Zero-free neighborhood around the unit circle for Kac polynomials

DOI10.1007/S10998-021-00409-7arXiv1807.11594MaRDI QIDQ6304841

Publication date: 30 July 2018

Abstract: In this paper, we study how the roots of the so-called Kac polynomial

W_{n} (z) = s u m_{k = 0}^{n - 1} x i_{k} z^{k}

are concentrating to the unit circle when its 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 are concentrating around the unit circle as

n o i n f t y

if and only if

m a t h b b E [l o g (1 + | x i_{0} |)] < i n f t y

. Under the condition of

m a t h b b E [x i_{0}^{2}] < i n f t y

, we show that there exists an annulus of width

m a t h r m O (n^{- 2} (l o g n)^{- 3})

around the unit circle which is free of roots with probability

1 - m a t h r m O ((l o g n)^{- 1 / 2})

. The proof relies on the so-called small ball probability inequalities and the least common denominator.

Mathematics Subject Classification ID

Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) (30C15) Polynomials in real and complex fields: location of zeros (algebraic theorems) (12D10) Stochastic processes (60G99) Random power series in one complex variable (30B20)

This page was built for publication: Zero-free neighborhood around the unit circle for Kac polynomials

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6304841)