### References & Citations

# Mathematics > Probability

# Title: Sums of random polynomials with differing degrees

(Submitted on 16 Oct 2021)

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. As $n \to \infty$, the limiting distribution for the zeros of the sum $p+q$ was by computed by Reddy and the second author [J. Math. Anal. Appl. 495 (2021) 124719]. In this paper, we generalize 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 $q$. Our methods also allow us to consider the sum of $m$ independent random polynomials with differing degrees when $m$ is fixed and the degree of the sum tends to infinity.

Link back to: arXiv, form interface, contact.