Computational Methods in Commutative Algebra and Algebraic Geometry: Vol 2 (Algorithms and Computation in Mathematics) - Hardcover

Eisenbud, David; Vasconcelos, Wolmer; Herzog, Jurgen

Hardcover

ISBN 10: 3540605207 ISBN 13: 9783540605201

Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K, 1997

Synopsis

This ACM volume in computational algebra deals with methods and techniques to tackle problems that can be represented by data structures which are essentially matrices with polynomial entries, mediated by the disciplines of commutative algebra and algebraic geometry. It relates discoveries by a growing, interdisciplinary, group of researchers in the past decade. It highlights the use of advanced techniques to bring down the cost of computation. The book includes concrete algorithms written in MACAULAY. It is intended for advanced students and researchers with interests both in algebra and computation. Many parts of it can be read by anyone with a basic abstract algebra course.

About this title

Review

From the reviews:

.,." Many parts of the book can be read by anyone with a basic abstract algebra course. It seems to the reviewer that it was one of the author's intentions to equip students who are interested in computational problems with the necessary algebraic background in pure mathematics and to encourage them to do further research in commutative algebra and algebraic geometry. But researchers will also benefit from this exposition. They will find an up-to-date description of the related research. ... The reviewer recommends the book to anybody who is interested in commutative algebra and algebraic geometry and its computational aspects." (P.Schenzel, Mathematical Reviews 2002)

.,." I would describe this book as a sophisticated notebook, with plenty of suggestions, examples and cross references, reporting on the work of Vasconcelos himself and of many others. ... It is a welcome new and deep exploration into commutative algebra and its relations with algebraic geometry. It is full of results, from simple tricks to more elaborate constructions, all having in common a computational and constructive nature. It will be a gold mine especially for those commutative algebraists who share with Vasconcelos the tast for a homological point of view. ..." (E.Sernesi, Jahresberichte der DMV 1999, Vol. 101, Issue 4)

.,." Das Buch ist ... kein Lehrbuch im herk??mmlichen Verst??ndnis, obwohl f??r Graduiertenseminare bei entsprechenden Vorkenntnissen durchaus geeignet (und vom Autor auf verschiedenen Sommerschulen auch bereits eingesetzt). Es beginnt im wesentlichen dort, wo in den oben genannten grundlegenderen Monographien insbesondere konstruktive Aspekte nichtweiter vertieft werden und ist damit ein Mosaikstein im wirklichen Sinne.

Es ist zugleich keine Monographie im herk??mmlichen Verst??ndnis, die sich an einem engen Thema ersch??pfend abarbeitet, sondern mehr eine Fundgrube von Ideen und Ans??tzen, in denen die verschiedensten Techniken der kommutativen Algebra bis hin zu tiefliegenden homologischen Methoden zusammenspielen. Sie sind, wie vom Autor gewohnt, oft unterschiedlich detailliert ausgearbeitet, beleuchten aber an vielen Stellen Querverbindungen, die man in dieser Konstellation in anderen Arbeiten selten findet.

F??r ein detailliertes Studium insbesondere von Fragestellungen, die in den sp??teren Kapiteln aufgegriffen werden, wird der Leser deshalb kaum um die Konsultation der entsprechenden Zeitschriftenaufs??tze herumkommen, sofern er sie nicht sowieso schon kennt. Als guter Leitfaden f??r ein solches Unterfangen ist das vorliegnede Buch allerdings bestens geeignet und deshalb jedem, der sich ernsthaft mit konstruktiven Methoden in der kommutativen Algebra besch??ftigen m??chte, zu empfehlen. ..." (H.-G. Gr??be, Computeralgebra Rundbrief GI-DMV-GAMM, Nr. 22, M??rz 1998)

From the Back Cover

This book gives an account of recent developments on the interplay between theoretical aspects of commutative algebra and algebraic geometry and computational issues in algebra. A great deal of emphasis is given to the fact that the non-elementary complexity of the underlying fundamental algorithms and data structures (e.g. factorization, Gröbner bases, matrices with polynomial entries) require that the cost of computation be borne largely by theoretical means. The material is focused on the explicit construction of basic objects of algebrogeometric interest - primary decomposition, integral closure, computation of ideal transforms and cohomology, among others. It looks also at various numerical signatures of rings and modules such as those obtained from their Hilbert functions. Another feature is an analysis of nonlinear systems of polynomial equations with the view as to how best deliver the equations to numerical solvers. There are numerous pointers to the current literature, which together with the exercises and a selected set of challenge questions round the text.

From the reviews of the hardcover edition:

"... Many parts of the book can be read by anyone with a basic abstract algebra course. It seems to the reviewer that it was one of the author's intentions to equip students who are interested in computational problems with the necessary algebraic background in pure mathematics and to encourage them to do further research in commutative algebra and algebraic geometry. But researchers will also benefit from this exposition. They will find an up-to-date description of the related research. ... The reviewer recommends the book to anybody who is interested in commutative algebra and algebraic geometry and its computational aspects." (P.Schenzel, Mathematical Reviews 2002)

Edition details

• Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
• Publication date: 1997
• Language: English
• ISBN 10: 3540605207
• ISBN 13: 9783540605201
• Binding: Hardcover
• Number of pages: 405