## Computational Commutative Algebra 2Hofstadter’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 ?rst? There we wrote that it took three years of intense work just to ?ll three centimeters of your bookshelf. The completion of this volume took four years and it is about four centimeters thick. Thus we have a con?rmed invariant which governs our writing: our velocity is one centimeter per year, after all e?ects 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 ?ll it with amusing quotes, varied jokes, funny word games, ?owery metaphors and occasional literary e?orts. There are two possible drawbacks of this method. Firstly, not everyone has the same sense of humour and not every metaphor works as intended. For instance, it is easy 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 di?cult to write humorously in a foreign language. |

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### Contents

1 | |

10 | |

Hilbert Functions | 173 |

Further Applications | 347 |

A The ABC of CoCoA | 535 |

More on CoCoA Programming 543 | 542 |

Suggestions for Further Reading | 557 |

Notation | 565 |

571 | |

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affine algebraically true append Apply your function binomial Buchberger Algorithm called choose CoCoA function Commutative Algebra computing minimal contained continue with step Corollary deﬁned Deﬁnition deg-ordered degree dim(P/I Example Exercise exists ﬁeld ﬁltration ﬁnitely finitely generated graded ﬁrst G Zm g1 g graded free P-module graded free resolution graded P-module graded submodule grading given Gröbner basis Hence Hilbert function Hilbert polynomial Hilbert series homogeneous ideal homogeneous polynomial homomorphism Ihom implies indeterminates integer function isomorphism K-algebra K-vector leading term Lemma let f Lex-segment linear Macaulay bases Matm,n(Z matrix minimal graded free minimal homogeneous module term ordering monoid monomial ideal order ideal polynomial f polynomial ring primary decomposition prime ideal projective point set Proof Proposition prove residue classes respect SAGBI bases satisﬁes sequence standard graded subalgebra subset syzygy Theorem toric ideal tuple Tutorial vector space Write a CoCoA