Large level crossings of a random polynomial

We know the expected number of times that a polynomial of degree n with independent random real coefficients asymptotically crosses the level K , when K is any real value such that ( K 2 / n ) → 0 as n → ∞ . The present paper shows that, when K is allowed to be large, this expected number of crossings reduces to only one. The coefficients of the polynomial are assumed to be normally distributed. It is shown that it is sufficient to let K ≥ exp ( n f ) where f is any function of n such that f → ∞ as n → ∞ .