Positive density of integer polynomials with some prescribed properties

2015/04/20 by Artūras Dubickas, Min Sha, Dubickas, Artūras +1 · 3 citations
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #Primary 11C08 #Secondary 11B37

paper · pdf · doi:10.48550/arxiv.1504.05144

openalex publication_date 2015/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Counting polynomials with positive roots
• On Expansions of Monadic Second-Order Logic with Dynamical Predicates
• Three quantities arising from Bézout's identity and resultants of integer polynomials

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