Correlations between real and complex zeros of a random polynomial

TL;DR

This paper derives an explicit formula for the mixed correlation functions between real and complex zeros of a random polynomial with independent real-valued coefficients, enhancing understanding of their joint distribution.

Contribution

It provides a novel explicit formula for mixed correlation functions between real and complex zeros of random polynomials with independent coefficients.

Findings

Explicit formula for mixed (k,l)-correlation functions

Quantitative description of zero distributions

Improved understanding of real and complex zero interactions

Abstract

Consider a random polynomial $$ G(z):=\xi_0+\xi_1z+\dots+\xi_nz^n,\quad z\in\mathbb{C}, $$ 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 $G_n$.