Zeros of random Reinhardt polynomials

Abstract

For a strictly pseudoconvex Reinhardt domain with smooth boundary in and a positive smooth measure on the boundary of , we consider the ensemble of polynomials of degree N with the Gaussian probability measure which is induced by . Our aim is to compute the scaling limit distribution function and scaling limit pair correlation function for zeros near a point . First, we apply the stationary phase method to the Boutet de Monvel–Sjöstrand Theorem to get the asymptotic for the scaling limit partial Szegö kernel around . Then by using the Kac–Rice formula, we compute the scaling limit distribution and pair correlation functions.