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TitleComputations in Algebraic Geometry with Macaulay 2 [electronic resource] / edited by David Eisenbud, Michael Stillman, Daniel R. Grayson, Bernd Sturmfels
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2002
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Descript XV, 329 p. 1 illus. online resource


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. Reยญ cently developed algorithms have made theoretical aspects of the subject accessible to a broad range of mathematicians and scientists. The algorithยญ mic approach to the subject has two principal aims: developing new tools for research within mathematics, and providing new tools for modeling and solvยญ 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 experiยญ 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


I Introducing Macaulay 2 -- Ideals, Varieties and Macaulay 2 -- Projective Geometry and Homological Algebra -- Data Types, Functions, and Programming -- Teaching the Geometry of Schemes -- II Mathematical Computations -- Monomial Ideals -- From Enumerative Geometry to Solving Systems of Polynomial Equations -- Resolutions and Cohomology over Complete Intersections -- Algorithms for the Toric Hilbert Scheme -- Sheaf Algorithms Using the Exterior Algebra -- Needles in a Haystack: Special Varieties via Small Fields -- D-modules and Cohomology of Varieties

Mathematics Computer science -- Mathematics Algebraic geometry Computer software Combinatorics Mathematics Algebraic Geometry Mathematical Software Combinatorics Symbolic and Algebraic Manipulation


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