Zeros of a growing number of derivatives of random polynomials with independent roots
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- by Marcus Michelen and Xuan-Truong Vu;
- Proc. Amer. Math. Soc. 152 (2024), 2683-2696
- DOI: https://doi.org/10.1090/proc/16794
- Published electronically: April 29, 2024
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Abstract:
Let $X_1,X_2,\ldots$ be independent and identically distributed random variables in ${\mathbb {C}}$ chosen from a probability measure $\mu$ and define the random polynomial \begin{align*} P_n(z)=(z-X_1)\ldots (z-X_n)\,. \end{align*} We show that for any sequence $k = k(n)$ satisfying $k \leq \log n / (5 \log \log n)$, the zeros of the $k$th derivative of $P_n$ are asymptotically distributed according to the same measure $\mu$. This extends work of Kabluchko, which proved the $k = 1$ case, as well as Byun, Lee and Reddy [Trans. Amer. Math. Soc. 375, pp. 6311–6335] who proved the fixed $k$ case.References
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Bibliographic Information
- Marcus Michelen
- Affiliation: Department of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, Illinois 60607
- MR Author ID: 1312016
- Email: michelen@uic.edu, michelen.math@gmail.com
- Xuan-Truong Vu
- Affiliation: Department of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, Illinois 60607
- MR Author ID: 1414060
- Email: tvu25@uic.edu, truongvu.math@gmail.com
- Received by editor(s): January 12, 2023
- Received by editor(s) in revised form: August 19, 2023, and November 20, 2023
- Published electronically: April 29, 2024
- Additional Notes: Both authors were supported in part by NSF grants DMS-2137623 and DMS-2246624.
- Communicated by: Zhen-Qing Chen
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 2683-2696
- MSC (2020): Primary 60B99, 30C15
- DOI: https://doi.org/10.1090/proc/16794
- MathSciNet review: 4741259