Random Polynomials

PrefaceAcknowledgmentsChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 4 The Number and Expected Number of Real Zeros of Random Algebraic Polynomials 4.1 Introduction 4.2 Estimates Νn(Β, ω) 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 8 Convergence and Limit Theorems for Random Polynomials 8.1 Introduction 8.2 The Limiting Behavior of n-1Nn (B, ώ) 8.3 The Limiting Behavior of Fn,k(z, ω) and Nn,k(B, ώ) 8.4 Stability of the Zeros of Random Algebraic Polynomials 8.5 Some Limit Theorems for Random Algebraic Polynomials and Random Companion Matrices ReferencesAppendix Fortran ProgramsIndex