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Kreuzer M., Robbiano L. — Computational commutative algebra 1 |
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Предметный указатель |
Affine algebra 134
Affine set 144
Affine space 144
Affine variety 138 144
Algebra 18
Algebra, affine 134
Algebra, automorphism 231
Algebra, fundamental theorem of 137
Algebra, Nitely generated 23
Algebra, presentation 23
Algebraic closure of a field 137
Algebraic element 227
Algebraic relation 23 227
Algebraically closed field 137
Algorithm for solving systems effectively 257
Algorithm, Berlekamp 38
Algorithm, Buchberger 123
Algorithm, division 71
Algorithm, Euclidean 26
Algorithm, extended Buchberger 125
Algorithm, extended Euclidean 27
Algorithm, normal remainder 75
Alternating group 237
Annihilator 166
Annihilator computation 167
Annihilator of a cyclic module 167
Annihilator of a module 166
Antisymmetry 50 54
Associated elements 31
Associated matrix 182
Associated prime 175
Associated projective space 204
Associated toric ideal 222
Automorphism of a polynomial ring 232
Automorphism of an algebra 231
Basis of a module 19
Basis, canonical 20
Berlekamp algorithm 38
binomial 127
Binomial ideal 127
Buchberger triple 131
Buchberger's algorithm 123
Buchberger's algorithm, extended version 125
Buchberger's algorithm, optimization 124 131
Buchberger's Criterion 122 128
Cancellation law 42
Cancellation law, left 42
Cancellation law, right 42
Canonical basis 20
Cauchy — Schwarz inequality 269
Characterization of DegRevLex 65
Characterization of Groebner bases 111 122
Characterization of homogeneous prime ideals 80
Characterization of Lex 63
Characterization of Noetherian modules 112
Characterization of Noetherian monoids 42
Characterization of RevLex 64
Chinese remainder theorem 39 245
CoCoa 11
CoCoA introduction 275
CoCoA programming 283
Coefficient 22
Coefficient, leading 60
Cogenerator 47
Colon ideal 48 166
Colon ideal, computation 167 175 204
Colon module 169
Colon module, computation 170 175
Colon module, using elimination 199
Comaximal ideals 245
Commutative monoid 17
Complete relation 50
Complex of linear maps 194
Component elimination, module 202
Component elimination, ordering 202
Confluent rewrite relation 95
Content 33
Contravariant hom-functor 182
Convex hull 67
Convex set 67
Coprime elements 31
Correlation coefficient 268
Cost compatible term ordering 210
Cost function 209
Covariance 268
Covariant Hom-functor 182
Cyclic group 236
Cyclic module 19
Decomposition into homogeneous components 77
Degree in -graded modules 102
Degree of a polynomial 22
Degree of a spline 157
Degree of a term 22
Degree, compatible term ordering 52
Degree, shifting 78
Degree-lexicographic term ordering 51
Degree-reverse-lexicographic term ordering 51
Dense polynomial 262
Depth of a module 193
Dihedral group 237
Diophantine system 207
disconnected set 142
Division algorithm 71
Division algorithm, implementation 74
Domain, Euclidean 36
Domain, factorial 30
Domain, integral 18
Domain, principal ideal 19
Dual map 190
Dual projective space 205
Dual sequence 190
Effective roots 248
Efficient frontier 269
Efficient portfolio 269
Element, algebraic 227
Element, associated 31
Element, coprime 31
Element, defines a rewrite rule 92
Element, homogeneous 76
Element, irreducible 29
Element, prime 29
Element, reduces to another element 92
Element, reducible 29
Element, squarefree part 31
Element, transcendental 227
Elementary symmetric polynomial 66
Elimination computation 197 202
Elimination ideal 196
Elimination module 48 196 202
Elimination of module components 202
Elimination ordering 52 196 202
Enneper surface 233
Euclidean algorithm 26
Euclidean algorithm, extended 27
Euclidean Domain 36
Evaluation 23
Evaluation homomorphism 23
Exact sequence of Hom-modules 185
Exact sequence, defining syzygies 100
Expected return 268
Explicit membership 152
Ext-module 194
Extended division 214 220
Extended term 208
Factorial domain 30
Factorization 29
Fiber 234
