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Computational Methods in Commutative Algebra and Algebraic Geometry [Kõva köide]

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  • Formaat: Hardback, 405 pages, bibliography, index
  • Sari: Algorithms and Computation in Mathematics Vol 2
  • Ilmumisaeg: 31-Oct-1997
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3540605207
  • ISBN-13: 9783540605201
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  • Kõva köide
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Computational Methods in Commutative Algebra and Algebraic GeometrySuurem pilt
  • Formaat: Hardback, 405 pages, bibliography, index
  • Sari: Algorithms and Computation in Mathematics Vol 2
  • Ilmumisaeg: 31-Oct-1997
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3540605207
  • ISBN-13: 9783540605201
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783540605201.html
Märksõnad:
This ACM volume in computational algebra deals with methods and techniques to tackle problems that can be represented by data structures which are essentially matrices with polynomial entries, mediated by the disciplines of commutative algebra and algebraic geometry. It relates discoveries by a growing, interdisciplinary, group of researchers in the past decade. It highlights the use of advanced techniques to bring down the cost of computation. The book includes concrete algorithms written in MACAULAY. It is intended for advanced students and researchers with interests both in algebra and computation. Many parts of it can be read by anyone with a basic abstract algebra course.
Introduction 1(6)
1. Fundamental Algorithms 7(20) 1.1 Grobner Basics 8(4) 1.2 Division Algorithms 12(5) 1.3 Computation of Syzygies 17(4) 1.4 Hilbert Functions 21(4) 1.5 Computer Algebra Systems 25(2)
2. Toolkit 27(34) 2.1 Elimination Techniques 28(5) 2.2 Rings of Endomorphisms 33(2) 2.3 Noether Normalization 35(4) 2.4 Fitting Ideals 39(4) 2.5 Finite and Quasi-Finite Morphisms 43(3) 2.6 Flat Morphisms 46(9) 2.7 Cohen-Macaulay Algebras 55(6)
3. Principles of Primary Decomposition 61(36) 3.1 Associated Primes and Irreducible Decomposition 63(10) 3.2 Equidimensional Decomposition of an Ideal 73(6) 3.3 Equidimensional Decomposition Without Exts 79(2) 3.4 Mixed Primary Decomposition 81(4) 3.5 Elements of Factorizers 85(12)
4. Computing in Artin Algebras 97(24) 4.1 Structure of Artin Algebras 98(5) 4.2 Zero-Dimensional Ideals 103(4) 4.3 Idempotents versus Primary Decomposition 107(2) 4.4 Decomposition via Sampling 109(5) 4.5 Root Finders 114(7)
5. Nullstellensatze 121(22) 5.1 Radicals via Elimination 122(2) 5.2 Modules of Differentials and Jacobian Ideals 124(4) 5.3 Generic Socles 128(2) 5.4 Explicit Nullstellensatze 130(5) 5.5 Finding Regular Sequences 135(4) 5.6 Top Radical and Upper Jacobians 139(4)
6. Integral Closure 143(38) 6.1 Integrally Closed Rings 145(3) 6.2 Multiplication Rings 148(4) 6.3 S(2)-ification of an Affine Ring 152(8) 6.4 Desingularization in Codimension One 160(5) 6.5 Discriminants and Multipliers 165(3) 6.6 Integral Closure of an Ideal 168(7) 6.7 Integral Closure of a Morphism 175(6)
7. Ideal Transforms and Rings of Invariants 181(28) 7.1 Divisorial Properties of Ideal Transforms 182(3) 7.2 Equations of Blowup Algebras 185(9) 7.3 Subrings 194(7) 7.4 Rings of Invariants 201(8)
8. Computation of Cohomology 209(8) David Eisenbud 8.1 Eyeballing 210(2) 8.2 Local Duality 212(2) 8.3 Approximation 214(3)
9. Degrees of Complexity of a Graded Module 217(54) 9.1 Degrees of Modules 220(14) 9.2 Index of Nilpotency 234(5) 9.3 Qualitative Aspects of Noether Normalization 239(14) 9.4 Homological Degrees of a Module 253(10) 9.5 Complexity Bounds in Local Rings 263(8) Appendix 271(110) A. A Primer on Commutative Algebra 271(60) A.1 Noetherian Rings 271(7) A.2 Krull Dimension 278(6) A.3 Graded Algebras 284(3) A.4 Integral Extensions 287(7) A.5 Finitely Generated Algebras over Fields 294(5) A.6 The Method of Syzygies 299(12) A.7 Cohen-Macaulay Rings and Modules 311(8) A.8 Local Cohomology 319(7) A.9 Linkage Theory 326(5) B. Hilbert Functions 331(24) Jurgen Herzog B.1 G-Graded Rings and G-Filtrations 331(4) B.2 The Study of R via gr(F)(R) 335(5) B.3 The Hilbert-Samuel Function 340(4) B.4 Hilbert Functions, Resolutions and Local Cohomology 344(3) B.5 Lexsegment Ideals and Macaulay Theorem 347(3) B.6 The Theorems of Green and Gotzmann 350(5) C. Using Macaulay 2 355(26) David Eisenbud Daniel Grayson Michael Stillman C.1 Elementary Uses of Macaulay 2 356(14) C.2 Local Cohomology of Graded Modules 370(5) C.3 Cohomology of a Coherent Sheaf 375(6) Bibliography 381(10) Index 391