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 , … , t n ) (1,\,t,\,\ldots \,,t^{n}) projected onto the surface of the unit sphere, divided by π \pi . 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 Fubini-Study metric.