On the expected number of real roots of polynomials and exponential sums

The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the Bézout number. A similar result is known for random multi-homogeneous systems, invariant through a product of orthogonal groups. In this note, those results are generalized to certain families of sparse polynomial systems, with no orthogonal invariance assumed.     

On random algebraic polynomials

This paper provides asymptotic estimates for the expected number of real zeros and -level crossings of a random algebraic polynomial of the form , where are independent standard normal random variables and is a constant independent of . It is shown that these asymptotic estimates are much greater than those for algebraic polynomials of the form .

     

Expected number of real roots of random trigonometric polynomials

We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials

$X_n\left(t\right)=u+\frac{1}{\sqrt{n}}\sum _{k=1}^n(A_{k}cos\left(kt\right)+B_{k}sin\left(kt\right))\text{,}t\in [0,2\pi ]\text{,}u\in \mathbb{R}$

whose coefficients $A_k,B_k$, $k\in \mathbb{N}$, are independent identically distributed random variables with zero mean and unit variance. If $N_n[a,b]$ denotes the number of real roots of $X_n$ in an interval $[a,b]?[0,2\pi ]$, we prove that

$\underset{n\to \infty }{lim}\frac{\mathbb{E}{N}_n[a,b]}{n}=\frac{b?a}{\pi \sqrt{3}}exp(?\frac{u^2}{2})\text{.}$

     

On algebraic polynomials with random coefficients

The expected number of real zeros and maxima of the curve representing algebraic polynomial of the form

where , are independent standard normal random variables, are known. In this paper we provide the asymptotic value for the expected number of maxima which occur below a given level. We also show that most of the zero crossings of the curve representing the polynomial are perpendicular to the axis. The results show a significant difference in mathematical behaviour between our polynomial and the random algebraic polynomial of the form which was previously the most studied.

     

Asymptotic behavior of roots of random polynomial equations

We derive several new results on the asymptotic behavior of the roots of random polynomial equations, including conditions under which the distributions of the zeros of certain random polynomials tend to the uniform distribution on the circumference of a circle centered at the origin. We also derive a probabilistic analog of the Cauchy-Hadamand theorem that enables us to obtain the radius of convergence of a random power series.     

Gale duality bounds for roots of polynomials with nonnegative coefficients

We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most d. For this, we interpret the basis polynomials as vector fields in the real plane, and at each point in the plane analyze the combinatorics of the Gale dual vector configuration. This approach permits us to incorporate arbitrary linear equations and inequalities among the coefficients in a unified manner to obtain more precise bounds on the location of roots. We apply our technique to bound the location of roots of Ehrhart and chromatic polynomials. Finally, we give an explanation for the clustering seen in plots of roots of random polynomials.     

On the roots of Wiener polynomials of graphs

The Wiener polynomial of a connected graph $G$ is defined as $W(G;x)=\sum x^{d(u,v)}$, where $d(u,v)$ denotes the distance between $u$ and $v$, and the sum is taken over all unordered pairs of distinct vertices of $G$. We examine the nature and location of the roots of Wiener polynomials of graphs, and in particular trees. We show that while the maximum modulus among all roots of Wiener polynomials of graphs of order $n$ is $\left(\genfrac{}{}{0ex}{}{n}{2}\right)?1$, the maximum modulus among all roots of Wiener polynomials of trees of order $n$ grows linearly in $n$. We prove that the closure of the collection of real roots of Wiener polynomials of all graphs is precisely $(?\infty ,0]$, while in the case of trees, it contains $(?\infty ,?1]$. Finally, we demonstrate that the imaginary parts and (positive) real parts of roots of Wiener polynomials can be arbitrarily large.     

On the location of critical points of polynomials

Given a polynomial of degree and with at least two distinct roots let . For a fixed root we define the quantities and . We also define and to be the corresponding minima of and as runs over . Our main results show that the ratios and are bounded above and below by constants that only depend on the degree of . In particular, we prove that , for any polynomial of degree .

