An Introduction to Gröbner Bases

✍ Scribed by Ralf Fröberg

• Publisher: Wiley
• Year: 1997
• Tongue: English
• Leaves: 166
• Edition: 1
• Category: Library

✦ Synopsis

As algebra becomes more widely used in a variety of applications and computers are developed to allow efficient calculations in the field, so there becomes a need for new techniques to further this area of research. Gr?bner Bases is one topic which has recently become a very popular and important area of modern algebra. This book provides a concrete introduction to commutative algebra through Gr?bner Bases. The inclusion of exercises, lists of further reading and related literature make this a practical approach to introducing Gr?bner Bases. The author presents new concepts and results of recent research in the area allowing students and researchers in technology, computer science and mathematics to gain a basic understanding of the technique. A first course in algebra is the only prior knowledge required for this introduction. Chapter titles include: Monomial ldeas Gr?bner Bases Algebraic Sets Solving Systems of Polynomial Equations Applications of Gr?bner Bases Homogeneous Algebra Hilbert Series Variations of Gr?bner Bases Improvements to Buchberger's Algorithms Software

✦ Table of Contents

1.1 Definition of a Ring ......Page 1
Exercises ......Page 4
1.2.1 Integral Domains and Fields ......Page 5
1.2.3 A Finite Ring, Zn ......Page 6
1.2.4 Polynomial Rings ......Page 8
1.2.5 Zerodivisors ......Page 9
1.3.1 Definition of an Ideal ......Page 10
1.3.3 Principal Ideals and Euclid's Algorithm ......Page 11
1.3.4 Euclidean Rings ......Page 16
1.3.5 Ideals and their Calculus ......Page 17
1.4 Equivalence Relations ......Page 20
1.5 Field of Fractions of an Integral Domain ......Page 21
Exercises ......Page 22
1.6 Unigue Factorization Domains ......Page 23
1.7 Factor Rings and Homomorphisms ......Page 25
1.8 Prime Ideals and Maximal Ideals ......Page 29
1.9 Vector Spaces ......Page 31
Exercises ......Page 34
2 Monomial Ideals ......Page 36
2.2 Intersections o( Monomial Ideals ......Page 37
2.3 Quotient of Monomial Ideals ......Page 38
2.4 Prime Ideals ......Page 39
2.5 The Radical of a Monomial Ideal ......Page 40
3.1 Monomial Orderings ......Page 42
3.1.1 A Classification of Orderings ......Page 45
3.2.1 Dickson's Lemma ......Page 46
3.2.2 Applications of Dickson's Lemma ......Page 47
3.3 The Reduction Process ......Page 48
3.4 Definition of Gröbner Bases ......Page 49
3.5 Hilbert's Basis Theorem and Noetherian Rings ......Page 50
3.6 Gröbner Bases and Normal Forms ......Page 51
3.8 Construction of Gröbner Bases ......Page 52
3.9 Free Modules and Syzygies ......Page 54
3.10 Syzygies of Sequences of Monomials ......Page 55
3.12 A Criterion for Gröbner Basis ......Page 56
3.13 The Buchberger Algorithm ......Page 58
4.1 Algebraic Sets and Ideals ......Page 59
4.2 Hilbert's Nullstellensatz ......Page 64
4.3 A Dictionary: Algebraic Sets <-> Radical Ideals ......Page 66
5 Primary Decomposition of Ideals ......Page 68
6.1 Systems with Only One Solution ......Page 75
6.2.2 Decomposition of the Ring ......Page 76
6.3 Solving Zero-dimensional Systems ......Page 80
6.4 Systems of Higher Dimension ......Page 83
Exercises ......Page 84
7 Applications of Gröbner Bases ......Page 85
7.1.1 Ideal Membership ......Page 86
7.1.3 Subalgebra Mgnbtsalp ......Page 87
7.2 Calculation in Factor Rings of Polynomial Rings ......Page 89
7.3 Elimination ......Page 91
7.4.l Intersection of Ideals ......Page 92
7.4.2 Ideal Quotient ......Page 93
7.5 Supplementary Exercises ......Page 94
8.1 Homogeneous Ideals and Algebras ......Page 95
8.2.1 Homogenizing Polynomials ......Page 101
Exercises ......Page 102
8.2.4 Dehomogenizing Ideals ......Page 103
8.2.5 Homogenization versus Dehomogenization ......Page 104
8.3 Gröbner Bases for Homogenoous Ideals ......Page 107
Exercises ......Page 111
9 Projective Varieties ......Page 112
9.1 Projective Closure of an Algebraic Set ......Page 116
10 The Associated Graded Ring ......Page 118
11.1 Formal Power Seies ......Page 122
11.2 Hilbert Series ......Page 124
11.3 Geometric Meaning of Hilbert Scries ......Page 127
12.1 Gröbner Bases for Modules ......Page 133
12.2 More General Orderings within More General Rings ......Page 135
12.3 Gröbner Bases for Noncommutative Rings ......Page 136
Exercises ......Page 140
12.5 SAGBI Bases ......Page 141
13.1 Choice of Ordering ......Page 143
13.3 Unnecessary Reductions ......Page 144
13.6 Coefficients ......Page 145
13.7.1 Calculation of Hilbert Series ......Page 146
13.8 Change of Ordering ......Page 147
13.9 Tracing ......Page 148
14 Sotware ......Page 149
15 Hints to Some Exercises ......Page 151
16 Answers to Exercises ......Page 156
17.2 Articles ......Page 163