On the Average Number of Real Roots of a Random Algebraic Equation

There are many known asymptotic estimates of the expected number of zeros of a polynomial of degree n with independent random coefficients, for n → ∞. The present paper provides an estimate of the expected number of times that such a polynomial assumes the real value K, where K is not necessarily zero. The coefficients are assumed to be normally distributed. It is shown that the results are valid even for K → ∞, as long as K = O(√ n).