The complex zeros of random polynomials

Abstract

Mark Kac gave an explicit formula for the expectation of the number, ν n ( Ω ) {\nu _n}(\Omega ) , of zeros of a random polynomial, \[ P n ( z ) = ∑ j = 0 n − 1 η j z j , {P_n}(z) = \sum \limits _{j = 0}^{n - 1} {{\eta _j}{z^j}} , \] in any measurable subset Ω \Omega of the reals. Here, η 0 , … , η n − 1 {\eta _0}, \ldots ,{\eta _{n - 1}} are independent standard normal random variables. In fact, for each n > 1 n > 1 , he obtained an explicit intensity function g n {g_n} for which \[ E ν n ( Ω ) = ∫ Ω g n ( x ) d x . {\mathbf {E}}{\nu _n}(\Omega ) = \int _\Omega {{g_n}(x)\,dx.} \] Here, we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset Ω \Omega of the complex plane C \mathbb {C} . Namely, we show that \[ E ν n ( Ω ) = ∫ Ω h n ( x , y ) d x d y + ∫ Ω ∩ R g n ( x ) d x , {\mathbf {E}}{\nu _n}(\Omega ) = \int _\Omega {{h_n}(x,y)\,dxdy + \int _{\Omega \cap \mathbb {R}} {{g_n}(x)\,dx,} } \] where h n {h_n} is an explicit intensity function. We also study the asymptotics of h n {h_n} showing that for large n n its mass lies close to, and is uniformly distributed around, the unit circle.