1999
DOI: 10.1090/s0002-9939-99-04912-6
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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

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Cited by 19 publications
(19 citation statements)
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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%
See 1 more Smart Citation
“…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%
“…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…”
Section: The Proofsmentioning
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%
“…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%