Field 17
Field of algebraic numbers 98
Field, algebraic closure 137
| Field, algebraically closed 137
Field, finite 27
Field, graded 82
Field, having effective roots 248
Field, of definition 116
Field, of fractions 212
Field, perfect 37 247
Finiteness criterion 243
Flattening isomorphism 182
Folium of Descartes 201
FORM 77
Frobenius map 37
Functoriality of the Hom-module 182
Fundamental diagram of Hom-modules 187
Fundamental diagram of syzygy modules 103 121
Fundamental property of term orderings 56
Fundamental theorem of algebra 137
Gaubian elimination 119
Gaubian numbers 36 80
General linear group 236
Generation of leading term modules 88
Grabmannian 206
Graded field 82
Graded free module 78
Graded module 77
Graded residue class module 79
Graded residue class ring 79
Graded ring 76
Graded submodule 79
Grading by 77
Grading by 78
Grading on syzygy module 101
Grading standard 77
Graph 235
Graph, colouring 143
Greatest common divisor 31
Greatest common divisor, computation 165
Groebner basis 111
Groebner basis, characterization 111
Groebner basis, computation 129
Groebner basis, existence 112
Groebner basis, homogeneous 120
Groebner basis, invariance under field extension 116
Groebner basis, minimal 118
Groebner basis, reduced 115 120
Group 17
Group of matrices 236
Group of permutations 237
Group of reflections 237
Group of rotations 237
Group, alternating 237
Group, cyclic 236
Group, dihedral 237
Group, general linear 236
Heron's formula 196
Hilbert's basis theorem 113
Hilbert's Nullstellensatz 140
Hilbert's Nullstellensatz, field-theoretic version 136
Hilbert's Nullstellensatz, strong version 140
Hilbert's Nullstellensatz, weak version 139
Hilbert's syzygy theorem 159
Hom-functor, contravariant 182
Hom-functor, covariant 182
Hom-functor, exactness properties 185
Hom-module 181
Hom-module computation 186
Homogeneous component 77
Homogeneous element 76
Homogeneous form 77
Homogeneous Groebner basis 120
Homogeneous ideal 79
Homogeneous linear map 78
Homogeneous polynomial 77 83
Homogeneous prime ideal 80
Homogeneous system of generators 79
Homogenization 172
Homomorphism algebra 18
Homomorphism characteristic 18
Homomorphism evaluation 23
Homomorphism Frobenius 37
Homomorphism module 18
Homomorphism of graded modules 78
Homomorphism of graded rings 78
Homomorphism ring 18
Homomorphism structural 18
Homomorphism substitution 23 226
Hyperplane 204
Ideal 18
Ideal elimination 196
Ideal lattice 223
Ideal monomial 44
Ideal of algebraic relations 23 227
Ideal quotient 166
Ideal, binomial 127
Ideal, colon 48
Ideal, comaximal 245
Ideal, homogeneous 79
Ideal, irreducible 264
Ideal, maximal 19
Ideal, membership test 114
Ideal, primary 264
Ideal, prime 19
Ideal, principal 19
Ideal, radical 48 139
Ideal, stable 119
Ideal, toric 222
Ideal, vanishing 140
Ideal, zero-dimensional 244
Image of a linear map 178
Image of a morphism 235
Image of an algebra homomorphism 228
Implicitization 227
Increasingly ordered tuple 118
Integer programming 207 222
Integral domain 18
Interreduced tuple 118
Intersection computation 174 203
Intersection of several submodules 164
Intersection of two submodules 162
Intersection, using elimination 199
Invariant polynomial 237
Invariant theory 236
Irrational numbers 42
Irreducible element 29
Irreducible ideal 264
Irreducible w.r.t. a rewrite relation 92
Irredundant system of generators 154
Jacobian conjecture 233
Jacobian determinant 233
Kernel computation 191 227
Kernel of a linear map 178
Kernel of an algebra homomorphism 227
Klein Four group 236
Kronecker product 184
Krull — Azumaya lemma 76
Lagrange interpolation 156
Lagrange multipliers 270
Lattice 223
Lattice, ideal 223
Laurent polynomial 208
Leading coefficient 60
Leading form 102
Leading position 60
Leading power product 60
Leading term 60
Leading term ideal 62
Leading term ideal of a square 123
Leading term, term module 61
Least common multiple 31
Least common multiple, computation 165
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