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Computations in Algebraic Geometry with Macaulay 2

  • Textbook
  • © 2002

Overview

Editors:
  1. David Eisenbud

    1. Mathematical Sciences, Research Institute, Berkeley, USA

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    You can also search for this editor in PubMed   Google Scholar

  2. Michael Stillman

    1. Department of Mathematics, Cornell University, Ithaca, USA

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  3. Daniel R. Grayson

    1. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, USA

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  4. Bernd Sturmfels

    1. Department of Mathematics, University of California, Berkeley, USA

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  • Only textbook using Macaulay as a tool
  • Computational algebraic geometry presented in the optimal way by top researchers
  • Includes supplementary material: sn.pub/extras

Part of the book series: Algorithms and Computation in Mathematics (AACIM, volume 8)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xv
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  2. Introducing Macaulay 2

    1. Front Matter

      Pages 1-1
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    2. Ideals, Varieties and Macaulay 2

      • Bernd Sturmfels
      Pages 3-15

    3. Projective Geometry and Homological Algebra

      • David Eisenbud
      Pages 17-40

    4. Data Types, Functions, and Programming

      • Daniel R. Grayson, Michael E. Stillman
      Pages 41-53

    5. Teaching the Geometry of Schemes

      • Gregory G. Smith, Bernd Sturmfels
      Pages 55-70

  3. Mathematical Computations

    1. Front Matter

      Pages 71-71
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    2. Monomial Ideals

      • Serkan Hoşten, Gregory G. Smith
      Pages 73-100

    3. From Enumerative Geometry to Solving Systems of Polynomial Equations

      • Frank Sottile
      Pages 101-129

    4. Resolutions and Cohomology over Complete Intersections

      • Luchezar L. Avramov, Daniel R. Grayson
      Pages 131-178

    5. Algorithms for the Toric Hilbert Scheme

      • Michael Stillman, Bernd Sturmfels, Rekha Thomas
      Pages 179-214

    6. Sheaf Algorithms Using the Exterior Algebra

      • Wolfram Decker, David Eisenbud
      Pages 215-249

    7. Needles in a Haystack: Special Varieties via Small Fields

      • Frank-Olaf Schreyer, Fabio Tonoli
      Pages 251-279

    8. D-modules and Cohomology of Varieties

      • Uli Walther
      Pages 281-323

  4. Back Matter

    Pages 325-329
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Keywords

About this book

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.

Editors and Affiliations

  • Mathematical Sciences, Research Institute, Berkeley, USA

    David Eisenbud

  • Department of Mathematics, Cornell University, Ithaca, USA

    Michael Stillman

  • Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, USA

    Daniel R. Grayson

  • Department of Mathematics, University of California, Berkeley, USA

    Bernd Sturmfels

Bibliographic Information

  • Book Title: Computations in Algebraic Geometry with Macaulay 2

  • Editors: David Eisenbud, Michael Stillman, Daniel R. Grayson, Bernd Sturmfels

  • Series Title: Algorithms and Computation in Mathematics

  • DOI: https://doi.org/10.1007/978-3-662-04851-1

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2002

  • Hardcover ISBN: 978-3-540-42230-3Published: 25 September 2001

  • Softcover ISBN: 978-3-642-07592-6Published: 22 September 2011

  • eBook ISBN: 978-3-662-04851-1Published: 14 March 2013

  • Series ISSN: 1431-1550

  • Edition Number: 1

  • Number of Pages: XV, 329

  • Number of Illustrations: 1 b/w illustrations

  • Topics: Algebraic Geometry, Mathematical Software, Combinatorics, Symbolic and Algebraic Manipulation

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