Elimination Methods (Texts & Monographs in Symbolic Computation) - Brossura

Wang, D.

Brossura

ISBN 10: 3211832416 ISBN 13: 9783211832417

Casa editrice: Springer, 2000

Riassunto

This book provides a systematic and uniform presentation of elimination methods and the underlying theories, along the central line of decomposing arbitrary systems of polynomials into triangular systems of various kinds. Highlighting methods based on triangular sets, the book also covers the theory and techniques of resultants and Gröbner bases. The methods and their efficiency are illustrated by fully worked out examples and their applications to selected problems such as from polynomial ideal theory, automated theorem proving in geometry and the qualitative study of differential equations. The reader will find the formally described algorithms ready for immediate implementation and applicable to many other problems. Suitable as a graduate text, this book offers an indispensable reference for everyone interested in mathematical computation, computer algebra (software), and systems of algebraic equations.

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Recensione

"... the algorithms are presented in a ‘crescendo’ sequence, that is, the versions are improved step by step with the introduction of new techniques and concepts, which allows the readers to follow the construction of the ideas; furthermore, this style makes it a very good textbook ... it is a very good text to have on hand for anyone interested in the theories and techniques of elimination methods.” American Mathematical Society "... This book, therefore, is not only topical, but is also the first book that contains all major currently available algorithmic approaches on elimination theory in one conceptional frame ... The book is well suited both as a textbook and as a research reference for the state of the art in algorithmic elimination theory. Theorems and proofs are presented in such a way that the correctness of the algorithms based on them can easily be verified. Well-chosen examples make it easy to understand the concepts and the methods. The main contents of this book - altough not all details - should enter standard curricula on algebra." Association for Computing Machinery

Contenuti

Polynomial arithmetic and zeros.- 1.1 Polynomials.- 1.2 Greatest common divisor, pseudo-division, and polynomial remainder sequences.- 1.3 Resultants and subresultants.- 1.4 Field extension and factorization.- 1.5 Zeros and ideals.- 1.6 Hilbert’s Nullstellensatz.- Zero decomposition of polynomial systems.- 2.1 Triangular systems.- 2.2 Characteristic-set-based algorithm.- 2.3 Seidenberg’s algorithm refined.- 2.4 Subresultant-based algorithm.- Projection and simple systems.- 3.1 Projection.- 3.2 Zero decomposition with projection.- 3.3 Decomposition into simple systems.- 3.4 Properties of simple systems.- Irreducible zero decomposition.- 4.1 Irreducibility of triangular sets.- 4.2 Decomposition into irreducible triangular systems.- 4.3 Properties of irreducible triangular systems.- 4.4 Irreducible simple systems.- Various elimination algorithms.- 5.1 Regular systems.- 5.2 Canonical triangular sets.- 5.3 Gröbner bases.- 5.4 Resultant elimination.- Computational algebraic geometry and polynomial-ideal theory.- 6.1 Dimension.- 6.2 Decomposition of algebraic varieties.- 6.3 Ideal and radical ideal membership.- 6.4 Primary decomposition of ideals.- Applications.- 7.1 Solving polynomial systems.- 7.2 Automated geometry theorem proving.- 7.3 Automatic derivation of unknown relations.- 7.4 Other geometric applications.- 7.5 Algebraic factorization.- 7.6 Center conditions for certain differential systems.- Bibliographic notes.- References.

Dettagli dell'edizione

• Editore: Springer
• Data di pubblicazione: 2000
• Lingua: Inglese
• ISBN 10: 3211832416
• ISBN 13: 9783211832417
• Rilegatura: Copertina flessibile
• Numero di pagine: 264
• Contatto del produttore: non disponibile
• Persona responsabile: Springer Nature Customer Service Center GmbH; ProductSafety@springernature.com; ; Europaplatz 3,69115 Heidelberg, Germany; Heidelberg; 69115; Germania