Random Polynomials

Preface
Acknowledgments
Chapter 1 Introduction
1.1 Introduction
1.2 Origins of Some Random Algebraic Polynomials
1.3 Some Historical Remarks
1.4 Other Types of Random Polynomials
References
Chapter 2 Random Algebraic Polynomials: Basic Definitions and Properties
2.1 Introduction
2.2 Random Power Series and Random Algebraic Polynomials
2.3 Other Definitions of Random Algebraic Polynomials
2.4 Measurability of the Zeros of a Random Algebraic Polynomial
2.5 Measurability of the Number of Zeros of a Random Algebraic Polynomial
2.6 Some Properties of Random Algebraic Polynomials
References
Chapter 3 Random Matrices and Random Algebraic Polynomials
3.1 Introduction
3.2 Some Examples of Random Matrices
3.3 Random Characteristic Polynomials
3.4 Newton's Formula for Random Algebraic Polynomials
3.5 Random Companion Matrices
References
Chapter 4 The Number and Expected Number of Real Zeros of Random Algebraic Polynomials
4.1 Introduction
4.2 Estimates ?n(?, ?)
4.3 The Expected Number of Real Zeros of Random Algebraic Polynomials
4.4 The Average Number of Zeros of Random Algebraic Polynomials with Complex Coefficients
References
Chapter 5 The Number and Expected Number of Real Zeros of Other Random Polynomials
5.1 Introduction
5.2 The Number and Expected Number of Real Zeros of Trigonometric Polynomials
5.3 The Expected Number of Real Zeros of Random Hyperbolic Polynomials
5.4 The Expected Number of Real Zeros of Random Orthogonal Polynomials
5.5 Numerical Results
References
Chapter 6 The Variance of the Number of Real Zeros of Random Algebraic Polynomials
6.1 Introduction
6.2 The Main Theorem
6.3 Formula for the Variance
6.4 Some Lemmas
6.5 Proof of Theorem 6.2(a)
6.6 Proof of Theorem 6.2(b)
6.7 Some Computational Results
References
Chapter 7 Distributions of the Zeros of Random Algebraic Polynomials
7.1 Introduction
7.2 Distribution of the Real Zeros of Random Linear and Quadratic Equations
7.3 Distribution of the Zeros of a Random Polynomial with Complex Coefficients
7.4 Condensed Distribution of the Zeros of a Random Algebraic Polynomial
7.5 Distribution of the Number of Real Zeros
7.6 Some Numerical Results
7.7 On the Distribution of the Zeros of Random Algebraic Polynomials
References
Chapter 8 Convergence and Limit Theorems for Random Polynomials
8.1 Introduction
8.2 The Limiting Behavior of n-1Nn (B, ?)
8.3 The Limiting Behavior of Fn,k(z, ?) and Nn,k(B, ?)
8.4 Stability of the Zeros of Random Algebraic Polynomials
8.5 Some Limit Theorems for Random Algebraic Polynomials and Random Companion Matrices
References
Appendix Fortran Programs
Index