How many zeros of a random polynomial are real?

Abstract

We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve (1, t, …, tn) projected onto the surface of the unit sphere, divided by π. The probability density of the real zeros is proportional to how fast this curve is traced out. We then relax Kac’s assumptions by considering a variety of random sums, series, and distributions, and we also illustrate such ideas as integral geometry and the Fuhini-Studv metric. © 1995 American Mathematical Society.