Table Of ContentKreuzer·Robbiano
ComputationalCommutativeAlgebra2
·
Martin Kreuzer Lorenzo Robbiano
Computational
Commutative
Algebra 2
123
MartinKreuzer
Universita¨tDortmund
FachbereichMathematik
44221Dortmund,Germany
e-mail:[email protected]
LorenzoRobbiano
Universita`diGenova
DipartimentodiMatematica
ViaDodecaneso35
16146Genova,Italy
e-mail:[email protected]
LibraryofCongressControlNumber:00046340
Mathematics Subject Classification (2000): Primary: 13P10; Secondary: 68W30,
13A02,13D40,14Qxx
ISBN-10 3-540-25527-3 Springer-VerlagBerlinHeidelberg
ISBN-13 978-3-540-25527-7 Springer-VerlagBerlinHeidelbergNewYork
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Foreword
Hofstadter’s Law: It always takes longer than you think it will take,
even if you take into account Hofstadter’s Law.
(Douglas R. Hofstadter)
Dear Reader,
why did we begin the foreword of this second volume with the same quote
as the first? There we wrote that it took three years of intense work just to
fill three centimeters of your bookshelf. The completion of this volume took
four years and it is about four centimeters thick. Thus we have a confirmed
invariant which governs our writing: our velocity is one centimeter per year,
after all effects due to Hofstadter’s Law have been taken into account. When
westartedthisprojectinthelastmillennium,weplannedabookforlearning,
teaching, reading and, most of all, enjoying the topic at hand. Surely there
is no law which says that a mathematical book has to be dull, boring, dry,
or tedious. But how do you make it enjoyable?
Our approach has been to fill it with amusing quotes, varied jokes, funny
wordgames,flowerymetaphorsandoccasionalliteraryefforts.Therearetwo
possible drawbacks of this method. Firstly, not everyone has the same sense
ofhumourandnoteverymetaphorworksasintended.Forinstance,itiseasy
to joke about certain politicians, but what happens if they read this book?
And when we wrote of a small boat sailing slowly into the Brazilian sunset,
it was pointed out to us that this entails a geographical problem. Secondly,
it is very difficult to write humorously in a foreign language. In the foreword
of our first book, we acknowledged a modest lack of linguistic sophistication.
This time around, our scribbling has utterly changed for the worse. When
confronted with a concoction such as
... and if the whole section was a piece of cake for you, bring your bite to
bear on Tutorials 47 and 48 [...].
evenourhardest,steadfastestadvisorsJohnAbbottandTonyGeramitahad
to concede soft spots. The most enthusiastic remark we could wring from
themwasthatonecanfindalmosteverywordweuseinanEnglishdictionary.
But if a word in the dictionary were mispelled, how would we know? Should
we have taken more heed of the suggestions of our readers, one of whom
VI
recommended correcting “I is an ideal” to “I am an ideal”? And what
about definitions like “A flower is an educated weed”?
Anothersourcefortroublewerecommas.Theirdistributioninthisvolume
isratherrandom,becausewewerepuzzledbyrulessuchasthefollowingone:
“If you use then after an if, then don’t put a comma before then.”
I must thank my parents,
but mostly my mother and my father.
(“Spillo” Altobelli)
Aswiththefirstvolume,thesecondisreallyajointeffortofmanypeople.
We won’t bore you with platitudes about how many people contributed to
this book; we’ll just bore you with a complete list of those people. A special
mention goes to John Abbott who spent a lot of time, and almost all his pa-
tience, to help us improve our text. Moreover, he continually stimulated our
writing with profound and enlightening remarks. Further substantial proof-
reading was done by Valentina Bertella, Anna Bigatti, Achim Kehrein, and
Tony Geramita. We are also grateful to Laura Bazzotti, Henrik Bresinsky,
Aldo Conca, Giorgio Dalzotto, Giulio Genovese, Daniel Heldt, Eva Ludwig,
Matthias Machnik, and Maria Evelina Rossi for helpful remarks. In spite of
this extensive proofreading, the book still contains infinitely many errors.
