Counting integer polynomials with several roots of maximal modulus

In this paper, for positive integers $H$ and $k \leq n$, we obtain some estimates on the cardinality of the set of monic integer polynomials of degree $n$ and height bounded by $H$ with exactly $k$ roots of maximal modulus. These include lower and upper bounds in terms of $H$ for fixed $k$ and $n$. We also count reducible and irreducible polynomials in that set separately. Our results imply, for instance, that the number of monic integer irreducible polynomials of degree $n$ and height at most $H$ whose $n$ roots all have the same modulus is approximately $2H$ for odd $n$, while for even $n$ there are more than $H^{n/8}$ such polynomials.