Zero distribution of some shift polynomials

Given an entire function f of finite order ρ, let $g(f):={\sum }_{j=1}^k{b}_j(z)f(z+c_j)$ be a shift polynomial of f with small meromorphic coefficients $b_j$ in the sense of $O(r^{\lambda +\epsilon })+S(r,f)$, $\lambda <\rho$. Provided α, β, $b_0$ are similar small meromorphic functions, we consider zero distribution of $f^n{(g(f))}^s?b_0$, resp. of $g(f)?\alpha f^n?\beta$.     

Zero distribution of some difference polynomials

In this paper,suppose that a,c∈C{0},c_j∈C(j=1,2,···,n) are not all zeros and n≥2,and f (z) is a finite order transcendental entire function with Borel finite exceptional value or with infinitely many multiple zeros,the zero distribution of difference polynomials of f (z+c)-afⁿ(z) and f (z)f (z+c₁)···f (z+c_n) are investigated.A number of examples are also presented to show that our results are best possible in a certain sense.     

Zero distribution and division results for exponential polynomials

An exponential polynomial of order q is an entire function of the form

$$g(z) = {P_1}(z){e^{{Q_1}(z)}} + ...{P_k}(z){e^{{Q_k}(z)}},$$

where the coefficients P_j(z),Q_j(z) are polynomials in z such that

$$\max \{ deg({Q_j})\} = q.$$

It is known that the majority of the zeros of a given exponential polynomial are in domains surrounding finitely many critical rays. The shape of these domains is refined by showing that in many cases the domains can approach the critical rays asymptotically. Further, it is known that the zeros of an exponential polynomial are always of bounded multiplicity. A new sufficient condition for the majority of zeros to be simple is found. Finally, a division result for a quotient of two exponential polynomials is proved, generalizing a 1929 result by Ritt in the case q = 1 with constant coefficients. Ritt’s result is closely related to Shapiro’s conjecture that has remained open since 1958.

     

Zero subspaces of polynomials on

We provide two examples of complex homogeneous quadratic polynomials P on Banach spaces of the form ℓ₁(Γ). The first polynomial P has both separable and nonseparable maximal zero subspaces. The second polynomial P has the property that while the index-set Γ is not countable, all zero subspaces of P are separable.     

Zero sets of bivariate Hermite polynomials

We establish various properties for the zero sets of three families of bivariate Hermite polynomials. Special emphasis is given to those bivariate orthogonal polynomials introduced by Hermite by means of a Rodrigues type formula related to a general positive definite quadratic form. For this family we prove that the zero set of the polynomial of total degree n+m$n+m$ consists of exactly n+m$n+m$ disjoint branches and possesses n+m$n+m$ asymptotes. A natural extension of the notion of interlacing is introduced and it is proved that the zero sets of the family under discussion obey this property. The results show that the properties of the zero sets, considered as affine algebraic curves in R²${\mathbb{R}}^2$, are completely different for the three families analyzed.     

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 .

     

On random interpolation polynomials

Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 148–150, July, 1992.     

Zero sets of polynomials in several variables

Let , where n is odd. We show that there is an integer N = N(k, n) such that for every n-homogeneous polynomial there exists a linear subspace , such that P|_X ≡ 0. This quantitative estimate improves on previous work of Birch et al., who studied this problem from an algebraic viewpoint. The topological method of proof presented here also allows us to obtain a partial solution to the Gromov-Milman problem (in dimension two) on an isometric version of a theorem of Dvoretzky. Received: 28 September 2004     

Zero sets of polynomials in finite rings

In a field, of course, ann-th degree polynomial has at mostn zeros. Is there anything like this in rings? We find: ann-th degree polynomial in a finite ringA, ofr elements, that is not zero everywhere inA is non-zero at leastr/2 ⁿ times. This is sharp for alln, even in commutative rings; perhaps also in unitary rings, though examples are lacking beyond the cubic case. However, the best such result for commutative unitary rings (not determined pastn=3) is better; (2/2 ⁿ )r is proved, and the best coefficient is between that and its square root.     

On roots of random polynomials

We study the distribution of the complex roots of random polynomials of degree with i.i.d. coefficients. Using techniques related to Rice's treatment of the real roots question, we derive, under appropriate moment and regularity conditions, an exact formula for the average density of this distribution, which yields appropriate limit average densities. Further, using a different technique, we prove limit distribution results for coefficients in the domain of attraction of the stable law.

     

Zero asymptotic behaviour for orthogonal matrix polynomials

Weak-star asymptotic results are obtained for the zeros of orthogonal matrix polynomials (i.e., the zeros of their determinants) on ℝ from two different assumptions: first from the convergence of matrix coefficients occurring in the three-term recurrence for these polynomials; and, second, from conditions on the generating matrix measure. The matrix analogues of the Chebyshev polynomials of the first kind are also investigated. The research of the first and second authors has been supported by DGICYT ref. PB96-1321-C02-01, and the research of the third author was supported, in part, by the U.S. National Science Foundation under the grant DMS-9501130.     

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.     

Double roots of random littlewood polynomials

We consider random polynomials whose coefficients are independent and uniform on {-1, 1}. We prove that the probability that such a polynomial of degree n has a double root is o(n^-2) when n+1 is not divisible by 4 and asymptotic to \(1/\sqrt 3 \) otherwise. This result is a corollary of a more general theorem that we prove concerning random polynomials with independent, identically distributed coefficients having a distribution which is supported on {-1, 0, 1} and whose largest atom is strictly less than \(\frac{{8\sqrt 3 }}{{\pi {n^2}}}\). In this general case, we prove that the probability of having a double root equals the probability that either -1, 0 or 1 are double roots up to an o(n^-2) factor and we find the asymptotics of the latter probability.     

Limit distributions of random symmetrical polynomials

Conditions are studied which should be imposed on the coefficients of a homogeneous random polynomial of the fourth degree to provide its convergence to some nondegenerate random variable. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part II.     

Zero sets of polynomials: one versus two variables

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Zero sets of polynomials: one versus two variables

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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.