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Random polynomials. (English) Zbl 0615.60058
Probability and Mathematical Statistics. Orlando etc.: Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers). XV, 206 p. Cloth: $ 49.50; Paper: $ 29.95 (1986).
A polynomial with random coefficients is called a random polynomial. In fact many problems in applied mathematical sciences lead to random polynomials. For instance, in the study of difference and differential equations with random coefficients, in the spectral theory of random matrices, in the study of approximate solution of operator equations, in the study of polynomial regression equation in the method of least squares, in the analysis of capital and investment in mathematical economics, in the study of filtering problems, in the study of minimum phase in statistical communication theory etc., the study of random polynomials is the subject matter of interest and investigation. The book under review aims at providing a critical account about random polynomials.
This book contains eight chapters. Chapter 1 gives a brief account about the development of random polynomials. The authors have introduced the concept of random polynomials and studied their basic properties in Chapter 2. This chapter also contains a very short and elegant proof due to Kannan for the measurability of zeros of a random polynomial. Further it is proved that all realizations of a random algebraic polynomial are continuous. It is interesting to note that the martingale property of random polynomials is also proved.
Chapter 3 discusses random matrices and their associated random characteristic polynomials. The usefulness of Newton’s formula and companion matrices to random algebraic polynomials is presented. Chapter 4 deals with the number and expected number of real zeros of random algebraic polynomials. An in-depth survey of available results is neatly presented. This chapter contains the Kac-Rice formula for the average number of real zeros of a random algebraic polynomial with complex random coefficients. Several particular cases are discussed.
In chapter 5, the authors have discussed the number of real zeros of a random trigonometric polynomial and the average number of real zeros for hyperbolic and orthogonal polynomials. They have presented several computer generated results to illustrate the theory. Chapter 6 contains the asymptotic estimate of the variance of the number of real zeros of a random algebraic polynomial when the coefficients are dependent normal random variables. This chapter also contains some computational results based on the theoretical estimates and on computer generated samples of random algebraic polynomials.
A very important and difficult problem in the study of random polynomials is the determination of the probability distribution of the zeros of a random polynomial. Explicit results on this problem of determining the distribution function of solutions of random linear and quadratic equations are presented. There are several graphs based on the computer generated numerical results presented at the end of chapter 7.
Chapter 8 contains the limiting behaviour of (i) the random measures like the number of zeros in a Borel set of a complex plane, (ii) the zeros of random algebraic polynomials, (iii) the product of polynomials, (iv) random companion matrices and (v) sum of random companion matrices. There are three computer programs in the Appendix to calculate various statistical characteristics of random polynomials.
Each chapter in this book has a good collection of appropriate and important references. This book contains almost all available research materials on this topic. It contains all new ideas of random polynomials, original proofs and new proofs at some places. In conclusion, it is a worthwhile addition to the literature and it paves a way for further research in mathematics.
Reviewer: V.Thangaraj

MSC:
60H25 Random operators and equations (aspects of stochastic analysis)
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30B20 Random power series in one complex variable