Computations in Algebraic Geometry with Macaulay 2

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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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.

• Groebner bases
• algebraic geometry
• algorithms
• commutative algebra
• computer algebra system
• symbolic algebra
• syzygies
• combinatorics

• 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

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