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Solving Polynomial Equation Systems II: Macaulay's Paradigm and Gröbner Technology

Publisher: 
Cambridge University Press
Number of Pages: 
759
Price: 
150.00
ISBN: 
0-521-81156-2

See the review of volumes I and II on the details page for volume I.

Date Received: 
Friday, September 16, 2005
Reviewable: 
Yes
Include In BLL Rating: 
No
Teo Mora
Series: 
Encyclopedia of Mathematics and Its Applications 99
Publication Date: 
2005
Format: 
Hardcover
Category: 
Monograph
David P. Roberts
09/14/2006

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

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