A First Course in Computational Algebraic Geometry is designed for young students with some background in algebra who wish to perform their first experiments in computational geometry. Originating from a course taught at the African Institute for Mathematical Sciences, the book gives a compact presentation of the basic theory, with particular emphasis on explicit computational examples using the freely available computer algebra system, Singular. Readers will quickly gain the confidence to begin performing their own experiments.
This quick guide is designed for young students with some background in algebra who wish to perform their first experiments in computational geometry. It provides a compact presentation of the basic theory, with particular emphasis on explicit computational examples using the computer algebra system Singular.
Arvustused
'Decker and Pfister exposit the very rudiments of algebraic geometry in tandem with the workings of the program Singular. The student who masters this content could proceed in many directions Recommended. Upper-division undergraduates through researchers/faculty.' D. V. Feldman, Choice
Muu info
A quick guide to computing in algebraic geometry with many explicit computational examples introducing the computer algebra system Singular.
| Preface |
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vii | |
| Prologue: General Remarks on Computer Algebra Systems |
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1 | (10) |
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1 The Geometry-Algebra Dictionary |
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11 | (49) |
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1.1 Affine Algebraic Geometry |
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11 | (38) |
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1.1.1 Ideals in Polynomial Rings |
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11 | (3) |
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1.1.2 Affine Algebraic Sets |
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14 | (6) |
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1.1.3 Hilbert's Nullstellensatz |
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20 | (3) |
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1.1.4 Irreducible Algebraic Sets |
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23 | (2) |
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1.1.5 Removing Algebraic Sets |
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25 | (4) |
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29 | (3) |
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1.1.7 The Geometry of Elimination |
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32 | (5) |
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1.1.8 Noether Normalization and Dimension |
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37 | (8) |
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45 | (4) |
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1.2 Projective Algebraic Geometry |
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49 | (11) |
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1.2.1 The Projective Space |
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49 | (3) |
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1.2.2 Projective Algebraic Sets |
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52 | (2) |
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1.2.3 Affine Charts and the Projective Closure |
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54 | (3) |
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1.2.4 The Hilbert Polynomial |
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57 | (3) |
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60 | (35) |
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2.1 Standard Bases and Singular |
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60 | (15) |
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75 | (9) |
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75 | (1) |
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75 | (2) |
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77 | (1) |
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2.2.4 Ideal Intersections |
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78 | (1) |
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79 | (1) |
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2.2.6 Kernel of a Ring Map |
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79 | (1) |
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2.2.7 Integrality Criterion |
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80 | (2) |
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2.2.8 Noether Normalization |
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82 | (1) |
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2.2.9 Subalgebra Membership |
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83 | (1) |
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83 | (1) |
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2.3 Dimension and the Hilbert Function |
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84 | (6) |
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2.4 Primary Decomposition and Radicals |
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90 | (4) |
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2.5 Buchberger's Algorithm and Field Extensions |
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94 | (1) |
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95 | (6) |
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4 A Problem in Group Theory Solved by Computer Algebra |
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101 | (11) |
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4.1 Finite Groups and Thompson's Theorem |
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101 | (3) |
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4.2 Characterization of Finite Solvable Groups |
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104 | (8) |
| Bibliography |
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112 | (3) |
| Index |
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115 | |
Wolfram Decker heads Singular's core development team within the Department of Mathematics at the University of Kaiserslautern, together with Gert-Martin Greuel, Gerhard Pfister and Hans Schönemann. He is also coordinator of the nationwide German Research Council Priority Programme 'Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory'. Gerhard Pfister heads Singular's core development team within the Department of Mathematics at the University of Kaiserslautern, together with Wolfram Decker, Gert-Martin Greuel and Hans Schönemann.
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