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

CHAPTER 8 - Convergence and Limit Theorems for Random Polynomials

CHAPTER 7 - Distribution of the Zeros of Random Algebraic Polynomials

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

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

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.

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

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.

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

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.

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

CHAPTER 1 - Introduction

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.

CHAPTER 2 - Random Algebraic Polynomials: Basic Definitions and Properties