Zeros Of Random Polynomials Research Articles

Zeros of random polynomials and their higher derivatives

We consider zeros of higher derivatives of various random polynomials and show that their limiting empirical measures agree with those of roots of corresponding random polynomials. Examples of random polynomials include those whose roots are given by i.i.d. random variables and those whose zeros are nearly all deterministic except for a fixed number of random roots. As an application, we show that such a phenomenon holds for the random polynomials whose roots follow the distribution of the 2D Coulomb gas ensemble.

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.

Eigenvalues distribution for products of independent spherical ensembles

We consider the product of independent spherical ensembles. By the special structure of eigenvalues as a rotation-invariant determinant point process, we show that the empirical spectral distribution of the product converges, with probability one, to a non-random distribution. And the limiting eigenvalue distribution is a power of spherical law. We also present an interesting correspondence between the eigenvalues of three classes of random matrix ensembles and zeros of random polynomials.

Small Deviations for a Family of Smooth Gaussian Processes

We study the small deviation probabilities of a family of very smooth self-similar Gaussian processes. The canonical process from the family has the same scaling property as standard Brownian motion and plays an important role in the study of zeros of random polynomials.

Large Deviations of Empirical Measures of Zeros of Random Polynomials

Large Deviations of Empirical Measures of Zeros of Random Polynomials

Zeros of random polynomials on $\C^m$

For a regular compact set $K$ in $\mathbb{C}^m$ and a measure $\mu$ on $K$ satisfying the Bernstein-Markov inequality, we consider the ensemble $\mathcal{P}_N$ of polynomials of degree $N$, endowed with the Gaussian probability measure induced by $L^2(\mu)$. We show that for large $N$, the simultaneous zeros of $m$ polynomials in $\mathcal{P}_N$ tend to concentrate around the Silov boundary of $K$; more precisely, their expected distribution is asymptotic to $N^m \mu_{eq}$, where $\mu_{eq}$ is the equilibrium measure of $K$. For the case where $K$ is the unit ball, we give scaling asymptotics for the expected distribution of zeros as $N\to\infty$.\end

Real almost zeros of random polynomials with complex coefficients

We present a simple formula for the expected number of times that a complex‐valued Gaussian stochastic process has a zero imaginary part and the absolute value of its real part is bounded by a constant value M. We show that only some mild conditions on the stochastic process are needed for our formula to remain valid. We further apply this formula to a random algebraic polynomial with complex coefficients. We show how the above expected value in the case of random algebraic polynomials varies for different behaviour of M.

EQUILIBRIUM DISTRIBUTION OF ZEROS OF RANDOM POLYNOMIALS

We consider ensembles of random polynomials of the form p(z) = P N=1 ajPj where {aj} are independent complex normal random variables and where {Pj} are the orthonormal polynomials on the boundary of a bounded simply connected analytic plane domain ⊂ C relative to an analytic weight �(z)|dz| . In the simplest case where is the unit disk and � = 1, so that Pj(z) = z j , it is known that the average distribution of zeros is the uniform measure on S 1 . We show that for any analytic (,�), the zeros of random polynomials almost surely become equidistributed relative to the equilibrium measure on @ as N → ∞. We further show that on the length scale of 1/N, the correlations have a universal scaling limit independent of (,�).

The zeros of random polynomials: further results and applications

We consider the density of the zeros of random polynomials with nonzero mean correlated Gaussian coefficients. We show that for most communication problems the original result by Hammersley (1956) can be expressed in a simpler form. In order to illustrate the usefulness of the presented theory, two applications are considered. First, the zeros of frequency-selective Ricean fading channels are investigated. Second, the zeros of the transfer function corresponding to channel impulse response estimates obtained by a least-squares approach are calculated. For both applications implications for equalizer design are discussed.

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 polynomial g1coshx + 2g2cosh2x + …+ngncoshnx where gj, (j = 1, 2, …, n) are independent normally distributed random variables with mean zero and variance one. It is shown that for sufficiently large n this asymptotic value is (1/π)logn.

Zeros of random polynomials over finite fields

Let be the finite field with q elements (q a prime power), let r 1 and let X1, , Xr be independent indeterminates over . We choose an arbitrary and a d 1 and consider

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