CHAPTER 5 - The Number and Expected Number of Real Zeros of Other Random Polynomials

On the Expected Number of Real Zeros of Random Polynomials I. Coefficients with Zero Means

On the Expected Number of Real Zeros of Random Polynomials. II. Coefficients With Non-Zero Means

The study of random polynomials has a long and rich history. This paper studies random algebraic polynomials $$P_n(x) = a_0 + a_1 x + \cdots + a_{n-1} x^{n-1}$$ where the coefficients $$(a_k)$$ are correlated random variables taken as the increments $$X(k+1) - X(k)$$ , $$k\in \mathbb {N}$$ , of a fractional Brownian motion X of Hurst index $$0< H < 1$$ . This reduces to the classical setting of independent coefficients for $$H = 1/2$$ . We obtain that the average number of the real zeros of $$P_n(x)$$ is $$\sim K_H \log n$$ , for large n, where $$K_H = (1 + 2 \sqrt{H(1-H)})/\pi $$ [a generalisation of a classical result obtained by Kac (Bull Am Math Soc 49:314–320, 1943)]. Unexpectedly, the parameter H affects only the number of positive zeros, and the number of real zeros of the polynomials corresponding to fractional Brownian motions of indexes H and $$1-H$$ is essentially the same. The limit case $$H = 0$$ presents some particularities: the average number of positive zeros converges to a constant. These results shed some light on the nature of fractional Brownian motion, on the one hand, and on the behaviour of real zeros of random polynomials of dependent coefficients, on the other hand.

CHAPTER 1 - Introduction

CHAPTER 8 - Convergence and Limit Theorems for Random Polynomials

This paper provides an asymptotic estimate for the expected number of level crossings of a trigonometric polynomial TN(θ)=∑j=0N−1{αN−jcos(j+1/2)θ+βN−jsin(j+1/2)θ}, where αj and βj, j=0,1,2,…, N−1, are sequences of independent identically distributed normal standard random variables. This type of random polynomial is produced in the study of random algebraic polynomials with complex variables and complex random coefficients, with a self-reciprocal property. We establish the relation between this type of random algebraic polynomials and the above random trigonometric polynomials, and we show that the required level crossings have the functionality form of cos(N+θ/2). We also discuss the relationship which exists and can be explored further between our random polynomials and random matrix theory.

Let be a random id algebraic polynomial where {ai} is a sequence of independent identically distributed ( iid ) standard normal random variables. In this paper we have obtained a lower bound for the variance of the number of real zeros .of the random algebraic polynomials Qn(x). We have shown that the bound is for sufficiently large n. Our estimate is times that of Maslova (1974). We have also presented a graph and a comparison table illustrating the values of variances

New features on real zeros of random polynomials

CHAPTER 7 - Distribution of the Zeros of Random Algebraic Polynomials

We study the average number of intersecting points of a given curve with random hyperplanes in an n-dimensional Euclidean space. As noticed by A. Edelman and E. Kostlan, this problem is closely linked to finding the average number of real zeros of random polynomials. They showed that a real polynomial of degree n has on average log n + O(1) real zeros (M. Kac's theorem). This result leads us to the following problem: given a real sequence (α k ) k ∊N, study the average where ρ(fn) is the number of real zeros of fn(X) = α0+α1X+ … + αnXn. We give theoretical results for the Thue—Morse polynomials and numerical evidence for other polvnomials

CHAPTER 2 - Random Algebraic Polynomials: Basic Definitions and Properties

Proceedings of the London Mathematical SocietyVolume s3-18, Issue 2 p. 308-314 Articles Real Zeros of Random Polynomials. II B. F. Logan, B. F. Logan Bell Telephone Laboratories, Murray Hill, New JerseySearch for more papers by this authorL. A. Shepp, L. A. Shepp Bell Telephone Laboratories, Murray Hill, New JerseySearch for more papers by this author B. F. Logan, B. F. Logan Bell Telephone Laboratories, Murray Hill, New JerseySearch for more papers by this authorL. A. Shepp, L. A. Shepp Bell Telephone Laboratories, Murray Hill, New JerseySearch for more papers by this author First published: April 1968 https://doi.org/10.1112/plms/s3-18.2.308Citations: 27AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volumes3-18, Issue2April 1968Pages 308-314 RelatedInformation

An explicit formula for the correlation functions of real zeros of a random polynomial with arbitrary independent continuously distributed coefficients is derived.

CHAPTER 4 - The Number and Expected Number of Real Zeros of Random Algebraic Polynomials

The average number of real zeros of random n degree real polynomials is well known since M. Kac’s seminal article of 1943 [12] which states that it is logn + O(1). Some fifty years later, A. Edelman and E. Kostlan found a beautiful geometrical proof which allowed them to give many other related results [10]. Using their method we discuss the average number of real zeros of random real polynomials $$ \sum\limits_{j = 0}^n {A_j X^j } $$ where the A j ’s are independent Gaussian variables with mean 0 and with variance $$ \sigma ^2 \left( {A_j } \right) = \left( {\begin{array}{*{20}c} n j \end{array} } \right)n^{ - \beta j} $$ where β ∈ ℝ is a given parameter. The average number of real zeros in the interval (a, b) is shown to be $$ E\left( {n;a,b} \right) = \frac{1} {\pi }\sqrt n \left( {Arc\tan \frac{b} {{n^{\beta /2} }} - Arc\tan \frac{a} {{n^{\beta /2} }}} \right). $$ While discussing special polynomials we are led to show that under general conditions, polynomials of the type $$ \sum\limits_{i = 1}^k {A_i \left( X \right)\left( {a_i X + b_i } \right)^n } $$ have at most O(1) real zeros as n increases to infinity.