Randell Heyman

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

COUNTING IRREDUCIBLE BINOMIALS OVER FINITE FIELDS

n

On the number of Eisenstein polynomials of bounded height

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

Pairwise non-coprimality of triples

Let

Counting tuples restricted by coprimality conditions

q

On a sum involving the Euler function.

be a prime power. We estimate the number of tuples of degree bounded monic polynomials

Evaluationally coprime linear polynomials

(Q_1,\ldots,Q_v) \in (\mathbb{F}_q[z])^v

On the greatest common divisor of shifted sets

that satisfy given pairwise coprimality conditions. We show how this generalises from monic polynomials in finite fields to Dedekind domains with finite norms.