Zeroes of random Reinhardt polynomials

الأصل · EN

For a Reinhardt domain Ω with the smooth boundary in Cᵐ⁺¹ and a positive smooth measure μ on the boundary of Ω, we consider the ensemble Pₙ of polynomials of degree N with the Gaussian probability measure γₙ which is induced by L²(∂Ω,dμ). Our aim is to compute scaling limit distribution function and scaling limit pair correlation function between zeros when z∈∂Ω. First of all we apply stationary phase method to the Boutet de Monvel-Sjöstrand theorem to get the asymptotic for the partial szegö kernel, Sₙ(z,z), and then we compute the scaling limit partial szegö kernel in any direction in Cᵐ⁺¹, then by using well-known Kac-Rice formula we compute scaling limit distribution function and scaling limit pair correlation function between zeros.