Preface xi
Setting xiv
Part three: Gauss, Euclid, Buchberger: Elementary
Gröbner Bases 1
20 Hilbert 3
20.1 Affine Algebraic Varieties and Ideals 3
20.2 Linear Change of Coordinates 8
20.3 Hilbert’s Nullstellensatz 10
20.4 *Kronecker Solver 15
20.5 Projective Varieties and Homogeneous Ideals 22
20.6 *Syzygies and Hilbert Function 28
20.7 *More on the Hilbert Function 34
20.8 Hilbert’s and Gordan’s Basissätze 36
21 Gauss II 46
21.1 Some Heretical Notation 47
21.2 Gaussian Reduction 51
21.3 Gaussian Reduction and Euclidean Algorithm Revisited 63
22 Buchberger 72
22.1 From Gauss to Gröbner 75
22.2 Gröbner Basis 78
22.3 Toward Buchberger’s Algorithm 83
22.4 Buchberger’s Algorithm (1) 96
22.5 Buchberger’s Criteria 98
22.6 Buchberger’s Algorithm (2) 104
23 Macaulay I 109
23.1 Homogenization and Affinization 110
23.2 H-bases 114
23.3 Macaulay’s Lemma 119
23.4 Resolution and Hilbert Function for Monomial Ideals 122
23.5 Hilbert Function Computation: the ‘Divide-and-Conquer’ Algorithms 136
23.6 H-bases and Gröbner Bases for Modules 138
23.7 Lifting Theorem 142
23.8 Computing Resolutions 146
23.9 Macaulay’s Nullstellensatz Bound 152
23.10 *Bounds for the Degree in the Nullstellensatz 156
24 Gröbner I 170
24.1 Rewriting Rules 173
24.2 Gröbner Bases and Rewriting Rules 183
24.3 Gröbner Bases for Modules 188
24.4 Gröbner Bases in Graded Rings 195
24.5 Standard Bases and the Lifting Theorem 198
24.6 Hironaka’s Standard Bases and Valuations 203
24.7 *Standard Bases and Quotients Rings 218
24.8 *Characterization of Standard Bases in Valuation Rings 223
24.9 Term Ordering: Classification and Representation 234
24.10 *Gröbner Bases and the State Polytope 247
25 Gebauer and Traverso 255
25.1 Gebauer–Möller and Useless Pairs 255
25.2 Buchberger’s Algorithm (3) 264
25.3 Traverso’s Choice 271
25.4 Gebauer–Möller’s Staggered Linear Bases and Faugère’s F 5 274
26 Spear 289
26.1 Zacharias Rings 291
26.2 Lexicographical Term Ordering and Elimination Ideals 300
26.3 Ideal Theoretical Operation 304
26.4 *Multivariate Chinese Remainder Algorithm 313
26.5 Tag-Variable Technique and Its Application to Subalgebras 316
26.6 Caboara–Traverso Module Representation 321
26.7 *Caboara Algorithm for Homogeneous Minimal Resolutions 329
Part four: Duality 333
27 Noether 335
27.1 Noetherian Rings 337
27.2 Prime, Primary, Radical, Maximal Ideals 340
27.3 Lasker–Noether Decomposition: Existence 345
27.4 Lasker–Noether Decomposition: Uniqueness 350
27.5 Contraction and Extension 356
27.6 Decomposition of Homogeneous Ideals 364
27.7 *The Closure of an Ideal at the Origin 368
27.8 Generic System of Coordinates 371
27.9 Ideals in Noether Position 374
27.10 *Chains of Prime Ideals 378
27.11 Dimension 380
27.12 Zero-dimensional Ideals and Multiplicity 384
27.13 Unmixed Ideals 390
28 Möller I 393
28.1 Duality 393
28.2 Möller Algorithm 401
29 Lazard 414
29.1 The FGLM Problem 415
29.2 The FGLM Algorithm 418
29.3 Border Bases and Gröbner Representation 426
29.4 Improving Möller’s Algorithm 432
29.5 Hilbert Driven and Gröbner Walk 440
29.6 *The Structure of the Canonical Module 444
30 Macaulay II 451
30.1 The Linear Structure of an Ideal 452
30.2 Inverse System 456
30.3 Representing and Computing the Linear Structure of an Ideal 461
30.4 Noetherian Equations 466
30.5 Dialytic Arrays of M(r) and Perfect Ideals 478
30.6 Multiplicity of Primary Ideals 492
30.7 The Structure of Primary Ideals at the Origin 494
31 Gröbner II 500
31.1 Noetherian Equations 501
31.2 Stability 502
31.3 Gröbner Duality 504
31.4 Leibniz Formula 508
31.5 Differential Inverse Functions at the Origin 509
31.6 Taylor Formula and Gröbner Duality 512
32 Gröbner III 517
32.1 Macaulay Bases 518
32.2 Macaulay Basis and Gröbner Representation 521
32.3 Macaulay Basis and Decomposition of Primary Ideals 522
32.4 Horner Representation of Macaulay Bases 527
32.5 Polynomial Evaluation at Macaulay Bases 531
32.6 Continuations 533
32.7 Computing a Macaulay Basis 542
33 Möller II 549
33.1 Macaulay’s Trick 550
33.2 The Cerlienco–Mureddu Correspondence 554
33.3 Lazard Structural Theorem 560
33.4 Some Factorization Results 562
33.5 Some Examples 569
33.6 An Algorithmic Proof 574
Part five: Beyond Dimension Zero 583
34 Gröbner IV 585
34.1 Nulldimensionalen Basissätze 586
34.2 Primitive Elements and Allgemeine Basissatz 593
34.3 Higher-Dimensional Primbasissatz 598
34.4 Ideals in Allgemeine Positions 601
34.5 Solving 605
34.6 Gianni–Kalkbrener Theorem 608
35 Gianni–Trager–Zacharias 614
35.1 Decomposition Algorithms 615
35.2 Zero-dimensional Decomposition Algorithms 616
35.3 The GTZ Scheme 622
35.4 Higher-dimensional Decomposition Algorithms 631
35.5 Decomposition Algorithms for Allgemeine Ideals 634
35.5.1 Zero-dimensional Allgemeine Ideals 634
35.5.2 Higher-dimensional Allgemeine Ideals 637
35.6 Sparse Change of Coordinates 640
35.6.1 Gianni’s Local Change of Coordinates 641
35.6.2 Giusti–Heintz Coordinates 645
35.7 Linear Algebra and Change of Coordinates 650
35.8 Direct Methods for Radical Computation 654
35.9 Caboara–Conti–Traverso Decomposition Algorithm 658
35.10 Squarefree Decomposition of a
Zero-dimensional Ideal 660
36 Macaulay III 665
36.1 Hilbert Function and Complete Intersections 666
36.2 The Coefficients of the Hilbert Function 670
36.3 Perfectness 678
37 Galligo 686
37.1 Galligo Theorem (1): Existence of Generic Escalier 686
37.2 Borel Relation 697
37.3 *Galligo Theorem (2): the Generic Initial Ideal is Borel Invariant 706
37.4 *Galligo Theorem (3): the Structure of the Generic Escalier 710
37.5 Eliahou–Kervaire Resolution 714
38 Giusti 725
38.1 The Complexity of an Ideal 726
38.2 Toward Giusti’s Bound 728
38.3 Giusti’s Bound 733
38.4 Mayr and Meyer’s Example 735
38.5 Optimality of Revlex 741
Bibliography 749
Index758