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Computations in Algebraic Geometry with Macaulay 2 2002 ed. [Kõva köide]

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  • Formaat: Hardback, 329 pages, kõrgus x laius: 235x155 mm, kaal: 713 g, 1 Illustrations, black and white; XV, 329 p. 1 illus., 1 Hardback
  • Sari: Algorithms and Computation in Mathematics 8
  • Ilmumisaeg: 25-Sep-2001
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3540422307
  • ISBN-13: 9783540422303
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Computations in Algebraic Geometry with Macaulay 2 2002 ed.Suurem pilt
  • Formaat: Hardback, 329 pages, kõrgus x laius: 235x155 mm, kaal: 713 g, 1 Illustrations, black and white; XV, 329 p. 1 illus., 1 Hardback
  • Sari: Algorithms and Computation in Mathematics 8
  • Ilmumisaeg: 25-Sep-2001
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3540422307
  • ISBN-13: 9783540422303
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783540422303.html
Märksõnad:
Systems of polynomial equations arise throughout mathematics, science, and engineering. Algebraic geometry provides powerful theoretical techniques for studying the qualitative and quantitative features of their solution sets. Re­ cently developed algorithms have made theoretical aspects of the subject accessible to a broad range of mathematicians and scientists. The algorith­ mic approach to the subject has two principal aims: developing new tools for research within mathematics, and providing new tools for modeling and solv­ ing problems that arise in the sciences and engineering. A healthy synergy emerges, as new theorems yield new algorithms and emerging applications lead to new theoretical questions. This book presents algorithmic tools for algebraic geometry and experi­ mental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out. Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this book will encounter Macaulay 2 in the context of concrete applications and practical computations in algebraic geometry. The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems. A wide range of mathematical scientists should find these expositions valuable. This includes both the users of other programs similar to Macaulay 2 (for example, Singular and CoCoA) and those who are not interested in explicit machine computations at all.

Arvustused

"... Fazit: das Buch ist kein Lehrbuch im traditionellen Sinn. Sicherlich ist Teil I eine gelungene Einfuhrung, wenn man schon die elementaren Grundlagen der Algebraischen oder Analytischen Geometrie kennt. In Teil II ist das Buch aber eher wie ein Tagungsband, in dem einzelne Spezialisten ihre Themen vorstellen. Hier kann man sich etwas aussuchen, denn die Artikel sind unabhangig voneinander. ... Ich habe beim Lesen viele interessante Stellen gefunden, die man beim fluchtigen Durchblattern ubersehen kann. Man muss sich Zeit nehmen, dann wird der Band wirklich zum Gewinn fur alle, die Interesse an Algebraischer Geometrie haben." S.Muller-Stach, Jahresberichte der DMV 2002, Bd. 104, Heft 4

Preface v List of Contributors xv Part I Introducing Macaulay 2(323) Ideals, Varieties and Macaulay 2 3(14) Bernd Sturmfels A Curve in Affine Three-Space 3(1) Intersecting Our Curve With a Surface 4(2) Changing the Ambient Polynomial Ring 6(2) Monomials Under the Staircase 8(4) Pennies, Nickels, Dimes and Quarters 12(5) References 15(2) Projective Geometry and Homological Algebra 17(24) David Eisenbud The Twisted Cubic 18(2) The Cotangent Bundle of P3 20(4) The Cotangent Bundle of a Projective Variety 24(2) Intersections by Serres Method 26(2) A Mystery Variety in P3 28(13) Appendix A. How the ``Mystery Variety was Made 37(3) References 40(1) Data Types, Functions, and Programming 41(14) Daniel R. Grayson Michael E. Stillman Basic Data Types 41(3) Control Structures 44(2) Input and Output 46(2) Hash Tables 48(4) Methods 52(1) Pointers to the Source Code 53(2) References 53(2) Teaching the Geometry of Schemes 55(18) Gregory G. Smith Bernd Sturmfels Distinguished Open Sets 55(1) Irreducibility 56(2) Singular Points 58(2) Fields of Definition 60(1) Multiplicity 61(1) Flat Families 62(1) Bezouts Theorem 63(1) Constructing Blow-ups 64(1) A Classic Blow-up 65(3) Fano Schemes 68(5) References 70(3) Part II Mathematical Computations Monomial Ideals 73(28) Serkan Hosten Gregory G. Smith The Basics of Monomial Ideals 74(3) Primary Decomposition 77(6) Standard Pairs 83(6) Generic Initial Ideals 89(6) The Chain Property 95(6) References 99(2) From Enumerative Geometry to Solving Systems of Polynomial Equations 101(30) Frank Sottile Introduction 101(2) Solving Systems of Polynomials 103(9) Some Enumerative Geometry 112(2) Schubert Calculus 114(7) The 12 Lines: Reprise 121(10) References 128(3) Resolutions and Cohomology over Complete Intersections 131(48) Luchezar L. Avramov Daniel R. Grayson Matrix Factorizations 133(6) Graded Algebras 139(2) Universal Homotopies 141(4) Cohomology Operators 145(5) Computation of Ext Modules 150(7) Invariants of Modules 157(13) Invariants of Pairs of Modules 170(9) Appendix A. Gradings 176(1) References 177(2) Algorithms for the Toric Hilbert Scheme 179(36) Michael Stillman Bernd Sturmfels Rekha Thomas Generating Monomial Ideals 182(6) Polyhedral Geometry 188(5) Local Equations 193(6) The Coherent Component of the Toric Hilbert Scheme 199(16) Fourier-Motzkin Elimination 206(5) Minimal Presentation of Rings 211(2) References 213(2) Sheaf Algorithms Using the Exterior Algebra 215(36) Wolfram Decker David Eisenbud Introduction 215(3) Basics of the Bernstein-Gelfand-Gelfand Correspondence 218(4) The Cohomology and the Tate Resolution of a Sheaf 222(4) Cohomology and Vector Bundles 226(4) Cohomology and Monads 230(6) The Beilinson Monad 236(5) Examples 241(10) References 247(4) Needles in a Haystack: Special Varieties via Small Fields 251(30) Frank-Olaf Schreyer Fabio Tonoli How to Make Random Curves up to Genus 14 253(10) Comparing Greens Conjecture for Curves and Points 263(4) Pfaffian Calabi-Yau Threefolds in P6 267(14) References 277(4) D-modules and Cohomology of Varieties 281(44) Uli Walther Introduction 282(3) The Weyl Algebra and Grobner Bases 285(7) Bernstein-Sato Polynomials and Localization 292(12) Local Cohomology Computations 304(9) Implementation, Examples, Questions 313(12) References 321(4) Index 325