An Introduction to Gröbner Bases

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. Gröbner 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 Gröbner Bases. The inclusion of exercises, lists of further reading and related literature make this a practical approach to introducing Gröbner 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 * Gröbner Bases * Algebraic Sets * Solving Systems of Polynomial Equations * Applications of Gröbner Bases * Homogeneous Algebra * Hilbert Series * Variations of Gröbner Bases * Improvements to Buchberger's Algorithms * Software

Ralf Fröberg is the author of An Introduction to Gröbner Bases, published by Wiley.

Preface

ix

1 Rings, Fields, and Ideals

1

(36)

1.1 Definition of a Ring

1

(4)

Exercises

4

(1)

1.2 Rings and Fields

5

(5)

1.2.1 Integral Domains and Fields

5

(1)

1.2.2 A Convention

6

(1)

1.2.3 A Finite Ring, Z(n)

6

(2)

1.2.4 Polynomial Rings

8

(1)

1.2.5 Zerodivisors

9

(1)

1.3 Ideals and Rings

10

(10)

1.3.1 Definition of an Ideal

10

(1)

1.3.2 The Ideal Generated by a Set

11

(1)

1.3.3 Principal Ideals and Euclid's Algorithm

11

(5)

1.3.4 Euclidean Rings

16

(1)

1.3.5 Ideals and their Calculus

17

(3)

1.4 Equivalence Relations

20

(1)

1.5 Field of Fractions of an Integral Domain

21

(2)

Exercises

22

(1)

1.6 Unique Factorization Domains

23

(2)

Exercises

25

(1)

1.7 Factor Rings and Homomorphisms

25

(4)

1.8 Prime Ideals and Maximal Ideals

29

(2)

1.9 Vector Spaces

31

(6)

Exercises

34

(3)

2 Monomial Ideals

37

(6)

2.1 Sums and Products of Monomial Ideals

38

(1)

2.2 Intersections of Monomial Ideals

38

(1)

2.3 Quotients of Monomial Ideals

39

(1)

2.4 Prime Ideals

40

(1)

2.5 The Radical of a Monomial Ideals

41

(2)

3 Grobner Bases

43

(18)

3.1 Monomial Orderings

45

(2)

3.1.1 A Classification of Orderings

46

(1)

3.2 Dickson's Lemma and Some Applications

47

(2)

3.2.1 Dickson's Lemma

47

(1)

3.2.2 Applications of Dickson's Lemma

48

(1)

3.3 The Reduction Process

49

(1)

3.4 Definition of Grobner Bases

50

(1)

3.5 Hilbert's Basis Theorem and Noetherian Rings

51

(1)

3.6 Grobner Bases and Normal Forms

52

(1)

3.7 Reduced Grobner Bases

53

(1)

3.8 Construction of Grobner Bases

54

(1)

3.9 Free Modules and Syzygies

55

(1)

3.10 Syzygies of Sequences of Monomials

56

(1)

3.11 S-polynomials

57

(1)

3.12 A Criterion for Grobner Basis

57

(2)

3.13 The Buchberger Algorithm

59

(2)

4 Algebraic Sets

61

(10)

4.1 Algebraic Sets and Ideals

61

(5)

4.2 Hilbert's Nullstellensatz

66

(2)

4.3 A Dictionary: Algebraic Sets XXX Radical Ideals

68

(3)

5 Primary Decomposition of Ideals

71

(8)

6 Solving Systems of Polynomial Equations

79

(10)

6.1 Systems with Only One Solution

79

(1)

Exercises

80

(1)

6.2 Systems with Finitely Many Solutions

80

(4)

6.2.1 Decomposition of the Ideals

80

(1)

6.2.2 Decomposition of the Ring

81

(3)

6.3 Solving Zero-dimensional Systems

84

(3)

Exercises

86

(1)

6.4 Systems of Higher Dimension

87

(2)

Exercises

88

(1)

7 Applications of Grobner Bases

89

(10)

7.1 Membership Problems

90

(3)

7.1.1 Ideal Membership

90

(1)

7.1.2 Radical Membership

91

(1)

7.1.3 Subalgebra Membership

91

(2)

7.2 Calculation in Factor Rings of Polynomial Rings

93

(2)

7.3 Elimination

95

(1)

7.4 Ideal Operations

96

(2)

7.4.1 Intersection of Ideals

96

(1)

7.4.2 Ideal Quotient

97

(1)

7.5 Supplementary Exercises

98

(1)

8 Homogeneous Algebras

99

(18)

8.1 Homogeneous Ideals and Algebras

99

(6)

Exercises

105

(1)

8.2 Homogenizing and Dehomogenizing

105

(6)

8.2.1 Homogenizing Polynomials

105

(1)

Exercises

106

(1)

8.2.2 Homogenizing Ideals

107

(1)

Exercises

107

(1)

8.2.3 Dehomogenizing Polynomials

107

(1)

8.2.4 Dehomogenizing Ideals

107

(1)

8.2.5 Homogenization versus Dehomogenization

108

(3)

Exercises

111

(1)

8.3 Grobner Bases for Homogeneous Ideals

111

(6)

Exercises

115

(2)

9 Projective Varieties

117

(6)

9.1 Projective Closure of an Algebraic Set

121

(2)

10 The Associated Graded Ring

123

(4)

11 Hilbert Series

127

(12)

11.1 Formal Power Series

127

(2)

11.2 Hilbert Series

129

(3)

11.3 Geometric Meaning of Hilbert Series

132

(7)

12 Variations of Grobner Bases

139

(10)

12.1 Grobner Bases for modules

139

(2)

Exercises

141

(1)

12.2 More General Orderings within More General Rings

141

(1)

12.3 Grobner Bases for Noncommutative Rings

142

(3)

Exercises

146

(1)

12.4 Differential Grobner Bases

147

(1)

12.5 SAGBI Bases

147

(2)

13 Improvements to Buchberger's Algorithm

149

(6)

13.1 Choice of Ordering

149

(1)

13.2 Strategies

150

(1)

13.3 Unnecessary Reductions

150

(1)

13.4 Homogenizing

151

(1)

13.5 Factorizing

151

(1)

13.6 Coefficients

151

(1)

13.7 Using the Hilbert Series

152

(1)

13.7.1 Calculation of Hilbert Series

152

(1)

13.8 Change of Ordering

153

(1)

13.9 Tracing

154

(1)

14 Software

155

(2)

15 Hints to Some Exercises

157

(6)

16 Answers to Exercises

163

(8)

17 Bibliography

171

(4)

17.1 Books

171

(1)

17.2 Articles

171

(4)

Index

175

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

FREE SHIPPING

Amazon no longer offers textbook rentals. We do!

An Introduction to Gröbner Bases

Summary

Author Biography

Table of Contents

Supplemental Materials