On the number of real roots of a random algebraic equation

摘要

Let Nn(?‰) be the number of real roots of the random algebraic equation ?￡nv = 0 av??v (?‰)xv = 0, where the ??v(?‰)'s are independent, identically distributed random variables belonging to the domain of attraction of the normal law with mean zero and P{??v(?‰) a‰? 0} > 0; also the av 's are nonzero real numbers such that (kn/tn) = 0(log n) where kn = max0a‰¤va‰¤n |av| and tn = min0a‰¤va‰¤n |av|. It is shown that for any sequence of positive constants (?μn, n a‰￥ 0) satisfying ?μn a?’ 0 and ?μ2nlog n a?’ a?? there is a positive constant ?? so that for all n0 sufficiently large.