     

样本均值随机加权估计的渐近性质

本文给出了样本均值随机加权估计的条件分布以概率（几乎处处）收敛到正态分布的充要条件为X1属于正态吸收域D（2）（E（X~2_1）＜∞）；并且进一步证明了当样本具有缓慢变化的尾时，样本均值随机加权估计之条件分布以概率收敛到非退化分布．     

The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle

The orthogonal polynomials on the unit circle are defined by the recurrence relation

where for any k0. If we consider n complex numbers and , we can use the previous recurrence relation to define the monic polynomials Φ₀,Φ₁,…,Φ_n. The polynomial Φ_n(z)=Φ_n(z;α₀,…,α_n-2,α_n-1) obtained in this way is called the paraorthogonal polynomial associated to the coefficients α₀,α₁,…,α_n-1.We take α₀,α₁,…,α_n-2 i.i.d. random variables distributed uniformly in a disk of radius r<1 and α_n-1 another random variable independent of the previous ones and distributed uniformly on the unit circle. For any n we will consider the random paraorthogonal polynomial Φ_n(z)=Φ_n(z;α₀,…,α_n-2,α_n-1). The zeros of Φ_n are n random points on the unit circle.We prove that for any the distribution of the zeros of Φ_n in intervals of size near e^iθ is the same as the distribution of n independent random points uniformly distributed on the unit circle (i.e., Poisson). This means that, for large n, there is no local correlation between the zeros of the considered random paraorthogonal polynomials.     

Algebraic polynomials with non-identical random coefficients

There are many known asymptotic estimates for the expected number of real zeros of a random algebraic polynomial The coefficients are mostly assumed to be independent identical normal random variables with mean zero and variance unity. In this case, for all sufficiently large, the above expected value is shown to be . Also, it is known that if the have non-identical variance , then the expected number of real zeros increases to . It is, therefore, natural to assume that for other classes of distributions of the coefficients in which the variance of the coefficients is picked at the middle term, we would also expect a greater number of zeros than . In this work for two different choices of variance for the coefficients we show that this is not the case. Although we show asymptotically that there is some increase in the number of real zeros, they still remain . In fact, so far the case of is the only case that can significantly increase the expected number of real zeros.

     

An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC

Let ${\{{\phi }_i\}}_{i=0}^{\infty }$ be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure μ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say ${\mathbb{E}}_n(\mu )$, of random polynomials

$P_n(z):=\sum _{i=0}^n{\eta }_i{\phi }_i(z),$

where ${\eta }_0,\dots ,{\eta }_n$ are i.i.d. standard Gaussian random variables. When μ is the acrlength measure such polynomials are called Kac polynomials and it was shown by Wilkins that ${\mathbb{E}}_n(|\mathrm{d}\xi |)$ admits an asymptotic expansion of the form

${\mathbb{E}}_n(|\mathrm{d}\xi |)～\frac{2}{\pi }\log ?(n+1)+\sum _{p=0}^{\infty }{A}_p{(n+1)}^{?p}$

(Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where μ is absolutely continuous with respect to arclength measure and its Radon–Nikodym derivative extends to a holomorphic non-vanishing function in some neighborhood of the unit circle. In this case ${\mathbb{E}}_n(\mu )$ admits an analogous expansion with the coefficients $A_p$ depending on the measure μ for $p\ge 1$ (the leading order term and $A_0$ remain the same).     

From random matrices to quasi-periodic Jacobi matrices via orthogonal polynomials

We present an informal review of results on asymptotics of orthogonal polynomials, stressing their spectral aspects and similarity in two cases considered. They are polynomials orthonormal on a finite union of disjoint intervals with respect to the Szegö weight and polynomials orthonormal on with respect to varying weights and having the same union of intervals as the set of oscillations of asymptotics. In both cases we construct double infinite Jacobi matrices with generically quasi-periodic coefficients and show that each of them is an isospectral deformation of another. Related results on asymptotic eigenvalue distribution of a class of random matrices of large size are also shortly discussed.     

New integral mean estimates for polynomials

In this paper we prove someL ^P inequalities for polynomials, wherep is any positive number. They are related to earlier inequalities due to A Zygmund, N G De Bruijn, V V Arestov, etc. A generalization of a polynomial inequality concerning self-inversive polynomials, is also obtained.     

On the roots of the orthogonal polynomials and residual polynomials associated with a conjugate gradient method

In this paper we explore two sets of polynomials, the orthogonal polynomials and the residual polynomials, associated with a preconditioned conjugate gradient iteration for the solution of the linear system Ax = b. In the context of preconditioning by the matrix C, we show that the roots of the orthogonal polynomials, also known as generalized Ritz values, are the eigenvalues of an orthogonal section of the matrix C A while the roots of the residual polynomials, also known as pseudo-Ritz values (or roots of kernel polynomials), are the reciprocals of the eigenvalues of an orthogonal section of the matrix (C A)^?1. When C A is selfadjoint positive definite, this distinction is minimal, but for the indefinite or nonselfadjoint case this distinction becomes important. We use these two sets of roots to form possibly nonconvex regions in the complex plane that describe the spectrum of C A.