Counting integer reducible polynomials with bounded measure | Zendy

Artūras Dubickas | Zendy

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Counting integer reducible polynomials with bounded measure

Author(s) -

Artūras Dubickas

Publication year - 2016

Publication title -

applicable analysis and discrete mathematics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.69

H-Index - 26

eISSN - 2406-100X

pISSN - 1452-8630

DOI - 10.2298/aadm160714014d

Subject(s) - mathematics , monic polynomial , measure (data warehouse) , bounded function , integer (computer science) , degree (music) , discrete orthogonal polynomials , combinatorics , macdonald polynomials , wilson polynomials , difference polynomials , orthogonal polynomials , discrete mathematics , pure mathematics , polynomial , mathematical analysis , programming language , database , computer science , physics , acoustics

In this paper, we give an asymptotic formula for the number of integer reducible polynomials with fixed degree d ≥ 2 and Mahler measure bounded above by T and also for the number of such monic polynomials as T → ∞. We also consider the case of monic polynomials which have all their roots in the disc |z| ≤ R and find asymptotics for the number of such reducible polynomials too as R → ∞. In all cases the constants in the main terms are related to the constants of the corresponding counting formulas for the number of such irreducible polynomials due to Chern and Vaaler (in case of Mahler measure) and Akiyama and Pethő (in case of a disc).

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