This book combines, in a novel and general way, an extensive development of the theory of families of commuting matrices with applications to zero-dimensional commutative rings, primary decompositions and polynomial system solving. It integrates the Linear Algebra of the Third Millennium, developed exclusively here, with classical algorithmic and algebraic techniques. Even the experienced reader will be pleasantly surprised to discover new and unexpected aspects in a variety of subjects including eigenvalues and eigenspaces of linear maps, joint eigenspaces of commuting families of endomorphisms, multiplication maps of zero-dimensional affine algebras, computation of primary decompositions and maximal ideals, and solution of polynomial systems. This book completes a trilogy initiated by the uncharacteristically witty books Computational Commutative Algebra 1 and 2 by the same authors. The material treated here is not available in book form, and much of it is not available at all. The authors continue to present it in their lively and humorous style, interspersing core content with funny quotations and tongue-in-cheek explanations.

This book combines, in a novel and general way, an extensive development of the theory of families of commuting matrices with applications to zero-dimensional commutative rings, primary decompositions and polynomial system solving. It integrates the Linear Algebra of the Third Millennium, developed exclusively here, with classical algorithmic and algebraic techniques. Even the experienced reader will be pleasantly surprised to discover new and unexpected aspects in a variety of subjects including eigenvalues and eigenspaces of linear maps, joint eigenspaces of commuting families of endomorphisms, multiplication maps of zero-dimensional affine algebras, computation of primary decompositions and maximal ideals, and solution of polynomial systems. This book completes a trilogy initiated by the uncharacteristically witty books Computational Commutative Algebra 1 and 2 by the same authors. The material treated here is not available in book form, and much of it is not available at all. The authors continue to present it in their lively and humorous style, interspersing core content with funny quotations and tongue-in-cheek explanations.

Cryptography has become essential as bank transactions, credit card infor-mation, contracts, and sensitive medical information are sent through inse-cure channels. This book is concerned with the mathematical, especially algebraic, aspects of cryptography. It grew out of many courses presented by the authors over the past twenty years at various universities and covers a wide range of topics in mathematical cryptography. It is primarily geared towards graduate students and advanced undergraduates in mathematics and computer science, but may also be of interest to researchers in the area. Besides the classical methods of symmetric and private key encryption, the book treats the mathematics of cryptographic protocols and several unique topics such as Group-Based Cryptography Groebner Basis Methods in Cryptography Lattice-Based Cryptography

This introduction to polynomial rings, Gr bner bases and applications bridges the gap in the literature between theory and actual computation. It details numerous applications, covering fields as disparate as algebraic geometry and financial markets. To aid in a full understanding of these applications, more than 40 tutorials illustrate how the theory can be used. The book also includes many exercises, both theoretical and practical.

"The second volume of the authors' 'Computational commutative algebra'...covers on its 586 pages a wealth of interesting material with several unexpected applications. ... an encyclopedia on computational commutative algebra, a source for lectures on the subject as well as an inspiration for seminars. The text is recommended for all those who want to learn and enjoy an algebraic tool that becomes more and more relevant to different fields of applications." --ZENTRALBLATT MATH

"The second volume of the authors' 'Computational commutative algebra'...covers on its 586 pages a wealth of interesting material with several unexpected applications. ... an encyclopedia on computational commutative algebra, a source for lectures on the subject as well as an inspiration for seminars. The text is recommended for all those who want to learn and enjoy an algebraic tool that becomes more and more relevant to different fields of applications." --ZENTRALBLATT MATH

The main TOPIC of this book is that of Groebner bases and their applications. The main PURPOSE of this book is that of bridging the current gap in the literature between theory and real computation. The book can be used by teachers and students alike as a comprehensive guide to both the theory and the practice of Computational Commutative Algebra. It has been made as self-contained as possible, and thus is ideally suited as a textbook for graduate or advanced undergraduate courses. Numerous applications are described, covering fields as disparate as algebraic geometry and financial markets. To aid a deeper understanding of these applications there are 44 tutorials aimed at illustrating how the theory can be used in these cases. The computational aspects of the tutorials can be carried out with the computer algebra system CoCoA, an introduction to which appears in an appendix. Besides the tutorials there are plenty of exercises, some of a theoretical nature and others more practical.

Die vorliegende Arbeit entstand am Institut fur Bankinformatik an der Universitat Regensburg gemeinnutzige GmbH. Dieses Forschungsinstitut wurde von Prof. Dr. Dieter Bartmann 1993 gegrundet. Gesellschafter sind u. a. die Bayerische Landes- bank, die Bayerische Vereinsbank, die SchmidtBank und das Informatik-Zentrum- Bayern. Eine der Zielrichtungen besteht in der Institutionalisierung einer engen Verbundforschung mit der Praxis uber die Einrichtung entsprechender For- schungsgruppen, Kompetenzcenter genannt. Diese Kompetenzcenter stimmen die Projektinhalte jeweils eng mit den Projektpartnern aus der Praxis ab und kommu- nizieren diesen die Ergebnisse jeweils projektbegleitend uber entsprechende Workshops oder Publikationen. Fur diese Untersuchung massgebend war der Pro- jektplan des Kompetenzcenters Medien (CC Medien), das sich mit der langfristi- gen Entwicklung und Bedeutung der elektronischen Bank-Kunde-Schnittstelle befasst. Das Untersuchungsfeld des Autors im besonderen hat die technologische Entwicklung und Bedeutung elektronischer Bankvertriebswege und ihre Eignung zum Absatz von Bankprodukten zum Inhalt. An dieser Stelle moechte ich Herrn Prof. Bartmann fur die Schaffung dieser in der deutschen wirtschaftswissenschaftlichen Forschungslandschaft herausragenden Rahmenbedingungen ausdrucklich danken. Sie ermoeglichten letztendlich neben dem Entstehen dieser Arbeit ein fruhzeitiges und fortwahrendes Spiegeln der Er- gebnisse an den strategischen Anforderungen der bankbetrieblichen Praxis und damit auch den oft geforderten "Blick uber den rein theoretischen Tellerrand" hin zu neuen Erkenntnissen. Ich bin der UEberzeugung, dass so das Fundament fur eine fruchtbare und kritische Auseinandersetzung mit diesen dadurch "praxistaug- lichen" Resultaten gelegt ist.