## Abstract

In this paper, we study how the roots of the Kac polynomials $W_n(z)=\sum_{k=0}^{n-1}\xi_k z^k$ concentrate around the unit circle when the coefficients of $W_n$ are independent and identically distributed non-degenerate real random variables. It is well-known that the roots of a Kac polynomial concentrate around the unit circle as $n\to \infty$ if and only if $\mathbb{E}[\log(1 +|\xi_0|)] <\infty$. Under the finiteness of the second moment, we show that there exists an annulus of width $O(n^{-2}(log n)^{-3})$ around the unit circle which is free of roots with probability $1 - O((log n)^{-1/2})$. The proof relies on small ball probability inequalities and the least common denominator.

Original language | English |
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Journal | Periodica Mathematica Hungarica |

Pages (from-to) | 1-18 |

Number of pages | 18 |

ISSN | 0031-5303 |

DOIs | |

Publication status | Published - 7 Aug 2021 |

MoE publication type | A1 Journal article-refereed |

## Fields of Science

- 111 Mathematics
- Salem–Zygmund type inequalities
- Zeros of random polynomials
- 112 Statistics and probability
- Small ball probability
- Locally sub-Gaussian random variables
- Locally sub-Gaussian random variables
- Salem-Zygmund type inequalities
- Small ball probability
- Zeros of random polynomials
- Complex zeros