Positive density of integer polynomials with some prescribed properties

Arturas Dubickas; Min Sha

doi:10.1016/j.jnt.2015.07.017

Positive density of integer polynomials with some prescribed properties

Arturas Dubickas, Min Sha^*

^*Corresponding author for this work

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

4 Citations (Scopus)

Abstract

In this paper, we show that various kinds of integer polynomials with prescribed properties of their roots have positive density. For example, we prove that almost all integer polynomials have exactly one or two roots with maximal modulus. We also show that for any positive integer n and any set of n distinct points symmetric with respect to the real line, there is a positive density of integer polynomials of degree n, height at most H and Galois group S_n whose roots are close to the given n points.

Original language	English
Pages (from-to)	27-44
Number of pages	18
Journal	Journal of Number Theory
Volume	159
DOIs	https://doi.org/10.1016/j.jnt.2015.07.017
Publication status	Published - 1 Feb 2016
Externally published	Yes

Keywords

Dominant polynomial
Integer polynomial
Polynomial root
Positive density
Primary
Secondary

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10.1016/j.jnt.2015.07.017

Cite this

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abstract = "In this paper, we show that various kinds of integer polynomials with prescribed properties of their roots have positive density. For example, we prove that almost all integer polynomials have exactly one or two roots with maximal modulus. We also show that for any positive integer n and any set of n distinct points symmetric with respect to the real line, there is a positive density of integer polynomials of degree n, height at most H and Galois group Sn whose roots are close to the given n points.",

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author = "Arturas Dubickas and Min Sha",

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language = "English",

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journal = "Journal of Number Theory",

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AU - Sha, Min

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AB - In this paper, we show that various kinds of integer polynomials with prescribed properties of their roots have positive density. For example, we prove that almost all integer polynomials have exactly one or two roots with maximal modulus. We also show that for any positive integer n and any set of n distinct points symmetric with respect to the real line, there is a positive density of integer polynomials of degree n, height at most H and Galois group Sn whose roots are close to the given n points.

KW - Dominant polynomial

KW - Integer polynomial

KW - Polynomial root

KW - Positive density

KW - Primary

KW - Secondary

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DO - 10.1016/j.jnt.2015.07.017

M3 - Article

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VL - 159

SP - 27

EP - 44

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -

Positive density of integer polynomials with some prescribed properties

Abstract

Keywords

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