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＜電子ブック＞
Algebraic Complexity Theory / by Peter Bürgisser, Michael Clausen, Mohammad A. Shokrollahi
(Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics. ISSN:21969701 ; 315)

版	1st ed. 1997.
出版者	Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer
出版年	1997
本文言語	英語
大きさ	XXIII, 618 p : online resource
冊子体	Algebraic complexity theory / Peter Burgisser, Michael Clausen, M. Amin Shokrollahi ; with the collaboration of Thomas Lickteig
著者標目	*Bürgisser, Peter author Clausen, Michael author Shokrollahi, Mohammad A author SpringerLink (Online service)
件　名	LCSH:Discrete mathematics LCSH:Computer science LCSH:Mathematical logic LCSH:Algorithms LCSH:Algebraic geometry FREE:Discrete Mathematics FREE:Theory of Computation FREE:Mathematical Logic and Foundations FREE:Algorithms FREE:Algebraic Geometry
一般注記	1. Introduction -- I. Fundamental Algorithms -- 2. Efficient Polynomial Arithmetic -- 3. Efficient Algorithms with Branching -- II. Elementary Lower Bounds -- 4. Models of Computation -- 5. Preconditioning and Transcendence Degree -- 6. The Substitution Method -- 7. Differential Methods -- III. High Degree -- 8. The Degree Bound -- 9. Specific Polynomials which Are Hard to Compute -- 10. Branching and Degree -- 11. Branching and Connectivity -- 12. Additive Complexity -- IV. Low Degree -- 13. Linear Complexity -- 14. Multiplicative and Bilinear Complexity -- 15. Asymptotic Complexity of Matrix Multiplication -- 16. Problems Related to Matrix Multiplication -- 17. Lower Bounds for the Complexity of Algebras -- 18. Rank over Finite Fields and Codes -- 19. Rank of 2-Slice and 3-Slice Tensors -- 20. Typical Tensorial Rank -- V. Complete Problems -- 21. P Versus NP: A Nonuniform Algebraic Analogue -- List of Notation The algorithmic solution of problems has always been one of the major concerns of mathematics. For a long time such solutions were based on an intuitive notion of algorithm. It is only in this century that metamathematical problems have led to the intensive search for a precise and sufficiently general formalization of the notions of computability and algorithm. In the 1930s, a number of quite different concepts for this purpose were pro­ posed, such as Turing machines, WHILE-programs, recursive functions, Markov algorithms, and Thue systems. All these concepts turned out to be equivalent, a fact summarized in Church's thesis, which says that the resulting definitions form an adequate formalization of the intuitive notion of computability. This had and continues to have an enormous effect. First of all, with these notions it has been possible to prove that various problems are algorithmically unsolvable. Among of group these undecidable problems are the halting problem, the word problem theory, the Post correspondence problem, and Hilbert's tenth problem. Secondly, concepts like Turing machines and WHILE-programs had a strong influence on the development of the first computers and programming languages. In the era of digital computers, the question of finding efficient solutions to algorithmically solvable problems has become increasingly important. In addition, the fact that some problems can be solved very efficiently, while others seem to defy all attempts to find an efficient solution, has called for a deeper under­ standing of the intrinsic computational difficulty of problems Accessibility summary: This PDF is not accessible. It is based on scanned pages and does not support features such as screen reader compatibility or described non-text content (images, graphs etc). However, it likely supports searchable and selectable text based on OCR (Optical Character Recognition). Users with accessibility needs may not be able to use this content effectively. Please contact us at accessibilitysupport@springernature.com if you require assistance or an alternative format Inaccessible, or known limited accessibility No reading system accessibility options actively disabled Publisher contact for further accessibility information: accessibilitysupport@springernature.com HTTP:URL=https://doi.org/10.1007/978-3-662-03338-8

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