26 research outputs found

    Counting irreducible binomials over finite fields

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    We consider various counting questions for irreducible binomials over finite fields. We use various results from analytic number theory to investigate these questions.Comment: 11 page

    On Shifted Eisenstein Polynomials

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    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 nn polynomials that are not shifted Eisenstein polynomials is infinite. We conclude with some numerical results on the densities of shifted Eisenstein polynomials

    On the Number of Eisenstein Polynomials of Bounded Height

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    We obtain a more precise version of an asymptotic formula of A. Dubickas for the number of monic Eisenstein polynomials of fixed degree dd and of height at most HH, as Hβ†’βˆžH\to \infty. In particular, we give an explicit bound for the error term. We also obtain an asymptotic formula for arbitrary Eisenstein polynomials of height at most HH
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