Computations in Algebraic Geometry with Macaulay 2

ISBN-13: 9783540422303 / Angielski / Twarda / 2001 / 329 str.

D. Eisenbud; D. Grayson; M. Stillman

Opis

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."

Recenzje

"... Fazit: das Buch ist kein Lehrbuch im traditionellen Sinn. Sicherlich ist Teil I eine gelungene Einführung, 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 unabhängig voneinander. ... Ich habe beim Lesen viele interessante Stellen gefunden, die man beim flüchtigen Durchblättern übersehen kann. Man muss sich Zeit nehmen, dann wird der Band wirklich zum Gewinn für alle, die Interesse an Algebraischer Geometrie haben."

S.Müller-Stach, Jahresberichte der DMV 2002, Bd. 104, Heft 4

Spis treści

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.