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On shifted Eisenstein polynomials

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Abstract

We study polynomials with integer coefficients which become Eisenstein polynomials after the additive shift of a variable. We call such polynomials shifted Eisenstein polynomials. We determine an upper bound on the maximum shift that is needed given a shifted Eisenstein polynomial and also provide a lower bound on the density of shifted Eisenstein polynomials, which is strictly greater than the density of classical Eisenstein polynomials. We also show that the number of irreducible degree \(n\) polynomials that are not shifted Eisenstein polynomials is infinite. We conclude with some numerical results on the densities of shifted Eisenstein polynomials.

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Acknowledgments

The authors would like to acknowledge the assistance of Hilary Albert with the programming for Sect. 6. This work was supported in part by the ARC Grant DP130100237.

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Authors and Affiliations

  1. Department of Computing, Macquarie University, Sydney, NSW, 2109, Australia

    Randell Heyman & Igor E. Shparlinski

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  1. Randell Heyman
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  2. Igor E. Shparlinski
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Correspondence to Igor E. Shparlinski.

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Heyman, R., Shparlinski, I.E. On shifted Eisenstein polynomials. Period Math Hung 69, 170–181 (2014). https://doi.org/10.1007/s10998-014-0061-0

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