Counting and testing dominant polynomials

2014/07/10 by Artūras Dubickas, Min Sha, Dubickas, Artūras +1
Mathematics · #11R06 #Advanced Combinatorial Mathematics #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Primary 11C08 #Random Matrices and Applications #Secondary 11B37

paper · pdf · doi:10.48550/arxiv.1407.2789

openalex publication_date 2014/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we concentrate on counting and testing dominant polynomials with integer coefficients. A polynomial is called dominant if it has a simple root whose modulus is strictly greater than the moduli of its remaining roots. In particular, our results imply that the probability that the dominant root assumption holds for a random monic polynomial with integer coefficients tends to 1 in some setting. However, for arbitrary integer polynomials it does not tend to 1. For instance, the proportion of dominant quadratic integer polynomials of height H among all quadratic integer polynomials tends to (41+6 log 2)/72 as H → ∞. Finally, we will design some algorithms to test whether a given polynomial with integer coefficients is dominant or not without finding the polynomial roots.

Citations

• Root separation for irreducible integer polynomials

Related

• Counting polynomials with positive roots
• Counting integer polynomials with several roots of maximal modulus
• Some results on counting roots of polynomials and the Sylvester resultant
• The number of nonunimodular roots of a reciprocal polynomial
• On the greatest common divisor of integer parts of polynomials