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The density of shifted and affine Eisenstein polynomials


Authors: Giacomo Micheli and Reto Schnyder
Journal: Proc. Amer. Math. Soc. 144 (2016), 4651-4661
MSC (2010): Primary 11R45, 11R09, 11S05
DOI: https://doi.org/10.1090/proc/13097
Published electronically: April 27, 2016
MathSciNet review: 3544517
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Abstract: In this paper we provide a complete answer to a question by Heyman and Shparlinski concerning the natural density of polynomials which are irreducible by Eisenstein's criterion after applying some shift. The main tool we use is a local to global principle for density computations over a free $ \mathbb{Z}$-module of finite rank.


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Additional Information

Giacomo Micheli
Affiliation: Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
Address at time of publication: Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG
Email: giacomo.micheli@math.uzh.ch

Reto Schnyder
Affiliation: Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
Email: reto.schnyder@math.uzh.ch

DOI: https://doi.org/10.1090/proc/13097
Keywords: Natural Density, $p$-adic integers, Polynomials
Received by editor(s): July 21, 2015
Received by editor(s) in revised form: October 21, 2015, and January 14, 2016
Published electronically: April 27, 2016
Additional Notes: The first author was supported in part by Swiss National Science Foundation grant numbers 149716 and 161757
The second author was supported in part by Armasuisse and Swiss National Science Foundation grant number 149716
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2016 American Mathematical Society