Algebraic Complexity Theory

1. Low Degree

   1. Problems Related to Matrix Multiplication

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi

      Pages 425-453

   2. Lower Bounds for the Complexity of Algebras

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi

      Pages 455-488

   3. Rank over Finite Fields and Codes

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi

      Pages 489-504

   4. Rank of 2-Slice and 3-Slice Tensors

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi

      Pages 505-520

   5. Typical Tensorial Rank

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi

      Pages 521-540

2. Complete Problems

   1. Front Matter

      Pages 541-541

      PDF

   2. P Versus NP: A Nonuniform Algebraic Analogue

      • Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi

      Pages 543-576

3. Back Matter

   Pages 577-621

   PDF

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.

• Algebra
• Algebraic Problems
• Algorithms
• Computation trees
• Computational Complexity
• Computer
• Graph
• Matrix
• Straight line programs
• algorithm
• complexity
• complexity theory
• computer algebra
• geometry
• combinatorics
• algorithm analysis and problem complexity

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)

• 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

Policies and ethics