Approximately half of the roots of a random Littlewood polynomial are inside the disk

Oren Yakir

doi:10.4064/SM201117-28-1

Approximately half of the roots of a random Littlewood polynomial are inside the disk

Oren Yakir^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that for large n, all but o(2ⁿ) polynomials of the form P(z) = ∑_k=0ⁿ⁻¹ ±z^k have n/2+o(n) roots inside the unit disk. This solves a problem from Hayman’s 1967 book.

Original language	English
Pages (from-to)	227-240
Number of pages	14
Journal	Studia Mathematica
Volume	261
Issue number	2
DOIs	https://doi.org/10.4064/SM201117-28-1
State	Published - 2021

Keywords

Mahler measure
Point processes
Random polynomials

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10.4064/SM201117-28-1

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title = "Approximately half of the roots of a random Littlewood polynomial are inside the disk",

abstract = "We prove that for large n, all but o(2n) polynomials of the form P(z) = ∑k=0n−1 ±zk have n/2+o(n) roots inside the unit disk. This solves a problem from Hayman{\textquoteright}s 1967 book.",

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Approximately half of the roots of a random Littlewood polynomial are inside the disk

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Keywords

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