Computing in Algebraic Geometry: A Quick Start Using Singular

ISBN-13: 9783642067013 / Angielski / Miękka / 2010 / 328 str.

Wolfram Decker; Christoph Lossen

Opis

This book provides a quick access to computational tools for algebraic geometry, the mathematical discipline which handles solution sets of polynomial equations. Originating from a number of intense one week schools taught by the authors, the text is designed so as to provide a step by step introduction which enables the reader to get started with his own computational experiments right away. The authors present the basic concepts and ideas in a compact way.

Recenzje

From the reviews:

"Algebraic geometry generally studies the properties of solution sets of systems of polynomial equations without direct reference to the actual polynomials used in these systems. ... This is especially desirable for classwork where the development of the abstract machinery generally outlasts the patience of the students, except possibly the most motivated ones. ... However, the book can ... be used in an introductory algebraic geometry course where the students will have the advantage of experimenting with examples as their knowledge grows." (A. Sinan Sertöz, Mathematical Reviews, Issue 2007 b)

Spis treści

Introductory Remarks on Computer Algebra.- 1 Basic Notations and Ideas: A Historical Account.- 2 Basic Computational Problems and Their Solution.- 3 An Introduction to SINGULAR.- Practical Session I .- Practical Session II .- 4 Homological Algebra I .- 5 Homological Algebra II .- Practical Session III .- 6 Solving Systems of Polynomial Equations .- 7 Primary Decomposition and Normalization .- Practical Session .- 8 Algorithms for Invariant .- 9 Computing in Local Rings .- Practical Session V .- Appendix A Sheaf Cohomology and Beilinson Monads .- Appendix B Solutions to Exercises .- References .- Index

O autorze

Wolfram Decker is professor of mathematics at the Universität des Saarlandes, Saarbrücken, Germany. His fields of interest are algebraic geometry and computer algebra. From 1996-2004, he was the responsible overall organizer of the schools and conferences of two European networks in algebraic geometry, EuroProj and EAGER. He himself gave courses in a number of international schools on computer algebra methods in algebraic geometry, with theoretical and practical sessions: Zürich (Switzerland, 1994), Cortona (Italy, 1995), Nordfjordeid (Norway, 1999), Roma (Italy, 2001), Villa Hermosa (Mexico, 2002), Allahabad (India, 2003), Torino (Italy, 2004). He has managed several successful projects in computer algebra, involving undergraduate and graduate students, thus making contributions to two major computer algebra systems for algebraic geometers, SINGULAR and MACAULAY II.

Christoph Lossen is assistant professor (C2) of mathematics at the University of Kaiserslautern. His fields of interest are singularity theory and computer algebra. Since 2000, he is a member of the SINGULAR development team. He taught several courses on computer algebra methods with special emphasis on the needs of singularity theory, including international schools at Sao Carlos (Brazil, 2002), Allahabad (India, 2003) and Oberwolfach (Germany, 2003).