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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Real roots near the unit circle of random polynomials
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by Marcus Michelen PDF
Trans. Amer. Math. Soc. 374 (2021), 4359-4374 Request permission

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.
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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