On the number of polynomials of bounded height that satisfy Dumas's criterion

We study integer coefficient polynomials of fixed degree and maximum height $H$, that are irreducible by Dumas's criterion. We call such polynomials Dumas polynomials. We derive upper bounds on the number of Dumas polynomials, as $H$ approaches infinity. We also show that, for a fixed degree, the density of Dumas polynomials in all irreducible integer coefficient polynomials is strictly less than 1.