The density of shifted and affine Eisenstein polynomials
HTML articles powered by AMS MathViewer
- by Giacomo Micheli and Reto Schnyder
- Proc. Amer. Math. Soc. 144 (2016), 4651-4661
- DOI: https://doi.org/10.1090/proc/13097
- Published electronically: April 27, 2016
- PDF | Request permission
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.References
- Artūras Dubickas, Polynomials irreducible by Eisenstein’s criterion, Appl. Algebra Engrg. Comm. Comput. 14 (2003), no. 2, 127–132. MR 1995564, DOI 10.1007/s00200-003-0131-7
- Gotthold Eisenstein, Über die Irreducibilität und einige andere Eigenschaften der Gleichung, von welcher der Theilung der ganzen Lemniscate abhängt, J. Reine Angew. Math. 40 (1850), 185–188.
- Randell Heyman and Igor E. Shparlinski, On the number of Eisenstein polynomials of bounded height, Appl. Algebra Engrg. Comm. Comput. 24 (2013), no. 2, 149–156. MR 3063896, DOI 10.1007/s00200-013-0187-y
- Randell Heyman and Igor E. Shparlinski, On shifted Eisenstein polynomials, Period. Math. Hungar. 69 (2014), no. 2, 170–181. MR 3278954, DOI 10.1007/s10998-014-0061-0
- Bjorn Poonen and Michael Stoll, The Cassels-Tate pairing on polarized abelian varieties, Ann. of Math. (2) 150 (1999), no. 3, 1109–1149. MR 1740984, DOI 10.2307/121064
- Alain M. Robert, A course in $p$-adic analysis, Graduate Texts in Mathematics, vol. 198, Springer-Verlag, New York, 2000. MR 1760253, DOI 10.1007/978-1-4757-3254-2
- Theodor Schönemann, Von denjenigen Moduln, welche Potenzen von Primzahlen sind, J. Reine Angew. Math. 39 (1846), 160–179.
- William A. Stein and others, Sage Mathematics Software (Version 6.1.1), The Sage Development Team, 2014.
Bibliographic 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
- MR Author ID: 1078793
- 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
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4651-4661
- MSC (2010): Primary 11R45, 11R09, 11S05
- DOI: https://doi.org/10.1090/proc/13097
- MathSciNet review: 3544517