| Preface | p. xi |
| Setting | p. xiv |
| Gauss, Euclid, Buchberger: Elementary Grobner Bases | p. 1 |
| Hilbert | p. 3 |
| Affine Algebraic Varieties and Ideals | p. 3 |
| Linear Change of Coordinates | p. 8 |
| Hilbert's Nullstellensatz | p. 10 |
| *Kronecker Solver | p. 15 |
| Projective Varieties and Homogeneous Ideals | p. 22 |
| *Syzygies and Hilbert Function | p. 28 |
| *More on the Hilbert Function | p. 34 |
| Hilbert's and Gordan's Basissatze | p. 36 |
| Gauss II | p. 46 |
| Some Heretical Notation | p. 47 |
| Gaussian Reduction | p. 51 |
| Gaussian Reduction and Euclidean Algorithm Revisited | p. 63 |
| Buchberger | p. 72 |
| From Gauss to Grobner | p. 75 |
| Grobner Basis | p. 78 |
| Toward Buchberger's Algorithm | p. 83 |
| Buchberger's Algorithm (1) | p. 96 |
| Buchberger's Criteria | p. 98 |
| Buchberger's Algorithm (2) | p. 104 |
| Macaulay I | p. 109 |
| Homogenization and Affinization | p. 110 |
| H-bases | p. 114 |
| Macaulay's Lemma | p. 119 |
| Resolution and Hilbert Function for Monomial Ideals | p. 122 |
| Hilbert Function Computation: the 'Divide-and-Conquer' Algorithms | p. 136 |
| H-bases and Grobner Bases for Modules | p. 138 |
| Lifting Theorem | p. 142 |
| Computing Resolutions | p. 146 |
| Macaulay's Nullstellensatz Bound | p. 152 |
| *Bounds for the Degree in the Nullstellensatz | p. 156 |
| Grobner I | p. 170 |
| Rewriting Rules | p. 173 |
| Grobner Bases and Rewriting Rules | p. 183 |
| Grobner Bases for Modules | p. 188 |
| Grobner Bases in Graded Rings | p. 195 |
| Standard Bases and the Lifting Theorem | p. 198 |
| Hironaka's Standard Bases and Valuations | p. 203 |
| *Standard Bases and Quotients Rings | p. 218 |
| *Characterization of Standard Bases in Valuation Rings | p. 223 |
| Term Ordering: Classification and Representation | p. 234 |
| *Grobner Bases and the State Polytope | p. 247 |
| Gebauer and Traverso | p. 255 |
| Gebauer-Moller and Useless Pairs | p. 255 |
| Buchberger's Algorithm (3) | p. 264 |
| Traverso's Choice | p. 271 |
| Gebauer-Moller's Staggered Linear Bases and Faugere's F[subscript 5] | p. 274 |
| Spear | p. 289 |
| Zacharias Rings | p. 291 |
| Lexicographical Term Ordering and Elimination Ideals | p. 300 |
| Ideal Theoretical Operation | p. 304 |
| *Multivariate Chinese Remainder Algorithm | p. 313 |
| Tag-Variable Technique and Its Application to Subalgebras | p. 316 |
| Caboara-Traverso Module Representation | p. 321 |
| *Caboara Algorithm for Homogeneous Minimal Resolutions | p. 329 |
| Duality | p. 333 |
| Noether | p. 335 |
| Noetherian Rings | p. 337 |
| Prime, Primary, Radical, Maximal Ideals | p. 340 |
| Lasker-Noether Decomposition: Existence | p. 345 |
| Lasker-Noether Decomposition: Uniqueness | p. 350 |
| Contraction and Extension | p. 356 |
| Decomposition of Homogeneous Ideals | p. 364 |
| The Closure of an Ideal at the Origin | p. 368 |
| Generic System of Coordinates | p. 371 |
| Ideals in Noether Position | p. 374 |
| Chains of Prime Ideals | p. 378 |
| Dimension | p. 380 |
| Zero-dimensional Ideals and Multiplicity | p. 384 |
| Unmixed Ideals | p. 390 |
| Moller I | p. 393 |
| Duality | p. 393 |
| Moller Algorithm | p. 401 |
| Lazard | p. 414 |
| The FGLM Problem | p. 415 |
