On the distribution of the number of real zeros of a random polynomial

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

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

CHAPTER 7 - Distribution of the Zeros of Random Algebraic Polynomials

CHAPTER 8 - Convergence and Limit Theorems for Random Polynomials

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

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.

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.

For random coefficients aj and bj we consider a random trigonometric polynomial defined as . The expected number of real zeros of Tn(θ) in the interval (0, 2π) can be easily obtained. In this note we show that this number is in fact . However the variance of the above number is not known. This note presents a method which leads to the asymptotic value for the covariance of the number of real zeros of the above polynomial in intervals (0, π) and (π, 2π). It can be seen that our method in fact remains valid to obtain the result for any two disjoint intervals. The applicability of our method to the classical random trigonometric polynomial, defined as , is also discussed. Tn(θ) has the advantage on Pn(θ) of being stationary, with respect to θ, for which, therefore, a more advanced method developed could be used to yield the results.

We study statistical properties of zeros of random polynomials and random analytic functions associated with the pseudoeuclidean group of symmetries SU(1, 1), by utilizing both analytical and numerical techniques. We first show that zeros of the SU(1, 1) random polynomial of degree N are concentrated in a narrow annulus of the order of N−1 around the unit circle on the complex plane, and we find an explicit formula for the scaled density of the zeros distribution along the radius in the limit N→∞. Our results are supported through various numerical simulations. We then extend results of Hannay(1) and Bleher et al.(2) to derive different formulae for correlations between zeros of the SU(1, 1) random analytic functions, by applying the generalized Kac–Rice formula. We express the correlation functions in terms of some Gaussian integrals, which can be evaluated combinatorially as a finite sum over Feynman diagrams or as a supersymmetric integral. Due to the SU(1, 1) symmetry, the correlation functions depend only on the hyperbolic distances between the points on the unit disk, and we obtain an explicit formula for the two point correlation function. It displays quadratic repulsion at small distances and fast decay of correlations at infinity. In an appendix to the paper we evaluate correlations between the outer zeros |zj|>1 of the SU(1, 1) random polynomial, and we prove that the inner and outer zeros are independent in the limit when the degree of the polynomial goes to infinity.

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.

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

The problem of finding the probability distribution of the number of zeros in some real interval of a random polynomial whose coefficients have a given continuous joint density function is considered. An algorithm which enables one to express this probability as a multiple integral is presented. Formulas for the number of zeros of random quadratic polynomials and random polynomials of higher order, some coefficients of which are non-random and equal to zero, are derived via use of the algorithm. Finally, the applicability of these formulas in numerical calculations is illustrated.

Joint distribution of conjugate algebraic numbers: A random polynomial approach

CHAPTER 2 - Random Algebraic Polynomials: Basic Definitions and Properties

We consider a random self-similar polynomials where the coefficients form a sequence of independent normally distributed random variables. We study the behavior of the expected density of real zeros of these polynomials when the variances of the middle coefficients are substantially larger than the others. Numerical sets show the existence of a phase transition for a critical value of a parameter that defines the variance. We also discuss the case where the variances of the coefficients are decreasing, and obtain the asymptotic behavior of the expected number of real zeros of such polynomials.