Algebraic polynomials with non-identical random coefficients

Farahmand, Kambiz; Nezakati, Ahmad

doi:10.1090/s0002-9939-04-07501-x

Cited by 10 publications

(2 citation statements)

References 15 publications

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“…But if the mean of the coefficients is nonzero constant then this asymptotic value reduces to half. It is observed in some cases in [5] that the expected number of real zeros EN n (−∞, ∞) increases to √ n, when the coefficients are nonidentical i.e. var(a i ) = n j .…”

Section: Introductionmentioning

confidence: 99%

Real zeros of algebraic polynomials with nonidentical dependent random coefficients

Sahoo¹,

Maharana²

2019

Preprint

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The expected number of real zeros of a random algebraic polynomial a 0 + a 1 x + a 2 x 2 + a 3 x 3 + .... + a n−1 x n−1 depends on the types of random coefficients, with large n. In all works, the coefficients are either independent or dependent but varience of coefficients a i is one. In these cases the exepected number of real zeros is found out to be asymptotic to 2 π logn. In this article, we have considered the negatively correlated dependent random coefficients {a i } n−1 i=0 with varience σ 2i , for σ > 1 and coefficient of correlation ρ ij = −ρ |i−j| , where 0 < ρ < 1 3 . The expected number of real zeros is found to be 2 πσ logn, which is depended on σ.

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Section: Introductionmentioning

confidence: 99%

Real zeros of algebraic polynomials with nonidentical dependent random coefficients

Sahoo¹,

Maharana²

2019

Preprint

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“…The variance of the kth term of P n (t), which is n k , increases to its maximum at the middle term. This initiated another type of random algebraic polynomials with this property in [4]. However, so far the only class of polynomials with E N (−∞, ∞) ∼ √ n is in the form of P n (t) given in (1).…”

Section: Introductionmentioning

confidence: 99%

Real Zeros of Algebraic Polynomials With Stable Random Coefficients

Farahmand

2008

J. Aust. Math. Soc.

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We consider a random algebraic polynomial of the form P n,θ,α (t) = θ 0 ξ 0 + θ 1 ξ 1 t + · · · + θ n ξ n t n , where ξ k , k = 0, 1, 2, . . . , n have identical symmetric stable distribution with index α, 0 < α ≤ 2. First, for a general form of θ k,α ≡ θ k we derive the expected number of real zeros of P n,θ,α (t). We then show that our results can be used for special choices of θ k . In particular, we obtain the above expected number of zeros when θ k = n k 1/2 . The latter generate a polynomial with binomial elements which has recently been of significant interest and has previously been studied only for Gaussian distributed coefficients. We see the effect of α on increasing the expected number of zeros compared with the special case of Gaussian coefficients.

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