## On random polynomials with an intermediate number of real roots

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- by Marcus Michelen and Sean O’Rourke;
- Proc. Amer. Math. Soc.
**152**(2024), 4933-4942 - DOI: https://doi.org/10.1090/proc/16999
- Published electronically: September 17, 2024
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## Abstract:

For each $\alpha \in (0, 1)$, we construct a bounded monotone deterministic sequence $(c_k)_{k \geqslant 0}$ of real numbers so that the number of real roots of the random polynomial $f_n(z) = \sum _{k=0}^n c_k \varepsilon _k z^k$ is $n^{\alpha + o(1)}$ with probability tending to one as the degree $n$ tends to infinity, where $(\varepsilon _k)$ is a sequence of i.i.d. (real) random variables of finite mean satisfying a mild anti-concentration assumption. In particular, this includes the case when $(\varepsilon _k)$ is a sequence of i.i.d. standard Gaussian or Rademacher random variables. This result confirms a conjecture of O. Nguyen from 2019. More generally, our main results also describe several statistical properties for the number of real roots of $f_n$, including the asymptotic behavior of the variance and a central limit theorem.## References

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## Bibliographic Information

**Marcus Michelen**- Affiliation: Department of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, 851 S. Morgan St., Chicago, Illinois 60607
- MR Author ID: 1312016
- Email: michelen@uic.edu
**Sean O’Rourke**- Affiliation: Department of Mathematics, University of Colorado Boulder, 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): December 1, 2023
- Received by editor(s) in revised form: April 4, 2024, and May 28, 2024
- Published electronically: September 17, 2024
- Additional Notes: The first author was supported in part by NSF CAREER grant DMS-2336788 as well as grants DMS-2137623 and DMS-2246624.

The second author was supported in part by NSF CAREER grant DMS-2143142. - Communicated by: Zhen-Qing Chen
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**152**(2024), 4933-4942 - MSC (2020): Primary 60F05, 26C10
- DOI: https://doi.org/10.1090/proc/16999