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A note on the complex roots of complex random polynomials

✍ Scribed by A. Ramponi

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
96 KB
Volume
44
Category
Article
ISSN
0167-7152

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✦ Synopsis

By using the technique proposed in

), Trans. Amer. Math. Soc. 349, 2427 -2441]

, we derive an exact formula for the mean number of complex roots of a complex random polynomial. The explicit evaluation of the average density is obtained in the case of multivariate normal coe cients and its correspondence with the early Hammersley result is shown.

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This paper, for any constant K, provides an exact formula for the average density of the distribution of the complex roots of equation q z q z 2 0 1 2 n y 1 Γ„ 4 ny1 Γ„ 4 ny1 qΠΈΠΈΠΈ q z s K where s a q ib and a and b are sequences of independent identically and normally distributed random variables and

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An algorithm is suggested which performs fast calculations of all the roots of a polynomial with maximal computer accuracy using, as the only primary information, the coefficients and the degree of the polynomial. The algorithm combines global as well as local convergences, i.e. it ensures a rapid h