## Real roots near the unit circle of random polynomials

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- by Marcus Michelen PDF
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## Abstract:

Let $f_n(z) = \sum _{k = 0}^n \varepsilon _k z^k$ be a random polynomial where $\varepsilon _0,\ldots ,\varepsilon _n$ are i.i.d. random variables with $\mathbb {E} \varepsilon _1 = 0$ and $\mathbb {E} \varepsilon _1^2 = 1$. Letting $r_1, r_2,\ldots , r_k$ denote the real roots of $f_n$, we show that the point process defined by $\{|r_1| - 1,\ldots , |r_k| - 1 \}$ converges to a non-Poissonian limit on the scale of $n^{-1}$ as $n \to \infty$. Further, we show that for each $\delta > 0$, $f_n$ has a real root within $\Theta _{\delta }(1/n)$ of the unit circle with probability at least $1 - \delta$. This resolves a conjecture of Shepp and Vanderbei from 1995 by confirming its weakest form and refuting its strongest form.## References

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## Additional Information

**Marcus Michelen**- Affiliation: Department of Mathematics, Statistics, and Computer Science, The University of Illinois at Chicago, Chicago, Illinois 60607
- MR Author ID: 1312016
- Email: michelen.math@gmail.com
- Received by editor(s): October 28, 2020
- Published electronically: March 26, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 4359-4374 - MSC (2020): Primary 60G15; Secondary 60F05, 42A32, 26C10
- DOI: https://doi.org/10.1090/tran/8379
- MathSciNet review: 4251232