| Preface |
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ix | |
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Chapter 1 Polynomial rings and ideals |
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1 | (14) |
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1 | (5) |
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1.1.1 Definition of the polynomial ring |
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1 | (4) |
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1.1.2 Some basic properties of polynomial rings |
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5 | (1) |
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6 | (9) |
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1.2.1 Operations on ideals |
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6 | (1) |
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1.2.2 Residue class rings |
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7 | (1) |
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1.2.3 Monomial ideals and Dickson's lemma |
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8 | (2) |
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1.2.4 Operations on monomial ideals |
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10 | (2) |
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12 | (3) |
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15 | (18) |
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15 | (3) |
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2.1.1 Examples and basic properties of monomial orders |
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15 | (2) |
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2.1.2 Construction of monomial orders |
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17 | (1) |
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§2.2 Initial ideals and Grobner bases |
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18 | (4) |
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2.2.1 The basic definitions |
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18 | (2) |
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20 | (1) |
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2.2.3 Hilbert's basis theorem |
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21 | (1) |
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§2.3 The division algorithm |
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22 | (3) |
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§2.4 Buchberger's criterion |
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25 | (3) |
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§2.5 Buchberger's algorithm |
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28 | (1) |
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§2.6 Reduced Grobner bases |
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29 | (4) |
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30 | (3) |
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Chapter 3 First applications |
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33 | (18) |
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§3.1 Elimination of variables |
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33 | (1) |
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33 | (1) |
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3.1.2 The Elimination Theorem |
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34 | (1) |
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§3.2 Applications to operations on ideals |
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34 | (8) |
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3.2.1 Intersection of ideals |
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34 | (1) |
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35 | (1) |
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3.2.3 Saturation and radical membership |
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36 | (1) |
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3.2.4 K -algebra homomorphisms |
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37 | (3) |
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40 | (2) |
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§3.3 Zero dimensional ideals |
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42 | (4) |
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§3.4 Ideals of initial forms |
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46 | (5) |
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48 | (3) |
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Chapter 4 Grobner bases for modules |
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51 | (32) |
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51 | (2) |
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§4.2 Monomial orders and initial modules |
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53 | (3) |
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§4.3 The division algorithm and Buchberger's criterion and algorithm for modules |
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56 | (2) |
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58 | (25) |
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4.4.1 How to compute syzygy modules |
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58 | (5) |
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4.4.2 Systems of linear equations over the polynomial ring |
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63 | (3) |
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66 | (2) |
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4.4.4 Graded rings and modules |
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68 | (2) |
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4.4.5 Graded free resolutions |
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70 | (3) |
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4.4.6 Numerical data arising from graded resolutions |
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73 | (3) |
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76 | (4) |
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80 | (3) |
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Chapter 5 Grobner bases of toric ideals |
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83 | (16) |
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§5.1 Semigroup rings and toric ideals |
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83 | (4) |
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§5.2 Grobner bases of toric ideals |
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87 | (1) |
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§5.3 Simplicial complexes and squarefree monomial ideals |
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88 | (3) |
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§5.4 Normal semigroup rings |
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91 | (3) |
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§5.5 Edge rings associated with bipartite graphs |
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94 | (5) |
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97 | (2) |
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Chapter 6 Selected applications in commutative algebra and combinatorics |
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99 | (58) |
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99 | (6) |
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§6.2 Sortable sets of monomials |
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105 | (5) |
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§6.3 Generalized Hibi rings |
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110 | (3) |
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§6.4 Grobner bases for Rees rings |
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113 | (4) |
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6.4.1 The l-exchange property |
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113 | (2) |
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6.4.2 The Rees ring of generalized Hibi ideals |
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115 | (2) |
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§6.5 Determinantal ideals |
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117 | (10) |
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6.5.1 Determinantal ideals and their initial ideals |
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117 | (4) |
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6.5.2 The initial complex of a determinantal ideal |
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121 | (6) |
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§6.6 Sagbi bases and the coordinate ring of Grassmannians |
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127 | (8) |
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127 | (3) |
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6.6.2 The coordinate ring of Grassmannians |
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130 | (5) |
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§6.7 Binomial edge ideals |
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135 | (5) |
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§6.8 Connectedness of contingency tables |
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140 | (17) |
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6.8.1 Contingency tables and the X2-statistics |
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140 | (1) |
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141 | (3) |
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6.8.3 Contingency tables of shape 2 x n |
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144 | (8) |
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152 | (5) |
| Bibliography |
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157 | (4) |
| Index |
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161 | |
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