A First Course in Computational Algebraic Geometry

Preface page vii
Prologue:
General Remarks on Computer Algebra Systems 1
1 The Geometry–Algebra Dictionary 11
1.1 Affine Algebraic Geometry 11
1.1.1 Ideals in Polynomial Rings 11
1.1.2 Affine Algebraic Sets 14
1.1.3 Hilbert’s Nullstellensatz 20
1.1.4 Irreducible Algebraic Sets 23
1.1.5 Removing Algebraic Sets 25
1.1.6 Polynomial Maps 29
1.1.7 The Geometry of Elimination 32
1.1.8 Noether Normalization and Dimension 37
1.1.9 Local Studies 45
1.2 Projective Algebraic Geometry 49
1.2.1 The Projective Space 49
1.2.2 Projective Algebraic Sets 52
1.2.3 Affine Charts and the Projective Closure 54
1.2.4 The Hilbert Polynomial 57
2 Computing 60
2.1 Standard Bases and S INGULAR 60
2.2 Applications 75
2.2.1 Ideal Membership 75
2.2.2 Elimination 75
2.2.3 Radical Membership 77
2.2.4 Ideal Intersections 78
2.2.5 Ideal Quotients 79
2.2.6 Kernel of a Ring Map 79
2.2.7 Integrality Criterion 80
2.2.8 Noether Normalization 82
2.2.9 Subalgebra Membership 83
2.2.10 Homogenization 83
2.3 Dimension and the Hilbert Function 84
2.4 Primary Decomposition and Radicals 90
2.5 Buchberger’s Algorithm and Field Extensions 94
3 Sudoku 95
4 A Problem in Group Theory Solved by Computer Algebra 101
4.1 Finite Groups and Thompson’s Theorem 101
4.2 Characterization of Finite Solvable Groups 104
Bibliography 112
Index 115