Average Number of Real Zeros of Random Algebraic Polynomials Defined by the Increments of Fractional Brownian Motion

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

CHAPTER 8 - Convergence and Limit Theorems for Random Polynomials

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

CHAPTER 7 - Distribution of the Zeros of Random Algebraic Polynomials

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

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

New features on real zeros of random 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.

We extend the Kac-Rice formula for the expected number of real zeros of random algebraic polynomials on R1 with R1-valued random coefficients to complex zeros of random algebraic polynomials on C1 with C1-valued random coefficients. Our method directly extends to multivariable cases

CHAPTER 1 - Introduction

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

This paper provides a formula to be used for obtaining the variance of the number of real zeros of random algebraic polynomial . The expected number of real zeros of this type of polynomial is known. An easy modification of this formula leads to a formula for the covariance for the number of real zeros in any two disjoint intervals. Using the latter, we show the covariance of the number of real zeros, in any two disjoint interval that can be obtained. To this end, we assume a normal standard distribution for the coefficients a j 's, j = 0, 1, 2,…, n. Although we give a formula for the variance, the evaluation of the asymptotic value for the variance remains our main task for future work.

CHAPTER 6 - The Variance of the Number of Real Zeros of Random Algebraic Polynomials