(Proof by induction: You can always find another one.) We shoulder the full
responsibility for everything that is still wrong, in particular if we have any
words out.
Let us also not forget the many other people who have contributed to
this book in more indirect ways: all famous and infamous people (including
ourselves) whose quotes we abused, the soccer players of our favourite teams
Bayern Mu¨nchen and Juventus Turin who showed us that we were not the
onlyoneswhohadtheirupsanddowns,andthecentralbankersofthisworld
who furnished us with never-ending amazement about their uncanny ability
to create money out of nothing. Very instrumental in the writing of this
book was the support we received from our families. Our wives Bettina and
Gabriella and our children Chiara, Francesco, Katharina, Veronika, as well
as the latest addition, Martin junior, devoted a significant part of their lives
to aid us as much as possible.
A very special “thank you” goes to the Springer team, in particular to
Dr. Martin Peters and his assistant Ruth Allewelt, for their efficient and
professionalhelpintheeditorialandproductionpartofthisendeavour.Last,
but not least, we acknowledge the book itself: after we have spent the better
part of eight years putting much of what we know about computer algebra
intoit,thebookknowsmorethanwedosinceitalsorememberswhatwehave
alreadyforgotten.Mayitbeasenjoyableandinstructivetoyouasitistous!
Martin Kreuzer and Lorenzo Robbiano,
Dortmund and Genova, March 2005
Contents
Foreword ..................................................... V
Introduction.................................................. 1
0.1 What Is This Book About? .............................. 1
0.2 Now, What Is This Book Really About? ................... 2
0.3 What Is This Book Not About? .......................... 4
0.4 Are There any Applications of This Theory? ............... 5
0.5 How Was This Book Written?............................ 5
0.6 What Is a Tutorial?..................................... 7
0.7 What Is CoCoA?........................................ 7
0.8 And What Is This Book Good for? ....................... 8
0.9 Some Final Words of Wisdom............................ 9
4. The Homogeneous Case .................................. 11
4.1 Polynomial Rings Graded by Matrices..................... 15
4.1.A Gradings by Matrices............................. 16
4.1.B Graded Modules ................................. 20
4.1.C Gradings of Positive Type......................... 23
Tutorial 45. Homogeneous Maps and Toric Ideals ............ 29
Tutorial 46. Projective Varieties .......................... 30
4.2 Degree Forms and Macaulay Bases ....................... 32
4.2.A Positive Gradings ................................ 33
4.2.B Macaulay Bases.................................. 36
Tutorial 47. Computation of Macaulay Bases................ 40
Tutorial 48. Characterizations of Macaulay Bases ............ 42
4.3 Homogenization ........................................ 44
4.3.A Homogenization of Polynomials and Ideals .......... 45
4.3.B Macaulay Bases and Homogenization ............... 55
Tutorial 49. Homogenization of Modules ................... 60
Tutorial 50. The Homogeneous Part of an Ideal.............. 63
Tutorial 51. Implicitization and Homogenization ............. 64
Tutorial 52. Projective Closure ........................... 66
4.4 Term Orderings of DegRev Type ......................... 68
Tutorial 53. Reduced Gr¨obner Bases and Homogenization ..... 78
VIII Contents
Tutorial 54. Regular Sequences of Indeterminates ............ 80
Tutorial 55. Set-Theoretic Complete Intersections............ 82
4.5 Homogeneous Gro¨bner Bases............................. 85
4.5.A The Homogeneous Buchberger Algorithm ........... 87
4.5.B Truncated Gro¨bner Bases ......................... 92
Tutorial 56. Computation of Homogeneous Gr¨obner Bases ..... 96
Tutorial 57. Some Applications of Truncated Gr¨obner Bases ... 98
4.6 Minimal Homogeneous Systems of Generators.............. 100
Tutorial 58. Computing Some Minimal Systems of Generators.. 110
Tutorial 59. Optimizing the Homogeneous Buchberger Algorithm111
4.7 Minimal Homogeneous Presentations...................... 116
