Algebraic Complexity Theory

Transcription

1 Peter Biirgisser Michael Clausen M. Amin Shokrollahi Algebraic Complexity Theory With the Collaboration of Thomas Lickteig With 21 Figures Springer

2 Chapter 1. Introduction Exercises Open Problems Notes 23 Part I. Fundamental Algorithms Chapter 2. Efficient Polynomial Arithmetic Multiplication of Polynomials * Multiplication of Polynomials II * Multiplication of Several Polynomials Multiplication and Inversion of Power Series * Composition of Power Series Exercises Open Problems Notes :..: 58 Chapter 3. Efficient Algorithms with Branching Polynomial Greatest Common Divisors, * Local Analysis of the Knuth-Schonhage Algorithm.\ Evaluation and Interpolation * Fast Point Location in Arrangements of Hyperplanes * Vapnik-Chervonenkis Dimension and Epsilon-Nets Exercises Open Problems Notes 98 Part II. Elementary Lower Bounds Chapter 4. Models of Computation Straight-Line Programs and Complexity Computation Sequences * Autarky Ill

3 XX 4.4* Computation Trees * Computation Trees and Straight-line Programs Exercises Notes 124 Chapter 5. Preconditioning and Transcendence Degree Preconditioning Transcendence Degree * Extension to Linearly Disjoint Fields Exercises Open Problems Notes 142 Chapter 6. The Substitution Method Discussion of Ideas Lower Bounds by the Degree of Linearization * Continued Fractions, Quotients, and Composition Exercises Open Problems Notes 159 Chapter 7. Differential Methods Complexity of Truncated Taylor Series Complexity of Partial Derivatives Exercises Open Problems Notes 168 Part III. High Degree Chapter 8. The Degree Bound A Field Theoretic Version of the Degree Bound Geometric Degree and a Bezout Inequality The Degree Bound Applications * Estimates for the Degree * The Case of a Finite Field Exercises Open Problems Notes 205 Chapter 9. Specific Polynomials which Are Hard to Compute A Generic Computation Polynomials with Algebraic Coefficients Applications 218

4 XXI 9.4* Polynomials with Rapidly Growing Integer Coefficients * Extension to other Complexity Measures Exercises Open Problems Notes 243 Chapter 10. Branching and Degree Computation Trees and the Degree Bound Complexity of the Euclidean Representation * Degree Pattern of the Euclidean Representation Exercises Open Problems Notes 264 Chapter 11. Branching and Connectivity * Estimation of the Number of Connected Components Lower Bounds by the Number of Connected Components Knapsack and Applications to Computational Geometry Exercises Open Problems Notes 283 Chapter 12. Additive Complexity Introduction * Real Roots of Sparse Systems of Equations A Bound on the Additive Complexity Exercises Open Problems -, Notes 301 Part IV. Low Degree Chapter 13. Linear Complexity The Linear Computational Model First Upper and Lower Bounds * A Graph Theoretical Approach * Lower Bounds via Graph Theoretical Methods * Generalized Fourier Transforms Exercises Open Problems Notes 348 Chapter 14. Multiplicative and Bilinear Complexity Multiplicative Complexity of Quadratic Maps The Tensorial Notation 357

5 XXII 14.3 Restriction and Conciseness Other Characterizations of Rank Rank of the Polynomial Multiplication * The Semiring T Exercises Open Problems Notes 373 Chapter 15. Asymptotic Complexity of Matrix Multiplication The Exponent of Matrix Multiplication First Estimates of the Exponent Scalar Restriction and Extension Degeneration and Border Rank The Asymptotic Sum Inequality First Steps Towards the Laser Method * Tight Sets The Laser Method * Partial Matrix Multiplication * Rapid Multiplication of Rectangular Matrices Exercises Open Problems Notes 420 Chapter 16. Problems Related to Matrix Multiplication Exponent of Problems Triangular Inversion Z, /P-decomposition Matrix Inversion and Determinant * Transformation to Echelon Form * The Characteristic Polynomial * Computing a Basis for the Kernel * Orthogonal Basis Transform * Matrix Multiplication and Graph Theory Exercises Open Problems Notes 452 Chapter 17. Lower Bounds for the Complexity of Algebras First Steps Towards Lower Bounds Multiplicative Complexity of Associative Algebras * Multiplicative Complexity of Division Algebras * Commutative Algebras of Minimal Rank Exercises Open Problems Notes 485

6 XXIII Chapter 18. Rank over Finite Fields and Codes Linear Block Codes Linear Codes and Rank Polynomial Multiplication over Finite Fields * Matrix Multiplication over Finite Fields * Rank of Finite Fields Exercises Open Problems Notes 502 Chapter 19. Rank of 2-Slice and 3-Slice Tensors The WeierstraB-Kronecker Theory Rank of 2-Slice Tensors * Rank of 3-Slice Tensors Exercises Notes 519 Chapter 20. Typical Tensorial Rank Geometric Description Upper Bounds on the Typical Rank * Dimension of Configurations in Formats Exercises Open Problems * Appendix: Topological Degeneration Notes 539 Part V. Complete Problems Chapter 21. P Versus NP: A Nonuniform Algebraic Analogue Cook's Versus Valiant's Hypothesis \ p-definability and Expression Size Universality of the Determinant Completeness of the Permanent * The Extended Valiant Hypothesis Exercises Open Problems Notes 574 Bibliography 577 List of Notation 601 Index 609