by the way has the advantage that one can get in it easily at many places and may pick out for oneself the most interesting themes."
Reviewed by G.Kowol, Monatshefte der Mathematik 2003, Vol. 138, Issue 1
Grobner Bases, an Introduction 1(33) Arjeh M. Cohen Introduction 1(3) Monomials 4(3) The Buchberger Algorithm 7(8) Standard Monomials 15(2) Solving Polynomial Equations 17(6) Effectiveness of Polynomial Rings 23(11) Symbolic Recipes for Polynomial System Solving 34(32) Laureano Gonzalez-Vega Fabrice Rouillier Marie-Francoise Roy Introduction 34(1) General Systems of Equations 35(11) Algebraic Preliminaries 35(5) First Receipes for Polynomial System Solving 40(6) Linear Algebra, Traces, and Polynomial Systems 46(12) Eigenvalues and Polynomial Systems 46(2) Counting Solutions and Removing Multiplicities 48(3) Rational Univariate Representation 51(7) As Many Equations as Variables 58(8) Generalities on Complete Intersection Polynomial Systems 58(2) Receipes for Polynomial System Solving When the Number of Equations Equals the Number of Unknowns 60(1) Grobner Bases and Numerical Approximations 61(5) Lattice Reduction 66(12) Frits Beukers Introduction 66(1) Lattices 66(2) Lattice Reduction in Dimension 2 68(2) Lattice Reduction in Any Dimension 70(3) Implementations of the LLL--Algorithm 73(2) Small Linear Forms 75(3) Factorisation of Polynomials 78(13) Frits Beukers Introduction 78(1) Berlekamps Algorithm 78(3) Additional Algorithms 81(1) Polynomials with Integer Coefficients 82(2) Factorisation of Polynomials with Integer Coefficients, I 84(2) Factorisation of Polynomials with Integer Coefficients, II 86(2) Factorisation in K(X), K Algebraic Number Field 88(3) Computations in Associative and Lie Algebras 91(30) Gabor Ivanyos Lajos Ronyai Introduction 91(1) Basic Definitions and Structure Theorems 91(6) Computing the Radical 97(8) Applications to Lie Algebras 105(3) Finding the Simple Components of Semisimple Algebras 108(4) Zero Divisors in Finite Algebras 112(9) Symbolic Recipes for Real Solutions 121(47) Laureano Gonzalez-Vega Fabrice Rouillier Marie-Francoise Roy Guadalupe Trujillo Introduction 121(1) Real Root Counting: The Univariate Case 122(12) Computing the Number of Real Roots 123(1) Sylvester Sequence 124(4) Sylvester--Habicht Sequence 128(5) Some Recipes for Counting Real Roots 133(1) Real Root Counting: The Multivariate Case 134(5) The Sign Determination Scheme 139(3) Real Algebraic Numbers and Thom Codes 142(3) Quantifier Elimination 145(4) Appendix: Properties of the Polynomials in the Sylvester--Habicht Sequence 149(19) Definition and the Structure Theorem 150(4) Proof of the Structure Theorem 154(7) Sylvester--habicht Sequences and Cauchy Index 161(7) Grobner Bases and Integer Programming 168(16) Gunter M. Ziegler Introduction 168(1) What is Integer Programming? 168(1) A Buchberger Algorithm for Integer Programming 169(6) A Geometric Buchberger Algorithm 175(2) A Variant of the Buchberger Algorithm 177(3) Exercises 180(4) Working with Finite Groups 184(24) Hans Cuypers Leonard H. Soicher Hans Sterk Introduction 184(1) Permutation Groups 185(12) The Setting 185(2) Computing Orbits and Stabilizers 187(5) Computing Bases and Strong Generating Sets 192(2) Generators for Some Subgroups 194(3) Coset Enumeration 197(11) Introduction 197(1) Todd--Coxeter Coset Enumeration 197(11) Symbolic Analysis of Differential Equations 208(29) Marius van der Put Introduction 208(1) The Equation y = ƒ with ƒ ∈ C(x) 208(4) The Algorithm 210(2) The Equation y = ƒy with ƒ ∈ C(x)* 212(1) Rational Solutions of an Equation of Order n 213(3) Some Differential Galois Theory 216(6) Picard--Vessiot Theory 218(4) Order Two Equations Over C(x) 222(3) The Local Differential Galois Group 225(4) The Equation y = ry with r ∈ C(x), r ≠ 0 229(2) The Equation y = ry with r ∈ C(x, x--1) 231(6) Grobner Bases for Codes 237(23) Mario de Boer Ruud Pellikaan Introduction 237(1) Basic Facts from Coding Theory 237(3) Hamming Distance 238(1) Linear Codes 238(1) Weight Distribution 239(1) Automorphisms and Isometries of Codes 239(1) Determining the Minimum Distance 240(7) Exhaustive Search 240(1) Linear Algebra 241(1) Finite Geometry 242(1) Arrangements of Hyperplanes 243(2) Algebra 245(2) Cyclic Codes 247(4) The Mattson--Solomon Polynomial 248(2) Codewords of Minimal Weight 250(1) Codes from Varieties 251(9) Order and Weight Functions 252(2) A Bound on the Minimum Distance 254(6) Grobner Bases for Decoding 260(16) Mario de Boer Ruud Pellikaan Introduction 260(1) Decoding 260(2) Decoding Cyclic Codes with Grobner Bases 262(5) One--Step Decoding of Cyclic Codes 265(2) The Key Equation 267(4) The Algorithms of Euclid and Sugiyama 269(1) The Algorithm of Berlekamp--Massey 270(1) Grobner Bases and Arbitrary Linear Codes 271(5) Automatic Geometry Theorem Proving 276(21) Tomas Recio Hans Sterk, M. Pilar Velez Introduction 276(1) Approaches to Automatic Geometry Theorem Proving 277(1) Algebraic Geometry Formulation 277(5) Searching for Conditions 282(9) Searching for Extra Hypotheses 291(6) The Birkhoff Interpolation Problem 297(8) Maria-Jose Gonzalez-Lopez Laureano Gonzalez-Vega Introduction 297(1) Poised Matrices 297(4) Examples 301(2) Conclusions 303(2) The Inverse Kinematics Problem in Robotics 305(6) Maria-Jose Gonzalez-Lopex Laureano Gonzalez-Vega Introduction 305(1) The ROMIN Manipulator 305(2) The Elbow Manipulator 307(4) Quaternion Algebras 311(4) Gabor Ivanyos Lajos Ronyai Introduction 311(1) Four Dimensional Simple Algebras 311(1) Quaternion Algebras and Quardratic Forms 312(3) Explorations with the Icosahedral Group 315(8) Arjeh M. Cohen Hans Cuypers Remko Riebeek Introduction 315(1) Three--Dimensional Representations for W (H3) 316(2) Coset Enumeration 318(1) The Permutation Representation of W on the Cosets of I 318(5) The Small Mathieu Groups 323(15) Hans Cuypers Leonard H. Soicher Hans Sterk Introduction 323(1) The Affine Plane of Order 3 324(2) A 3-(10, 4, 1) Design and the Mathieu Group M10 326(2) The Groups M11 and M12 328(1) Two 2-Transitive Subgroups of M12 329(3) Graphs Which Are Locally the Incidence Graph of the Biplane 332(6) The Golay Codes 338(11) Mario de Boer Ruud Pellikaan Introduction 338(1) Minimal Weight Codewords of G11 338(3) Decoding of G23 with Grobner Bases 341(2) One--Step Decoding of G23 343(1) The Key Equation for G23 344(2) Exercises 346(3) Index 349
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