# Random Polynomials

## 1st Edition

### Probability and Mathematical Statistics: A Series of Monographs and Textbooks

**Authors:**A. T. Bharucha-Reid M. Sambandham

**Editors:**Z. W. Brinbaum E. Lukacs

**eBook ISBN:**9781483191461

**Imprint:**Academic Press

**Published Date:**21st April 1986

**Page Count:**222

## Description

Probability and Mathematical Statistics: A Series of Monographs and Textbooks: Random Polynomials focuses on a comprehensive treatment of random algebraic, orthogonal, and trigonometric polynomials.

The publication first offers information on the basic definitions and properties of random algebraic polynomials and random matrices. Discussions focus on Newton's formula for random algebraic polynomials, random characteristic polynomials, measurability of the zeros of a random algebraic polynomial, and random power series and random algebraic polynomials. The text then elaborates on the number and expected number of real zeros of random algebraic polynomials; number and expected number of real zeros of other random polynomials; and variance of the number of real zeros of random algebraic polynomials. Topics include the expected number of real zeros of random orthogonal polynomials and the number and expected number of real zeros of trigonometric polynomials.

The book takes a look at convergence and limit theorems for random polynomials and distribution of the zeros of random algebraic polynomials, including limit theorems for random algebraic polynomials and random companion matrices and distribution of the zeros of random algebraic polynomials.

The publication is a dependable reference for probabilists, statisticians, physicists, engineers, and economists.

## Table of Contents

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 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

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, 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

References

Appendix Fortran Programs

Index

## Details

- No. of pages:
- 222

- Language:
- English

- Copyright:
- © Academic Press 1986

- Published:
- 21st April 1986

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483191461

## About the Authors

### A. T. Bharucha-Reid

### M. Sambandham

## About the Editors

### Z. W. Brinbaum

### E. Lukacs

### Affiliations and Expertise

Bowling Green State University