An Approximate Formula for the Expected Number of Real Zeros of a Random Polynomial

We study the expected number of real zeros of an algebraic polynomial of degree n, whose coefficients are independent Cauchy distributed random variables. We present a formula for the expected number, which has the advantage of being easy to use numerically. This approach shows that the error term involved in the asymptotic estimate for the expected number of real zeros with this class of distribution for the coefficients is O(1).