4.7.A Existence and Uniqueness of Minimal Presentations .. 117
4.7.B Idealization of Graded Modules and Presentations.... 123
4.7.C Computation of Minimal Homogeneous Presentations. 130
Tutorial 60. Computing Some Idealizations ................. 143
Tutorial 61. Computing Some Minimal Presentations ......... 144
4.8 Minimal Graded Free Resolutions......................... 147
4.8.A Existence and Uniqueness of Minimal Free Resolutions148
4.8.B Computation of Minimal Graded Free Resolutions ... 154
Tutorial 62. The Hilbert-Burch Theorem ................... 167
Tutorial 63. Computing Some Graded Betti Numbers......... 169
5. Hilbert Functions......................................... 173
5.1 Basic Properties of Hilbert Functions ..................... 179
5.1.A Integer Functions of Polynomial Type .............. 180
5.1.B Hilbert Functions in the Standard Graded Case...... 185
Tutorial 64. Hilbert Functions and Graded Free Resolutions.... 191
5.2 Hilbert Series .......................................... 194
5.2.A Univariate Power Series........................... 195
5.2.B Hilbert Series in the Standard Graded Case ......... 202
Tutorial 65. Knight Moves............................... 205
Tutorial 66. Veronese Subrings ........................... 209
Tutorial 67. Powers of Polynomials and Ehrhart Functions..... 210
5.3 Computation of Hilbert Series............................ 214
Tutorial 68. Implementation of the Hilbert Series Algorithm ... 226
Tutorial 69. Hilbert Driven Gr¨obner Basis Computations I..... 228
5.4 Dimension, Multiplicity, and Hilbert Polynomials........... 232
5.4.A Dimension and Multiplicity of Standard Algebras .... 233
5.4.B Hilbert Polynomials in the Standard Graded Case.... 239
Tutorial 70. Computing the Dimension of a Module .......... 243
Tutorial 71. Chess Puzzles............................... 244
Tutorial 72. Photogrammetry ............................ 247
5.5 Bounds for Hilbert Functions ............................ 252
5.5.A Binomial Representations ......................... 254
5.5.B Lex-Segment Spaces and Ideals .................... 258
Contents IX
5.5.C The Theorems of Macaulay and Green.............. 262
Tutorial 73. Operations on Binomial Representations ......... 271
Tutorial 74. Bounds for Minimal Generators ................ 273
Tutorial 75. Gin for the Strongly Stable.................... 274
5.6 Affine Hilbert Functions and Krull Dimension.............. 278
5.6.A The Hilbert Function of an Affine Algebra .......... 279
5.6.B Primary Decomposition in Noetherian Rings ........ 284
5.6.C The Krull Dimension of an Affine Algebra........... 291
Tutorial 76. The Multiplicity of an Affine Algebra............ 297
Tutorial 77. Primary Decomposition of Monomial Ideals....... 298
5.7 Independent Sets of Indeterminates ....................... 301
5.7.A The Combinatorial Dimension of an Affine Algebra... 301
5.7.B Transcendence Degrees ........................... 308
Tutorial 78. Noether Normalization ....................... 312
Tutorial 79. Primary Decompositions II .................... 315
5.8 General Hilbert Functions ............................... 321
5.8.A Rings of Multivariate Laurent Series................ 323
5.8.B Hilbert Functions in the General Case .............. 325
5.8.C Change of Gradings .............................. 332
Tutorial 80. Hilbert Driven Gr¨obner Basis Computations II .... 335
Tutorial 81. Rees Rings ................................. 339
Tutorial 82. Segre Products and Hadamard Series............ 341
Tutorial 83. A Toy Example ............................. 344
6. Further Applications ..................................... 347
6.1 Toric Ideals and Hilbert Bases ........................... 351
6.1.A Toric Ideals ..................................... 352
6.1.B Hilbert Bases.................................... 358
Tutorial 84. Magic Squares .............................. 367
Tutorial 85. Computing the Gaps ......................... 370
Tutorial 86. Matrices of Positive Type ..................... 372
6.2 Liftings of Ideals and Distractions ........................ 375
Tutorial 87. SuperG Bases............................... 384
6.3 Finite Sets of Points .................................... 387
