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Kreuzer M., Robbiano L. — Computational commutative algebra 1
Kreuzer M., Robbiano L. — Computational commutative algebra 1

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Название: Computational commutative algebra 1

Авторы: Kreuzer M., Robbiano L.

Аннотация:

The main topic of this book is that of Groebner bases and their applications. The main purpose of this book is that of bridging the current gap in the literature between theory and real computation. The book can be used by teachers and students alike as a comprehensive guide to both the theory and the practice of Computational Commutative Algebra. It has been made as self-contained as possible, and thus is ideally suited as a textbook for graduate or advanced undergraduate courses. Numerous applications are described, covering fields as disparate as algebraic geometry and financial markets. To aid a deeper understanding of these applications, there are 44 tutorials aimed at illustrating how the theory can be used in these cases. The computational aspects of the tutorials can be carried out with the computer algebra system CoCoA, an introduction to which appears in an appendix. Besides the tutorials there are plenty of exercises, some of a theoretical nature and others more practical.


Язык: en

Рубрика: Математика/Алгебра/Вычислительная алгебра/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 309

Добавлена в каталог: 12.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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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 $\mathbb{T}^n\langle e_1,...,e_r\rangle$-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 $p^{th}$ 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 $p^{th}$ 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 $\mathbb{T}^n$      77
Grading by $\mathbb{T}^n\langle e_1,...,e_r\rangle$      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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