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 $n \rightarrow \infty$. 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 $K \rightarrow \infty$, as long as $K = O(\sqrt n)$.