On the lower bound of the number of real roots of a random algebraic equation with infinite variance. II

Samal, G.; Mishra, M. N.

doi:10.1090/S0002-9939-1972-0315785-5

On the lower bound of the number of real roots of a random algebraic equation with infinite variance. II
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by G. Samal and M. N. Mishra PDF

Proc. Amer. Math. Soc. 36 (1972), 557-563 Request permission

Abstract:

Let ${N_n}$ be the number of real roots of a random algebraic equation $\sum \nolimits _{v = 0}^n {{\xi _v}{x^v} = 0}$ where the ${\xi _v}$’s are independent random variables with a common characteristic function \[ \exp ( - C|t{|^\alpha }),\quad \alpha > 1,\] and C, a positive constant. Then for $n > {n_0}$, \[ {N_n} > (\mu \log n)/(\log \log n)\] outside a set of measure at most \[ \mu ’/{\{ \log ((\log {n_0})/(\log \log {n_0}))\} ^{\alpha - 1}}.\]

References

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On the number of real roots of a random algebraic equation

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G. Samal, On the number of real roots of a random algebraic equation, Proc. Cambridge Philos. Soc. 58 (1962), 433–442. MR 139221
G. Samal and M. N. Mishra, On the lower bound of the number of real roots of a random algebraic equation with infinite variance, Proc. Amer. Math. Soc. 33 (1972), 523–528. MR 295411, DOI 10.1090/S0002-9939-1972-0295411-4