Elimination Practice: Software Tools and Applications (With CD-Rom)

✍ Scribed by Dongming Wang

• Publisher: World Scientific Publishing Company
• Year: 2004
• Tongue: English
• Leaves: 233
• Category: Library

✦ Synopsis

With a software library included, this book provides an elementary introduction to polynomial elimination in practice. The library Epsilon, implemented in Maple and Java, contains more than 70 well-documented functions for symbolic elimination and decomposition with polynomial systems and geometric reasoning. The book presents the functionality, implementation, and performance of Epsilon and demonstrates the usefulness of the elimination tool by a number of selected applications, together with many examples and illustrations. The reader will find Epsilon an efficient tool, applicable to a wide range of problems in science, engineering, and industry, and this book an accessible exposition and a valuable reference for elimination theory, methods, and practice.

✦ Table of Contents

Contents......Page 8
1.1 Solving Polynomial Equations......Page 11
1.2 Proving Geometric Theorems......Page 14
1.3 Locating Singular Points......Page 19
1.4 Decomposing Algebraic Curves......Page 23
2.1 What Is EPSILON and What It Does......Page 25
2.2 Design and Implementation......Page 26
2.3 EPSILON Functions......Page 28
2.4 Some Experiments......Page 34
2.5 Interface Availability and Installation......Page 36
3.1 Introduction......Page 38
3.2 CharSets Functions......Page 39
3.3 Differential Module......Page 45
3.4 Implementation Strategies......Page 50
3.5 Experiments and Remarks......Page 55
4.1 Introduction......Page 60
4.2 TriSys Functions......Page 61
4.3 Differential Triangular Series......Page 65
4.4 SiSys Functions......Page 67
4.5 Implementation Issues and Comparisons......Page 70
5.1 Specification of Geometric Theorems......Page 76
5.2 Basic Translations......Page 78
5.3 Proving Geometric Theorems Automatically......Page 80
5.4 Automated Generation of Diagrams and Documents......Page 82
5.5 Nondegeneracy Conditions and Miscellaneous Functions......Page 83
5.6 Implementation Strategies......Page 86
5.7 Experiments with Algebraic Provers......Page 90
6.1 Implementations of Triangular Sets......Page 92
6.2 Grobner Bases Packages......Page 95
6.3 Computing Resultants and Subresultants......Page 99
6.4 Miscellaneous Functions......Page 101
7.1 General Principles......Page 105
7.2 Solving Zero-dimensional Systems......Page 107
7.3 Solving Systems of Positive Dimension......Page 117
7.4 Solving Parametric Systems......Page 120
8.1 A Simple Algebraic Approach......Page 125
8.2 Proving Theorems via Zero Decomposition......Page 131
8.3 Illustration with Examples......Page 135
8.4 Discovering Geometric Theorems......Page 145
9.1 Deriving Locus Equations......Page 154
9.2 Implicitizing Parametric Objects......Page 166
9.3 Computing Offsets......Page 171
9.4 Blending Algebraic Surfaces......Page 176
9.5 Decomposing Algebraic Varieties......Page 182
10.1 Computation with Polynomial Ideals......Page 187
10.2 Factorization of Polynomials......Page 191
10.3 Qualitative Study of Differential Equations......Page 195
10.4 Automated Reasoning in Differential Geometry......Page 203
Appendix A Polynomial Systems: 50 Test Examples......Page 210
Appendix B Algebraic Factorization: 55 Examples......Page 212
References......Page 217
Index......Page 225