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
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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
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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