6.3.A Affine Point Sets................................. 389
6.3.B Projective Point Sets ............................. 395
6.3.C Hilbert Functions of Points........................ 404
Tutorial 88. The Cayley-Bacharach Property................ 408
Tutorial 89. Generic Sets of Points ........................ 411
Tutorial 90. Error-Correcting Codes....................... 415
6.4 Border Bases .......................................... 419
6.4.A Existence and Uniqueness of Border Bases .......... 421
6.4.B Characterizations of Border Bases.................. 430
Tutorial 91. Module Structures on Vector Spaces ............ 442
Tutorial 92. Design of Experiments........................ 447
X Contents
6.5 Filtrations............................................. 453
6.5.1 General Filtrations ............................... 454
6.5.B Adic Filtrations and Tangent Cones ................ 460
Tutorial 93. Mora’s Algorithm ........................... 465
Tutorial 94. Hilbert Functions of Primary Ideals ............. 470
Tutorial 95. Singularities ................................ 472
6.6 SAGBI Bases .......................................... 477
6.6.A Definition and Basic Properties of SAGBI Bases ..... 478
6.6.B Characterization of SAGBI Bases .................. 486
6.6.C Computation of SAGBI Bases ..................... 494
Tutorial 96. Variations on the SAGBI Theme ............... 499
Tutorial 97. Gr¨obner and SAGBI Bases Under Composition.... 503
Tutorial 98. Molien’s Theorem ........................... 507
6.7 Automatic Theorem Proving............................. 509
6.7.A The Tribulations of Automation ................... 511
6.7.B Algebraically True Statements ..................... 516
6.7.C Optimal Hypothesis Ideals and Minimal Conditions .. 521
Tutorial 99. To Prove or Not to Prove ..................... 528
A. The ABC of CoCoA ....................................... 535
B. One Graphical Interface for Everybody ................... 537
C. More on CoCoA Programming ............................ 543
D. Suggestions for Further Reading.......................... 557
E. Hints for Selected Exercises .............................. 561
Notation...................................................... 565
Bibliography.................................................. 571
Index......................................................... 575
Introduction
Writing, for him who knows it
is better than all other professions;
it pleases more than bread and beer,
more than clothing and ornament.
(20th Dynasty, Egypt)
0.1 What Is This Book About?
The title of this book is “Computational Commutative Algebra 2”. It is the
naturalcontinuationof“ComputationalCommutativeAlgebra1”writtenby
us in the last millennium. In the introduction to the first volume we wrote
that the fundamental ideas of Computational Commutative Algebra were
deeply rooted in the development of mathematics in the 20th century, and
that we planned to be back with more in the not so distant future. So, here
we are!
Butwhydidittakesolong?Welovedeadlines,inparticularthewhoosh-
ing sound they make as they fly by. Therefore we successfully missed every
deadline we had set for ourselves. Clearly, one reason is that writing this
book pleased us more than pasta and wine. Another reason is that laying
the foundations for Computational Commutative Algebra in the third mil-
lennium turned out to be more of a job than we had bargained for. Many
sectionsofthisbookarelikesmallresearchpapersbecausewetriedtopresent
the material in the style that pleases us so much and differs markedly from
existing literature.
The result is a volume having almost twice the size of the first: some of
its 23 sections are as big as a whole chapter, some of its 55 tutorials are as
big as a whole section, and some exercises are as big as a tutorial. Together
the two books form a compilation of about 900 pages which includes a vast
collection of 99 tutorials to keep you busy for a long, long time. To sustain
your interest, we have tried to continue the presentation in the lively style
of the first volume. Alas, we have to inform you that this is absolutely and
definitively the second and last volume of the trilogy.
