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An asymptotic expansion for the expected number of real zeros of a random polynomial. (English) Zbl 0656.60062
Let \(f(z)=\sum^{n}_{k=0}a_ kz^ k\) be a random algebraic polynomial of degree n, where \(a_ k\), \(k=0,1,2,...,n\) are independent random variables normally distributed with mean zero and variance 1. Let E N(-\(\infty,\infty)\) be the expected number of real zeros of f(z). Then an asymptotic expansion of E N(-\(\infty,\infty)\) is of the form \[ E N(- \infty,\infty)=(2/\pi)\log (n+1)+\sum^{\infty}_{p=0}Ap(n+1)^{-p}, \] in which \(A_ 0=0\cdot 625735818\), \(A_ 1=0\), \(A_ 2=-0\cdot 24261274\), \(A_ 3=0\), \(A_ 4=-0\cdot 08794067\), \(A_ 5=0\).
Reviewer: A.M.Sambandham

MSC:
60G99 Stochastic processes
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