A note on the expectation of zeros of random harmonic polynomials: The Kac model

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

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

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.

Li and Wei (Proc Am Math Soc 137:195–204, 2009) studied the density of zeros of Gaussian harmonic polynomials with independent Gaussian coefficients. They derived a formula for the expected number of zeros of random harmonic polynomials as well as asymptotics for the case that the polynomials are drawn from the Kostlan ensemble. In this paper we extend their work to cover the case that the polynomials are drawn from the Weyl ensemble by deriving asymptotics for this class of harmonic polynomials.

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.

Joseph L. Walsh: Selected papers

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.

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.

We obtain exact analytical expressions for correlations between real zeros of the Kac random polynomial. We show that the zeros in the interval (−1, 1) are asymptotically independent of the zeros outside of this interval, and that the straightened zeros have the same limit-translation-invariant correlations. Then we calculate the correlations between the straightened zeros of theO(1) random polynomial.

In this note, we find the distibution of the number of real zeros of a random polynomial. We also derive a formula for the expected number of complex zeros lying in a given domain of the complex plane. Bibliography: 7 titles.