Sums of random polynomials with differing degrees

TL;DR

This paper extends previous work on the zeros of sums of random polynomials by analyzing cases where the polynomials have different degrees and characterizing the limiting distribution of zeros based on the measures and degree ratios.

Contribution

It generalizes the limiting distribution results to polynomials with different degrees and provides a complete description for various measure conditions.

Findings

Limiting distribution characterized by the maximum of scaled logarithmic potentials.

Results apply to measures with and without logarithmic moments.

Provides explicit description of zero distributions for diverse measure pairs.

Abstract

Let $\mu$ and $\nu$ be probability measures in the complex plane, and let $p$ and $q$ be independent random polynomials of degree $n$, whose roots are chosen independently from $\mu$ and $\nu$, respectively. Under assumptions on the measures $\mu$ and $\nu$, the limiting distribution for the zeros of the sum $p+q$ was by computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021) 124719] as $n \to \infty$. In this paper, we generalize and extend this result to the case where $p$ and $q$ have different degrees. In this case, the logarithmic potential of the limiting distribution is given by the pointwise maximum of the logarithmic potentials of $\mu$ and $\nu$, scaled by the limiting ratio of the degrees of $p$ and $q$. Additionally, our approach provides a complete description of the limiting distribution for the zeros of $p + q$ for any pair of measures $\mu$ and $\nu$, with different limiting behavior shown in the case when at least one of the measures fails to have a logarithmic moment.