In this fourth and final volume the author extends Buchberger's Algorithm in three different directions. First, he extends the theory to group rings and other Ore-like extensions, and provides an operative scheme that allows one to set a Buchberger theory over any effective associative ring. Second, he covers similar extensions as tools for discussing parametric polynomial systems, the notion of SAGBI-bases, Groebner bases over invariant rings and Hironaka's theory. Finally, Mora shows how Hilbert's followers - notably Janet, Gunther and Macaulay - anticipated Buchberger's ideas and discusses the most promising recent alternatives by Gerdt (involutive bases) and Faugere (F4 and F5). This comprehensive treatment in four volumes is a significant contribution to algorithmic commutative algebra that will be essential reading for algebraists and algebraic geometers.

Coding theory and cryptography allow secure and reliable data transmission, which is at the heart of modern communication. Nowadays, it is hard to find an electronic device without some code inside. Grobner bases have emerged as the main tool in computational algebra, permitting numerous applications, both in theoretical contexts and in practical situations.

This book is the first book ever giving a comprehensive overview on the application of commutative algebra to coding theory and cryptography. For example, all important properties of algebraic/geometric coding systems (including encoding, construction, decoding, list decoding) are individually analysed, reporting all significant approaches appeared in the literature. Also, stream ciphers, PK cryptography, symmetric cryptography and Polly Cracker systems deserve each a separate chapter, where all the relevant literature is reported and compared. While many short notes hint at new exciting directions, the reader will find that all chapters fit nicely within a unified notation."

The second volume of this comprehensive treatise focusses on Buchberger theory and its application to the algorithmic view of commutative algebra. In distinction to other works, the presentation here is based on the intrinsic linear algebra structure of Groebner bases, and thus elementary considerations lead easily to the state-of-the-art in issues of implementation. The same language describes the applications of Groebner technology to the central problems of commutative algebra. The book can be also used as a reference on elementary ideal theory and a source for the state-of-the-art in its algorithmization. Aiming to provide a complete survey on Groebner bases and their applications, the author also includes advanced aspects of Buchberger theory, such as the complexity of the algorithm, Galligo's theorem, the optimality of degrevlex, the Gianni-Kalkbrener theorem, the FGLM algorithm, and so on. Thus it will be essential for all workers in commutative algebra, computational algebra and algebraic geometry.

With the advent of computers, theoretical studies and solution methods for polynomial equations have changed dramatically. Many classical results can be more usefully recast within a different framework which in turn lends itself to further theoretical development tuned to computation. This first book in a trilogy is devoted to the new approach. It is a handbook covering the classical theory of finding roots of a univariate polynomial, emphasizing computational aspects, especially the representation and manipulation of algebraic numbers, enlarged by more recent representations like the Duval Model and the Thom Codification. Mora aims to show that solving a polynomial equation really means finding algorithms that help one manipulate roots rather than simply computing them; to that end he also surveys algorithms for factorizing univariate polynomials.

This book constitutes the strictly refereed proceedings of the 12th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-12, held in Toulouse, France, June 1997.
The 27 revised full papers presented were carefully selected by the program committee for inclusion in the volume. The papers address a broad range of current issues in coding theory and computer algebra spanning polynomials, factorization, commutative algebra, real geometry, group theory, etc. on the mathematical side as well as software systems, telecommunication, complexity theory, compression, signal processing, etc. on the computer science and engineering side.

This book constitutes the proceedings of the 11th International Conference on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-11, held in Paris, France in July 1995.
The volume presents five invited papers and 32 full revised research papers selected from a total of 68 submissions; it is focussed on research directed to the exploitation of algebraic techniques and methodologies for the application in coding and computer algebra. Among the topics covered are coding, cryptoloy, communication, factorization of polynomials, Gr bner bases, computer algebra, algebraic algorithms, symbolic computation, algebraic manipulation.

This volume is the proceedings of the 10th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 10), held in Puerto Rico, May 1993. The aim of the AAECC meetings is to attract high-level research papers and to encourage cross-fertilization among different areas which share the use of algebraic methods and techniques for applications in the sciences of computing, communications, and engineering. The AAECC symposia are mainly devoted to research in coding theory and computer algebra. The theoryof error-correcting codes deals with the transmission of information in the presence of noise. Coding is the systematic use of redundancy in theformation of the messages to be sent so as to enable the recovery of the information present originally after it has been corrupted by (not too much)noise. Computer algebra is devoted to the investigation of algorithms, computational methods, software systems and computer languages, oriented to scientific computations performed on exact and often symbolic data, by manipulating formal expressions by means of the algebraic rules they satisfy. Questions of complexity and cryptography are naturally linked with both coding theory and computer algebra and represent an important share of the area covered by AAECC.

The AAECC conferences focus on the algebraic aspects of modern computer science, which includes the most up-to-date and advanced topics. The topic of error-correcting codes is one where theory and implementation are unifiedinto a subject both of mathematical beauty and of practical importance. Algebraic algorithms are not only interesting theoretically but also important in computer and communication engineering and many other fields. This volume contains the proceedings of the 9th AAECC conference, held in New Orleans, LA, in October 1991. Researchers from Europe, America, Japan and other regions of the world presented papers at the conference. The papers present new results of recent theoretical and application-oriented research in the field.

In 1988, for the first time, the two international conferences AAECC-6 and ISSAC'88 (International Symposium on Symbolic and Algebraic Computation, see Lecture Notes in Computer Science 358) have taken place as a Joint Conference in Rome, July 4-8, 1988. The topics of the two conferences are in fact widely related to each other and the Joint Conference presented a good occasion for the two research communities to meet and share scientific experiences and results. The proceedings of the AAECC-6 are included in this volume. The main topics are: Applied Algebra, Theory and Application of Error-Correcting Codes, Cryptography, Complexity, Algebra Based Methods and Applications in Symbolic Computing and Computer Algebra, and Algebraic Methods and Applications for Advanced Information Processing. Twelve invited papers on subjects of common interest for the two conferences are divided between this volume and the succeeding "Lecture Notes" volume devoted to ISSACC'88. The proceedings of the 5th conference are published as Vol. 356 of the "Lecture Notes in Computer " "Science."

Coding theory and cryptography allow secure and reliable data transmission, which is at the heart of modern communication. Nowadays, it is hard to find an electronic device without some code inside. Grobner bases have emerged as the main tool in computational algebra, permitting numerous applications, both in theoretical contexts and in practical situations.

This book is the first book ever giving a comprehensive overview on the application of commutative algebra to coding theory and cryptography. For example, all important properties of algebraic/geometric coding systems (including encoding, construction, decoding, list decoding) are individually analysed, reporting all significant approaches appeared in the literature. Also, stream ciphers, PK cryptography, symmetric cryptography and Polly Cracker systems deserve each a separate chapter, where all the relevant literature is reported and compared. While many short notes hint at new exciting directions, the reader will find that all chapters fit nicely within a unified notation."

This third volume of four finishes the program begun in Volume 1 by describing all the most important techniques, mainly based on Groebner bases, which allow one to manipulate the roots of the equation rather than just compute them. The book begins with the 'standard' solutions (Gianni-Kalkbrener Theorem, Stetter Algorithm, Cardinal-Mourrain result) and then moves on to more innovative methods (Lazard triangular sets, Rouillier's Rational Univariate Representation, the TERA Kronecker package). The author also looks at classical results, such as Macaulay's Matrix, and provides a historical survey of elimination, from Bezout to Cayley. This comprehensive treatment in four volumes is a significant contribution to algorithmic commutative algebra that will be essential reading for algebraists and algebraic geometers.