Real Zeros Of Random Polynomials Research Articles

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

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.

Correlation Functions of Real Zeros of Random Polynomials

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

New features on real zeros of random polynomials

Let Q n ( x ) = ∑ k = 0 n A k x k be a random algebraic polynomial in which the coefficients A 0 , A 1 , A 2 , … form a sequence of independent normally distributed random variables. In this work we study the behavior of the expected density of real zeros of Q n ( x ) for the case that the variances of the middle coefficients are substantially large, say Var ( A k ) = ρ ( k − n / 2 ) 2 . We find some new and interesting features about the distribution and the expected number of real zeros of such a polynomial for different values of ρ . We also consider the case where the variances of the coefficients are decreasing as Var ( A k ) = e − k 2 / 2 n 7 / 4 , and we show that the asymptotic behavior of the expected number of real zeros of Q n ( x ) is of order n 3 / 8 .

Integral Geometry and Real Zeros of Thue–Morse 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

On real zeros of random polynomials with hyperbolic elements

This paper provides the asymptotic estimate for the expected number of real zeros of a random hyperbolic polynomialg1coshx+2g2cosh2x+…+ngncoshnxwheregj,(j=1,2,…,n)are independent normally distributed random variables with mean zero and variance one. It is shown that for sufficiently largenthis asymptotic value is(1/π)logn.

A lower bound for the variance of real zeros of random 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

Some computations of expected number of real zeros of random polynomials

Some computations of expected number of real zeros of 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 I. Coefficients with Zero Means

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

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

Real Zeros of Random Polynomials. II

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

Real Zeros of Random Polynomials

Real Zeros of Random Polynomials