J Austral Math Soc Ser A 54 pp86--96, 1993.

On the Number of Real Roots of a Random Algebraic Equation

D. Pratihari, R. K. Panda and B. P. Pattanaik

Abstract

Let N_n(w) be the number of real roots of the random algebraic equation åⁿ_v=0a_vx_v(w)x^v = 0, where the x_v(w)'s are independent, identically distributed random variables belonging to the domain of attraction of the normal law with mean zero and P{x_v(w) ¹ 0} > 0; also the a_v's are nonzero real numbers such that (k_n/t_n) = O(log n) where k_n = max_{0 £ v £ n} |a_v| and t_n = min_{0 £ v £ n} |a_v|. It is shown that for any sequence of positive constants (e_n, n ³ 0) satisfying e_n ® 0 and e_n²log n ® ¥ there is a positive constant m so that

Pr{infn > n0 Nn(w)/log n < en} < m(en0 log n0)-1

for all n₀ sufficiently large.

1991 AMS Subject Classification: 60B99

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Authors

D. Pratihari

R. K. Panda

B. P. Pattanaik

College of Basic Sciences & Humanities, Bhubaneswar - 751 003, Orissa, INDIA.