  In this paper, we concentrate on counting and testing dominant polynomials
with integer coefficients. A polynomial is called dominant if it has a simple
root whose modulus is strictly greater than the moduli of its remaining roots.
In particular, our results imply that the probability that the dominant root
assumption holds for a random monic polynomial with integer coefficients tends
to 1 in some setting. However, for arbitrary integer polynomials it does not
tend to 1. For instance, the proportion of dominant quadratic integer
polynomials of height $H$ among all quadratic integer polynomials tends to
$(41+6 \log 2)/72$ as $H \to \infty$. Finally, we will design some algorithms
to test whether a given polynomial with integer coefficients is dominant or not
without finding the polynomial roots.
