Expected number of real zeros of a random polynomial with independent identically distributed symmetric long-tailed coefficients

摘要

We show that the expected number of real zeros of the nth degree polynomial with real independent identically distributed coefficients with common characteristic function φ(z) =e-A(ln |1/z|)-a for 0 < |z| < 1 and φ(0) = 1, φ(z) ≡ 0 for 1 [1] |z| < ∞, with 1 < a and A a(a-1), is (a-1)/(a-1/2)log n asymptotically as n→∞.