Sums of random polynomials with independent roots

TL;DR

This paper investigates the asymptotic distribution of zeros of sums of independent random polynomials with roots drawn from specified measures, revealing a connection to free probability theory.

Contribution

It provides the limiting distribution of zeros for sums of independent random polynomials as degrees grow, linking it to free additive convolution.

Findings

Limiting zero distribution described by maximum of logarithmic potentials.

Extension to sums of multiple polynomials with fixed number of terms.

Connection established between polynomial zeros and free probability theory.

Abstract

We consider the zeros of the sum of independent random polynomials as their degrees tend to infinity. Namely, let $p$ and $q$ be two independent random polynomials of degree $n$, whose roots are chosen independently from the probability measures $\mu$ and $\nu$ in the complex plane, respectively. We compute the limiting distribution for the zeros of the sum $p+q$ as $n$ tends to infinity. The limiting distribution can be described by its logarithmic potential, which we show is the pointwise maximum of the logarithmic potentials of $\mu$ and $\nu$. More generally, we consider the sum of $m$ independent degree $n$ random polynomials when $m$ is fixed and $n$ tends to infinity. Our results can be viewed as describing a version of the free additive convolution from free probability theory for zeros of polynomials.