Correlations between real and complex zeros of a random polynomial

Abstract

Consider a random polynomial $$G(z):={\xi }_0+{\xi }_{1}z+\cdots +{\xi }_n{z}^n,z\in \mathbb{C},$$ where ${\xi }_0,{\xi }_1,\dots ,{\xi }_n$ are independent real-valued random variables with probability density functions $f_0,\dots ,f_n$. We give an explicit formula for the mixed $(k,l)$-correlation function ${\rho }_{k,l}:{\mathbb{R}}^k\times {\mathbb{C}}_{+}^l\to {\mathbb{R}}_{+}$ between $k$ real and $l$ complex zeros of $G_n$.

References in corpus (5)

• Asymptotic distribution of complex zeros of random analytic functions
• On the density function of the distribution of real algebraic numbers
• Real zeroes of random polynomials, II: Descartes' rule of signs and anti-concentration on the symmetric group
• Discriminant and root separation of integral polynomials
• Expected number of real zeros for random Freud orthogonal polynomials