Zeros Of Random Algebraic 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.

Expected discrepancy for zeros of random algebraic polynomials

We study asymptotic clustering of zeros of random polynomials, and show that the expected discrepancy of roots of a polynomial of degree n, with not necessarily independent coefficients, decays like p logn/n. Our proofs rely on discrepancy results for deterministic polynomials, and order statistics of a random variable. We also consider the expected number of zeros lying in certain subsets of the plane, such as circles centered on the unit circumference, and polygons inscribed in the unit circumference.

Covariance of the Number of Real Zeros of a Random Algebraic Polynomial with Binomial Elements

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.

Strong result for real zeros of random algebraic polynomials

An estimate is given for the lower bound of real zeros of random algebraic polynomials whose coefficients are non-identically distributed dependent Gaussian random variables. Moreover, our estimated measure of the exceptional set, which is independent of the degree of the polynomials, tends to zero as the degree of the polynomial tends to infinity.

Real Zeros of Random Algebraic Polynomials

Real Zeros of Random Algebraic Polynomials

An extention of the Kac-Rice formula for the average number of zeros of random algebraic polynomials

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

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

Previous article Next article On the Expected Number of Real Zeros of Random Polynomials. II. Coefficients With Non-Zero MeansI. A. Ibragimov and N. B. MaslovaI. A. Ibragimov and N. B. Maslovahttps://doi.org/10.1137/1116052PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] I. A. Ibragimov and , N. B. Maslova, On the expected number of real zeros of random polynomials, I. Coefficients with zero means, Theory Prob. Applications, 16 (1971), 228–248 10.1137/1116023 0277.60051 LinkGoogle Scholar[2] J. L. Doob, Stochastic processes, John Wiley & Sons Inc., New York, 1953viii+654 MR0058896 0053.26802 Google Scholar[3] I. A. Ibragimov and , Yu. V. Linnik, Independent Stationary Sequences of Random Variables, “Nauka”, Moscow, 1965, (In Russian.) English translation to appear, Wolters–Noordhoff, 1971 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Random trigonometric polynomials: Universality and non-universality of the variance for the number of real rootsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques, Vol. 58, No. 3 | 1 Aug 2022 Cross Ref An asymptotic expansion for the expected number of real zeros of Kac–Geronimus polynomialsRocky Mountain Journal of Mathematics, Vol. 51, No. 4 | 1 Aug 2021 Cross Ref Real roots of random polynomials with coefficients of polynomial growth: a comparison principle and applicationsElectronic Journal of Probability, Vol. 26, No. none | 1 Jan 2021 Cross Ref On Real Zeros of Self-Similar Random Gaussian Polynomials with Decreasing Variances: Apparition of a Phase TransitionBulletin of the Iranian Mathematical Society, Vol. 45, No. 1 | 11 July 2018 Cross Ref Распределение нулей невырожденных функций на коротких отрезкахЧебышевский сборник, Vol. 19, No. 1 | 14 October 2018 Cross Ref Roots of random polynomials with coefficients of polynomial growthThe Annals of Probability, Vol. 46, No. 5 | 1 Sep 2018 Cross Ref Expected Number of Real Zeros of Gaussian Self-Reciprocal Random Algebraic PolynomialsIranian Journal of Science and Technology, Transactions A: Science, Vol. 42, No. 1 | 3 February 2018 Cross Ref Expected Number Of Real Zeros Of a Random Algebraic PolynomialMaterials Today: Proceedings, Vol. 5, No. 1 | 1 Jan 2018 Cross Ref On the number of real roots of random polynomialsCommunications in Contemporary Mathematics, Vol. 18, No. 04 | 1 Aug 2016 Cross Ref CLT for the zeros of classical random trigonometric polynomialsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques, Vol. 52, No. 2 | 1 May 2016 Cross Ref Real roots of random polynomials: expectation and repulsionProceedings of the London Mathematical Society, Vol. 111, No. 6 | 23 November 2015 Cross Ref Local Universality of Zeroes of Random PolynomialsInternational Mathematics Research Notices, Vol. 2015, No. 13 | 6 June 2014 Cross Ref Algebraic polynomials with symmetric random coefficientsRocky Mountain Journal of Mathematics, Vol. 44, No. 2 | 1 Apr 2014 Cross Ref Asymptotics of the variance of the number of real roots of random trigonometric polynomialsScience China Mathematics, Vol. 55, No. 11 | 10 November 2012 Cross Ref Around the circular lawProbability Surveys, Vol. 9, No. none | 1 Jan 2012 Cross Ref Real Zeros of Algebraic Polynomials with Dependent Random CoefficientsStochastic Analysis and Applications, Vol. 28, No. 3 | 15 Apr 2010 Cross Ref Algebraic polynomials with random non-symmetric coefficientsStatistics & Probability Letters, Vol. 78, No. 11 | 1 Aug 2008 Cross Ref An Example of a Random Polynomial with Unusual Zeros BehaviorTheory of Probability & Its Applications, Vol. 50, No. 3 | 15 August 2006AbstractPDF (138 KB)Real zeros of random algebraic polynomials with binomial elementsJournal of Applied Mathematics and Stochastic Analysis, Vol. 2006 | 1 Jan 2006 Cross Ref The expected number of zeros of a random system of $p$-adic polynomialsElectronic Communications in Probability, Vol. 11, No. none | 1 Jan 2006 Cross Ref Large level crossings of random polynomialsStochastic Analysis and Applications, Vol. 20, No. 2 | 15 February 2007 Cross Ref Real zeros of a random hyperbolic polynomialStochastic Analysis and Applications, Vol. 17, No. 4 | 1 Jan 1999 Cross Ref On the expected number of zero crossings of a random algebraic polynomialNonlinear Analysis: Theory, Methods & Applications, Vol. 30, No. 2 | 1 Dec 1997 Cross Ref The level crossings of random polynomialsApplied Mathematics Letters, Vol. 9, No. 1 | 1 Jan 1996 Cross Ref Crossings of a random trigonometric polynomialApplied Mathematics Letters, Vol. 6, No. 1 | 1 Jan 1993 Cross Ref Real zeros of random algebraic polynomialsProceedings of the American Mathematical Society, Vol. 113, No. 4 | 1 January 1991 Cross Ref The growth of the expected number of real zeros of a random polynomialJournal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, Vol. 46, No. 1 | 9 April 2009 Cross Ref The growth of the expected number of real zeros of a random polynomial with dependent coefficientsMathematical Proceedings of the Cambridge Philosophical Society, Vol. 104, No. 3 | 24 October 2008 Cross Ref Volume 16, Issue 3| 1971Theory of Probability & Its Applications History Submitted:10 November 1969Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1116052Article page range:pp. 485-493ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics