ACAC Seminar Abstract

ACAC Seminar Abstract

ACAC Seminars

ACAC Seminar Abstract

Counting polynomials that satisfy the Eisenstein criterion

Speaker: Randell Heyman
Date, Time: Fri, 15 Mar 2013 14:30

The Eisenstein criterion is a well-known sufficient condition for the irreducibility of polynomials with integer (or rational) coefficients. But what is the chance that an irreducible polynomial can be shown to be irreducible by the Eisenstein criterion? In the last 13 years there has been progress in quantifying the probabilities that certain sets of polynomials satisfy the Eisenstein criterion. The talk will give some improved results for monic polynomials and some new results for the set of all polynomials of a maximum height, leading to a formula when the height approaches infinity. This involves a careful use of the inclusion/exclusion principle and also properties of arithmetic functions.
We will also cover some results on counting polynomials that can be "shifted." That is, they satisfy the Eisenstein criterion after an additive shift of the variable.
This is joint work with Igor Shparlinski.

Back to the top of this page