Counting and testing dominant polynomials

Arturas Dubickas, Min Sha*

*Corresponding author for this work

Research output: Contribution to journalArticle

7 Citations (Scopus)

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 what is known as 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 + 6log 2)/72 as H → ∞. Finally, we design some algorithms to test whether a given polynomial with integer coefficients is dominant without finding the polynomials roots.

Original languageEnglish
Pages (from-to)312-325
Number of pages14
JournalExperimental Mathematics
Volume24
Issue number3
DOIs
Publication statusPublished - 3 Jul 2015
Externally publishedYes

Keywords

  • Bistritz stability criterion
  • Dominant polynomial
  • linear recurrence sequence
  • Sturm's theorem

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