A note on the complex roots of complex random polynomials

Ramponi, Alessandro

doi:10.1016/s0167-7152(99)00007-3

Cited by 16 publications

(9 citation statements)

References 6 publications

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“…This, therefore, will be the case for Theorem 2. We first evaluate the terms on the right-hand side of (8). To this end, we note that if we let t α = p/(1 − p) then the value of π k given in (2) becomes…”

Section: Random Polynomials With Binomial Elementsmentioning

confidence: 99%

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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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Section: Random Polynomials With Binomial Elementsmentioning

confidence: 99%

“…82 K. Farahmand [2] Until recently the above classes of polynomials were most commonly studied, and their various characteristics, such as their level crossings and maxima (minima), were considered. Motivated by some physical applications, stated in Ramponi [8], Edelman and Kostlan [2] introduced polynomials of the form…”

Section: Introductionmentioning

confidence: 99%

Real Zeros of Algebraic Polynomials With Stable Random Coefficients

Farahmand

2008

J. Aust. Math. Soc.

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“…Kac's results were for the mean number of real zeros when the coefficients are independent, identically and normally distributed, and Hammersley gave a substantial, albeit complicated, generalization. Recent activity in this field is described by Friedman (1990), Farahmand (1998) and Ramponi (1999).…”

Section: Applied Probabilitymentioning

confidence: 99%

John Michael Hammersley. 21 March 1920 — 2 May 2004

Grimmett

Welsh²

2007

Biogr. Mems Fell. R. Soc.

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John Hammersley was a pioneer among mathematicians, who defied classification as pure or applied; when introduced to guests at Trinity College, Oxford, he would say he did ‘difficult sums’. He believed passionately in the importance of mathematics with strong links to real–life situations and in a system of mathematical education in which the solution of problems takes precedence over the generation of theory. He will be remembered for his work on percolation theory, subadditive stochastic processes, self–avoiding walks and Monte Carlo methods, and, by those who knew him, for his intellectual integrity and his ability to inspire and to challenge. Quite apart from his extensive research achievements, for which he earned a reputation as an outstanding problem–solver, he was a leader in the movement of the 1950s and 1960s to rethink the content of school mathematics syllabuses.

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“…Physical applications of these polynomials can be found in Ramponi [15]. The mathematical behaviour of P n (x) defined in (1.2) was presented for the first time by Edelman and Kostlan in their interesting work [5], which includes many of the original approaches to the study of random algebraic polynomials.…”

Section: Introductionmentioning

confidence: 99%