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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.

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Sums of random polynomials with differing degrees
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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.
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Bibliographic Information
  • Isabelle Kraus
  • Affiliation: Department of Mathematics, University of Colorado, Campus Box 395, Boulder, Colorado 80309-0395
  • MR Author ID: 1265976
  • Email: isabelle.kraus@colorado.edu
  • 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
  • Email: sean.d.orourke@colorado.edu
  • Received by editor(s): July 25, 2022
  • Received by editor(s) in revised form: October 12, 2023
  • Published electronically: March 20, 2024
  • Additional Notes: The second author was supported in part by NSF grants DMS-2137623 and DMS-2246624.
    The third author was supported in part by NSF grant DMS-1810500.
  • © Copyright 2024 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 377 (2024), 3325-3355
  • MSC (2020): Primary 30C15, 60B10
  • DOI: https://doi.org/10.1090/tran/9128