26 research outputs found
Counting irreducible binomials over finite fields
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
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 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
We obtain a more precise version of an asymptotic formula of A. Dubickas for
the number of monic Eisenstein polynomials of fixed degree and of height at
most , as . 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