# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## Sums of random polynomials with differing degreesHTML articles powered by AMS MathViewer

by Isabelle Kraus, Marcus Michelen and Sean O’Rourke
Trans. Amer. Math. Soc. 377 (2024), 3325-3355
DOI: https://doi.org/10.1090/tran/9128
Published electronically: March 20, 2024

## 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 computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021), p. 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.
References
Similar Articles
• Retrieve articles in Transactions of the American Mathematical Society with MSC (2020): 30C15, 60B10
• Retrieve articles in all journals with MSC (2020): 30C15, 60B10
Bibliographic Information
• Isabelle Kraus
• Affiliation: Department of Mathematics, University of Colorado, Campus Box 395, Boulder, Colorado 80309-0395
• MR Author ID: 1265976
• Marcus Michelen
• Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago
• MR Author ID: 1312016
• Email: michelen.math@gmail.com
• Sean O’Rourke
• Affiliation: Department of Mathematics, University of Colorado, Campus Box 395, Boulder, Colorado 80309-0395
• MR Author ID: 897261
• ORCID: 0000-0002-3805-1298