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| Chapter 1.Introduction 1.1 Exercises 1.2 Open Problems 1.3 Notes PartⅠ.Fundamental Algorithms Chapter 2. Efficient Polynomial Arithmetic 2.1 Multiplication of Polynomials I 2.2* Multiplication of Polynomials II 2.3* Multiplication of Several Polynomials 2.4 Multiplication and Inversion of Power Series 2.5* Composition of Power Series 2.6 Exercises 2.7 Open Problems 2.8 Notes Chapter 3. Efficient Algorithms with Branching 3.1 Polynomial Greatest Common Divisors 3.2* Local Analysis of the Knuth-Schonhage Algorithm 3.3 Evaluation and Interpolation 3.4* Fast Point Location in Arrangements of Hyperplanes 3.5* Vapnik-Chervonenkis Dimension and Epsilon-Nets 3.6 Exercises 3.7 Open Problems 3.8 Notes PartⅡ.Elementary Lower Bounds Chapter 4. Models of Computation 4.1 Straight-Line Programs and Complexity 4.2 Computation Sequences 4.3* Autarky 4.4* Computation Trees 4.5* Computation Trees and Straight-line Programs 4.6 Exercises 4.7 Notes Chapter 5. Preconditioning and Transcendence Degree 5.1 Preconditioning 5.2 Transcendence Degree 5.3* Extension to Linearly Disjoint Fields 5.4 Exercises 5.5 Open Problems 5.6 Notes Chapter 6. The Substitution Method 6.1 Discussion of Ideas 6.2 Lower Bounds by the Degree of Linearization 6.3* Continued Fractions, Quotients, and Composition 6.4 Exercises 6.5 Open Problems 6.6 Notes Chapter 7. Differential Methods 7.1 Complexity of Truncated Taylor Series 7.2 Complexity of Partial Derivatives 7.3 Exercises 7.4 Open Problems 7.5 Notes Part Ⅲ.High Degree Chapter 8. The Degree Bound 8.1 A Field Theoretic Version of the Degree Bound 8.2 Geometric Degree and a Bezout Inequality 8.3 The Degree Bound 8.4 Applications 8.5* Estimates for the Degree 8.6* The Case of a Finite Field 8.7 Exercises 8.8 Open Problems 8.9 Notes Chapter 9. Specific Polynomials which Are Hard to Compute 9.1 A Genetic Computation 9.2 Polynomials with Algebraic Coefficients 9.3 Applications 9.4* Polynomials with Rapidly Growing Integer Coefficients 9.5* Extension to other Complexity Measures 9.6 Exercises 9.7 Open Problems 9.8 Notes Chapter 10. Branching and Degree 10.1 Computation Trees and the Degree Bound 10.2 Complexity of the Euclidean Representation 10.3* Degree Pattern of the Euclidean Representation 10.4 Exercises 10.5 Open Problems 10.6 Notes Chapter 11. Branching and Connectivity 11.1 Estimation of the Number of Connected Component 11.2 Lower Bounds by the Number of Connected Components 11.3 Knapsack and Applications to Computational Geometry 11.4 Exercises 11.5 Open Problems 11.6 Notes Chapter 12. Additive Complexity 12.1 Introduction 12.2* Real Roots of Sparse Systems of Equations 12.3 A Bound on the Additive Complexity 12.4 Exercises 12.5 Open Problems 12.6 Notes Part Ⅳ.Low Degree Chapter 13. Linear Complexity 13.1 The Linear Computational Model 13.2 First Upper and Lower Bounds 13.3* A Graph Theoretical Approach 13.4* Lower Bounds via Graph Theoretical Methods 13.5* Generalized Fourier Transforms 13.6 Exercises 13.7 Open Problems 13.8 Notes Chapter 14. Multiplicative and Bilinear Complexity 14.1 Multiplicative Complexity of Quadratic Maps 14.2 The Tensorial Notation 14.3 Restriction and Conciseness 14.4 Other Characterizations of Rank 14.5 Rank of the Polynomial Multiplication 14.6 The Semiring T 14.7 Exercises 14.8 Open Problems 14.9 Notes Chapter 15. Asymptotic Complexity of Matrix Multiplication Chapter 16. Problems Related to Matrix Multiplication Chapter 17. Lower Bounds for the Complexity of Algebras Chapter 18. Rank over Finite Fields and Codes Chapter 19. Rank of 2-Slice and 3-Slice Tensors Chapter 20. Typical Tensorial Rank Part Ⅴ.Complete Problems Chapter 21. P Versus NP：A Nonuniform Algebraic Analogue Bibliography List of Notation Index |
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