The density of shifted and affine Eisenstein polynomials

Micheli, Giacomo; Schnyder, Reto

doi:10.1090/proc/13097

The density of shifted and affine Eisenstein polynomials
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by Giacomo Micheli and Reto Schnyder PDF

Proc. Amer. Math. Soc. 144 (2016), 4651-4661 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.