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Algebraic Complexity Theory

  • Book
  • © 1997

Overview

Authors:
  1. Peter Bürgisser

    1. Institut für Mathematik Abt. Angewandte Mathematik, Universität Zürich-Irchel, Zürich, Switzerland

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  2. Michael Clausen

    1. Institut für Informatik V, Universität Bonn, Bonn, Germany

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  3. Mohammad Amin Shokrollahi

    1. Institut für Informatik V, Universität Bonn, Bonn, Germany
      International Computer Science Institute, Berkeley, USA

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Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 315)

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Table of contents (21 chapters)

  1. Front Matter

    Pages I-XXIII
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  2. Introduction

    1. Introduction

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi
      Pages 1-24

  3. Fundamental Algorithms

    1. Front Matter

      Pages 25-25
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    2. Efficient Polynomial Arithmetic

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi
      Pages 27-59

    3. Efficient Algorithms with Branching

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi
      Pages 61-100

  4. Elementary Lower Bounds

    1. Front Matter

      Pages 101-101
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    2. Models of Computation

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi
      Pages 103-124

    3. Preconditioning and Transcendence Degree

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi
      Pages 125-142

    4. The Substitution Method

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi
      Pages 143-160

    5. Differential Methods

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi
      Pages 161-168

  5. High Degree

    1. Front Matter

      Pages 169-169
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    2. The Degree Bound

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi
      Pages 171-206

    3. Specific Polynomials which Are Hard to Compute

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi
      Pages 207-244

    4. Branching and Degree

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi
      Pages 245-264

    5. Branching and Connectivity

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi
      Pages 265-286

    6. Additive Complexity

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi
      Pages 287-301

  6. Low Degree

    1. Front Matter

      Pages 303-303
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    2. Linear Complexity

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi
      Pages 305-349

    3. Multiplicative and Bilinear Complexity

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi
      Pages 351-374

    4. Asymptotic Complexity of Matrix Multiplication

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi
      Pages 375-423
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Keywords

About this book

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.

Reviews

P. Bürgisser, M. Clausen, M.A. Shokrollahi, and T. Lickteig

Algebraic Complexity Theory

"The book contains interesting exercises and useful bibliographical notes. In short, this is a nice book."—MATHEMATICAL REVIEWS

From the reviews:

"This book is certainly the most complete reference on algebraic complexity theory that is available hitherto. … superb bibliographical and historical notes are given at the end of each chapter. … this book would most certainly make a great textbook for a graduate course on algebraic complexity theory. … In conclusion, any researchers already working in the area should own a copy of this book. … beginners at the graduate level who have been exposed to undergraduate pure mathematics would find this book accessible." (Anthony Widjaja, SIGACT News, Vol. 37 (2), 2006)

Authors and Affiliations

  • Institut für Mathematik Abt. Angewandte Mathematik, Universität Zürich-Irchel, Zürich, Switzerland

    Peter Bürgisser

  • Institut für Informatik V, Universität Bonn, Bonn, Germany

    Michael Clausen, Mohammad Amin Shokrollahi

  • International Computer Science Institute, Berkeley, USA

    Mohammad Amin Shokrollahi

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