| The FGLM Algorithm | p. 418 |
| Border Bases and Grobner Representation | p. 426 |
| Improving Moller's Algorithm | p. 432 |
| Hilbert Driven and Grobner Walk | p. 440 |
| The Structure of the Canonical Module | p. 444 |
| Macaulay II | p. 451 |
| The Linear Structure of an Ideal | p. 452 |
| Inverse System | p. 456 |
| Representing and Computing the Linear Structure of an Ideal | p. 461 |
| Noetherian Equations | p. 466 |
| Dialytic Arrays of M[superscript (r)] and Perfect Ideals | p. 478 |
| Multiplicity of Primary Ideals | p. 492 |
| The Structure of Primary Ideals at the Origin | p. 494 |
| Grobner II | p. 500 |
| Noetherian Equations | p. 501 |
| Stability | p. 502 |
| Grobner Duality | p. 504 |
| Leibniz Formula | p. 508 |
| Differential Inverse Functions at the Origin | p. 509 |
| Taylor Formula and Grobner Duality | p. 512 |
| Grobner III | p. 517 |
| Macaulay Bases | p. 518 |
| Macaulay Basis and Grobner Representation | p. 521 |
| Macaulay Basis and Decomposition of Primary Ideals | p. 522 |
| Horner Representation of Macaulay Bases | p. 527 |
| Polynomial Evaluation at Macaulay Bases | p. 531 |
| Continuations | p. 533 |
| Computing a Macaulay Basis | p. 542 |
| Moller II | p. 549 |
| Macaulay's Trick | p. 550 |
| The Cerlienco-Mureddu Correspondence | p. 554 |
| Lazard Structural Theorem | p. 560 |
| Some Factorization Results | p. 562 |
| Some Examples | p. 569 |
| An Algorithmic Proof | p. 574 |
| Beyond Dimension Zero | p. 583 |
| Grobner IV | p. 585 |
| Nulldimensionalen Basissatze | p. 586 |
| Primitive Elements and Allgemeine Basissatz | p. 593 |
| Higher-Dimensional Primbasissatz | p. 598 |
| Ideals in Allgemeine Positions | p. 601 |
| Solving | p. 605 |
| Gianni-Kalkbrener Theorem | p. 608 |
| Gianni-Trager-Zacharias | p. 614 |
| Decomposition Algorithms | p. 615 |
| Zero-dimensional Decomposition Algorithms | p. 616 |
| The GTZ Scheme | p. 622 |
| Higher-dimensional Decomposition Algorithms | p. 631 |
| Decomposition Algorithms for Allgemeine Ideals | p. 634 |
| Zero-dimensional Allgemeine Ideals | p. 634 |
| Higher-dimensional Allgemeine Ideals | p. 637 |
| Sparse Change of Coordinates | p. 640 |
| Gianni's Local Change of Coordinates | p. 641 |
| Giusti-Heintz Coordinates | p. 645 |
| Linear Algebra and Change of Coordinates | p. 650 |
| Direct Methods for Radical Computation | p. 654 |
| Caboara-Conti-Traverso Decomposition Algorithm | p. 658 |
| Squarefree Decomposition of a Zero-dimensional Ideal | p. 660 |
| Macaulay III | p. 665 |
| Hilbert Function and Complete Intersections | p. 666 |
| The Coefficients of the Hilbert Function | p. 670 |
| Perfectness | p. 678 |
| Galligo | p. 686 |
| Galligo Theorem (1): Existence of Generic Escalier | p. 686 |
| Borel Relation | p. 697 |
| Galligo Theorem (2): the Generic Initial Ideal is Borel Invariant | p. 706 |
| Galligo Theorem (3): the Structure of the Generic Escalier | p. 710 |
| Eliahou-Kervaire Resolution | p. 714 |
| Giusti | p. 725 |
| The Complexity of an Ideal | p. 726 |
| Toward Giusti's Bound | p. 728 |
| Giusti's Bound | p. 733 |
| Mayr and Meyer's Example | p. 735 |
| Optimality of Revlex | p. 741 |
| Bibliography | p. 749 |
| Index | p. 758 |
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