On random algebraic polynomials

Farahmand, Kambiz

doi:10.1090/s0002-9939-99-04912-6

Cited by 19 publications

(19 citation statements)

References 18 publications

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“…Random polynomials of the form (1.1) have been studied by many authors, and it is known that, under mild conditions on the distribution of the coefficients of P n , these empirical measures converge to µ T , the normalized arc-length measure, in the weak* topology (or weakly, in the language of probability theory) as n → ∞. For the history of the subject and a list of references we refer the reader to the books [5,12]; we shall discuss some recent results shortly.…”

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confidence: 99%

“…We give several examples of such statements below. We point out that for some special families of coefficients, explicit formulas for the density of zeros in a given set are available, see [5], [12], and the references therein. The following result states that the expected number of zeros in any compact set that does not meet T is of the order O(log n) as n → ∞.…”

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confidence: 99%

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Expected discrepancy for zeros of random algebraic polynomials

Pritsker

Sola

2014

Proc. Amer. Math. Soc.

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Abstract. 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 √ log n/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.

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confidence: 99%

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confidence: 99%

Expected discrepancy for zeros of random algebraic polynomials

Pritsker

Sola

2014

Proc. Amer. Math. Soc.

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“…We will now establish the result given in (5). To this end, for 0 < r < 1, using the above form of the intensity function and then changing to polar coordinates yields…”

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confidence: 74%

“…For other early results in random polynomials, we refer the reader to the books by Bharucha-Reid and Sambandham [3] and Farahmand [5].…”

Section: Introductionmentioning

confidence: 99%

Zeros of Complex Random Polynomials Spanned by Bergman Polynomials

Landi,

Johnson,

Moseley

et al. 2020

Preprint

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We study the expected number of zeros ofwhere {η k } are complex-valued i.i.d standard Gaussian random variables, and {p k (z)} are polynomials orthogonal on the unit disk. When p k (z) = (k + 1)/πz k , k ∈ {0, 1, . . . , n}, we give an explicit formula for the expected number of zeros of P n (z) in a disk of radius r ∈ (0, 1) centered at the origin. From our formula we establish the limiting value of the expected number of zeros, the expected number of zeros in a radially expanding disk, and show that the expected number of zeros in the unit disk is 2n/3. Generalizing our basis functions {p k (z)} to be regular in the sense of Ullman-Stahl-Totik, and that the measure of orthogonality associated to polynomials is absolutely continuous with respect to planar Lebesgue measure, we give the limiting value of the expected number of zeros of P n (z) in a disk of radius r ∈ (0, 1) centered at the origin, and show that asymptotically the expected number of zeros in the unit disk is 2n/3.

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“…The problem of determining the expectation E[N n (a, b)], which was first studied by Bloch and Pólya [13] in the 1930s, has now a long story. Many classical results with numerous references related to the subject can be found in the books by Bharucha-Reid and Sambandham [9], and by Farahmand [25]. Recall that when the coefficients ω j are normally distributed, the expected number of real zeros may be computed by the Kac-Rice formula, with seminal contributions of Kac [28,29] and Rice [39,40].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The number of real zeros of elliptic polynomials

Nguyen¹

2021

Preprint

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The aim of this paper is to further explore the number Nn(a, b) of real zeros of elliptic polynomials of degree n on any interval (a, b), where a and b may depend on n. We first obtain an asymptotic series for the variance of Nn(a, b) and the leading terms of the cumulants and central moments of Nn(a, b) in the large n limit. These terms play an important role in understanding the limiting law as well as the large n behavior of Nn(a, b). As a consequence, we then examine conditions on the interval (a, b) under which Nn(a, b) satisfies a central limit theorem and a strong law of large numbers.

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On random algebraic polynomials

Abstract: Abstract. This paper provides asymptotic estimates for the expected number of real zeros and K-level crossings of a random algebraic polynomial of the form a 0 n−1 0

Cited by 19 publications

References 18 publications

Expected discrepancy for zeros of random algebraic polynomials

Expected discrepancy for zeros of random algebraic polynomials

Zeros of Complex Random Polynomials Spanned by Bergman Polynomials

The number of real zeros of elliptic polynomials

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