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COMPUTATIONAL COMMUTATIVE AND NON-COMMUTATIVE ALGEBRAIC GEOMETRY NATO Science Series A series presenting the results of scientific meetings supported under the NATO Science Programme. The series is published by IOS Press and Springer Science and Business Media in conjunction with the NATO Public Diplomacy Division. Sub-Series I. Life and Behavioural Sciences IOS Press II. Mathematics, Physics and Chemistry Springer Science and Business Media III. Computer and Systems Sciences IOS Press IV. Earth and Environmental Sciences Springer Science and Business Media V. Science and Technology Policy IOS Press The NATO Science Series continues the series of books published formerly as the NATO ASI Series. The NATO Science Programme offers support for collaboration in civil science between scientists of countries of the Euro-Atlantic Partnership Council. The types of scientific meeting generally supported are “Advanced Study Institutes” and “Advanced Research Workshops”, although other types of meeting are supported from time to time. The NATO Science Series collects together the results of these meetings. The meetings are co-organized by scientists from NATO countries and scientists from NATO’s Partner countries – countries of the CIS and Central and Eastern Europe. Advanced Study Institutes are high-level tutorial courses offering in-depth study of latest advances in a field. Advanced Research Workshops are expert meetings aimed at critical assessment of a field, and identification of directions for future action. As a consequence of the restructuring of the NATO Science Programme in 1999, the NATO Science Series has been re-organized and there are currently five sub-series as noted above. Please consult the following web sites for information on previous volumes published in the series, as well as details of earlier sub-series: http://www.nato.int/science http://www.springeronline.nl http://www.iospress.nl http://www.wtv-books.de/nato_pco.htm Series III: Computer and SystemsSciences - Vol. 196 ISSN: 1387-6694 Computational Commutative and Non-Commutative Algebraic Geometry Edited by Svetlana Cojocaru Institute of Mathematics and Computer Science, Chisinau, Moldova Gerhard Pfister Technische Universität Kaiserslautern, Fachbereich Mathematik, Kaiserslautern, Germany Victor Ufnarovski Centre for Mathematical Sciences, Mathematics, Lund Institute of Technology, Lund University, Lund, Sweden Amsterdam• Berlin • Oxford • Tokyo • Washington, DC Published in cooperation with NATO Public Diplomacy Division Proceedings of the NATO Advanced Research Workshop on Computational Commutative and Non-Commutative Algebraic Geometry Chisinau, Moldova 6–11 June 2004 © 2005 IOS Press. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without prior written permission from the publisher. ISBN 1-58603-505-3 Library of Congress Control Number: 2005923358 Publisher IOS Press Nieuwe Hemweg 6B 1013 BG Amsterdam Netherlands fax: +31 20 687 0019 e-mail: [email protected] Distributor in the UK and Ireland Distributor in the USA and Canada IOS Press/Lavis Marketing IOS Press, Inc. 73 Lime Walk 4502 Rachael Manor Drive Headington Fairfax, VA 22032 Oxford OX3 7AD USA England fax: +1 703 323 3668 fax: +44 1865 750079 e-mail: [email protected] LEGAL NOTICE The publisher is not responsible for the use which might be made of the following information. PRINTED IN THE NETHERLANDS Computational Commutative and Non-Commutative Algebraic Geometry v S. Cojocaru et al. (Eds.) IOS Press, 2005 © 2005 IOS Press. All rights reserved. Preface A NATO Advanced Research Workshop “Computational Commutative and Non- Commutative Algebraic Geometry” was held in Chishinau, Moldova from 6 to 11 June, 2004. The financial support was offered by the NATO grant PST.ARW.980392. The workshop was organized by Gerhard Pfister and Svetlana Cojocaru with assistance of members of the Organizing Committee: Jürgen Herzog, Lorenzo Robbiano, Victor Ufnarovski. There were 25 lectures given by the participants and open discussions on the subjects presented. The central theme of this workshop was the interplay between commutative and noncommutative algebraic geometry, with its theoretical and computational aspects. The Scien- tific Program emphasizes current trends in Commutative and Non-Commutative Algebraic Geometry and Algebra. The contributors to this volume review the state of the art and pre- sent new evolution and progress reflecting the topics discussed in the lectures. The volume addresses in the first place researchers and graduate students. The editors would like to thank the contributors to this volume. Thanks are due to the NATO Public Diplomacy Divi- sion for their generous financial support. Special thanks to Tatiana Verlan, who had the task to correct and prepare the manuscript. December 2004, Svetlana Cojocaru, Gerhard Pfister, Victor Ufnarovski This page intentionally left blank vii Contents Preface v Noncommutative Algebraic Geometry and Algebra The Structure of Simp<∞ (A) for Finitely Generated k-algebras A 3 O. Laudal On Preimages of Ideals in Certain Non–commutative Algebras 44 V. Levandovskyy Commutative Algebraic Geometry and Algebra Maximal Cohen-Macaulay Modules and Stable Vector Bundles 65 C. Baciu On the Contact with the Polar 74 F. Delgado and F. Hernando On Some Generalization of Schubert’s Varieties 79 Y.A. Drozd and P.S. Kolomiets Lifting an Ideal from a Tight Bourbaki Sequence and Maximal Cohen–Macaulay Modules 90 V. Ene and D. Popescu Some Applications of Resolution of Singularities from the Practical Point of View 104 A. Frühbis-Krüger and G. Pfister Finite Free Resolutions 118 J. Herzog Computing the Integral Closure of an Ideal Using its Rees Algebra 145 T. Hirsch Modular Strata of Deformation Functors 156 B. Martin and T. Hirsch Algebraic Dependence of Polynomials After O. Perron and Some Applications 167 A. Płoski A Remark on MCM-modules of Rank One Over a Smooth Plane Cubic 174 V. Vuletescu viii Computer Algebra Problems in Interaction with the Computer Algebra System BERGMAN 185 J. Backelin, S. Cojocaru, A. Colesnicov, L. Malahova and V. Ufnarovski Involutive Algorithms for Computing Gröbner Bases 199 V.P. Gerdt Idealization of Modules in Computer Algebra 226 M. Kreuzer and L. Robbiano Algorithmic Problems in SAGBI Basis Calculations 244 D. Ronnovius and V. Ufnarovski Topology and Differential Equations On Group Topologies on an Abelian Group Preceding One Another 251 V. Arnautov and K. Filippov Commutative and Non-Commutative Algebras in the Study of Differential Equations 268 M. Popa and E. Starus Projective Planes and Quasigroups About the Description of All Projective Planes of the Prime Power Order 289 E. Kuznetsov On Simple n-ary Medial Quasigroups 305 V. Shcherbacov Author Index 325 Noncommutative Algebraic Geometry and Algebra This page intentionally left blank