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

Samal, G.; Mishra, M. N.

doi:10.1090/S0002-9939-1972-0295411-4

On the lower bound of the number of real roots of a random algebraic equation with infinite variance
HTML articles powered by AMS MathViewer

by G. Samal and M. N. Mishra PDF

Proc. Amer. Math. Soc. 33 (1972), 523-528 Request permission

Abstract:

Let ${N_n}$ be the number of real roots of a random algebraic equation $\Sigma _0^n{\xi _v}{x^v} = 0$, where the coefficients ${\xi _v}$’s are independent random variables with common characteristic function $\exp ( - C|t{|^\alpha })$, C being a positive constant and $\alpha \geqq 1$. It is proved that \[ {N_n} \geqq (\mu \log n)/(\log \log n).\] The measure of the exceptional set tends to zero as n tends to infinity.

References

B. V. Gnedenko and A. N. Kolmogorov, Limit distributions for sums of independent random variables, Addison-Wesley Publishing Co., Inc., Cambridge, Mass., 1954. Translated and annotated by K. L. Chung. With an Appendix by J. L. Doob. MR 0062975

On the number of real roots of a random algebraic equation

35

G. Samal, On the number of real roots of a random algebraic equation, Proc. Cambridge Philos. Soc. 58 (1962), 433–442. MR 139221