Algorithms and Computation in Mathematics

By: David Eisenbud (Editor), Daniel R. Grayson (Editor), Mike Stillman (Editor)

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Hardcover | 25 September 2001

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350 Pages

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24.13 x 15.88 x 2.54

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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.

Industry Reviews

"... 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

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You Can Find This Book In

Non-Fiction Mathematics Geometry Algebraic Geometry Medicine Medicine in General Computing & I.T.Business Applications Mathematical & Statistical Software Applied Mathematics Computer Programming & Software Development Databases

Data Capture & Analysis Combinatorics & Graph Theory Computer Science Mathematical Theory of Computation Maths for Computer Scientists Algebra

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Paperback

Published: 22nd September 2011

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Preface	p. v
List of Contributors	p. xv
Introducing Macaulay 2
Ideals, Varieties and Macaulay 2	p. 3
A Curve in AAEne Three-Space	p. 3
Intersecting Our Curve With a Surface	p. 4
Changing the Ambient Polynomial Ring	p. 6
Monomials Under the Staircase	p. 8
Pennies, Nickels, Dimes and Quarters	p. 12
References	p. 15
Projective Geometry and Homological Algebra	p. 17
The Twisted Cubic	p. 18
The Cotangent Bundle of P3	p. 20
The Cotangent Bundle of a Projective Variety	p. 24
Intersections by SerreÆs Method	p. 26
A Mystery Variety in P3	p. 28
How the ôMystery Varietyö was Made	p. 37
References	p. 40
Data Types, Functions, and Programming	p. 41
Basic Data Types	p. 41
Control Structures	p. 44
Input and Output	p. 46
Hash Tables	p. 48
Methods	p. 52
Pointers to the Source Code	p. 53
References	p. 53
Teaching the Geometry of Schemes	p. 55
Distinguished Open Sets	p. 55
Irreducibility	p. 56
Singular Points	p. 58 xii
Fields of De nition	p. 60
Multiplicity	p. 61
Flat Families	p. 62
BezoutÆs Theorem	p. 63
Constructing Blow-ups	p. 64
A Classic Blow-up	p. 65
Fano Schemes	p. 68
References	p. 70
Mathematical Computations
Monomial Ideals	p. 73
The Basics of Monomial Ideals	p. 74
Primary Decomposition	p. 77
Standard Pairs	p. 83
Generic Initial Ideals	p. 89
The Chain Property	p. 95
References	p. 99
From Enumerative Geometry to Solving Systems of Polynomial Equations	p. 101
Introduction	p. 101
Solving Systems of Polynomials	p. 103
Some Enumerative Geometry	p. 112
Schubert Calculus	p. 114
The 12 Lines: Reprise	p. 121
References	p. 128
Resolutions and Cohomology over Complete Intersections	p. 131
Matrix Factorizations	p. 133
Graded Algebras	p. 139
Universal Homotopies	p. 141
Cohomology Operators	p. 145
Computation of Ext Modules	p. 150
Invariants of Modules	p. 157
Invariants of Pairs of Modules	p. 170
Gradings	p. 176
References	p. 177
Algorithms for the Toric Hilbert Scheme	p. 179
Generating Monomial Ideals	p. 182
Polyhedral Geometry	p. 188
Local Equations	p. 193
The Coherent Component of the Toric Hilbert Scheme	p. 199
Fourier-Motzkin Elimination	p. 206
Minimal Presentation of Rings	p. 211
References	p. 213
Sheaf Algorithms Using the Exterior Algebra	p. 215
Introduction	p. 215
Basics of the Bernstein-Gel'fand-Gel'fand Correspondence	p. 218
The Cohomology and the Tate Resolution of a Sheaf	p. 222
Cohomology and Vector Bundles	p. 226
Cohomology and Monads	p. 230
The Beilinson Monad	p. 236
Examples	p. 241
References	p. 247
Needles in a Haystack: Special Varieties via Small Fields	p. 251
How to Make Random Curves up to Genus 14	p. 253
Comparing GreenÆs Conjecture for Curves and Points	p. 263
PfaAEan Calabi-Yau Threefolds in P6	p. 267
References	p. 277
D-modules and Cohomology of Varieties	p. 281
Introduction	p. 282
The Weyl Algebra and Grobner Bases	p. 285
Bernstein-Sato Polynomials and Localization	p. 292
Local Cohomology Computations	p. 304
Implementation, Examples, Questions	p. 313
References	p. 321
Index	p. 325
Table of Contents provided by Publisher. All Rights Reserved.

Algorithms and Computation in Mathematics

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

Other Editions and Formats

Paperback

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