Abstract
We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height {mathcal {H}} with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to {mathcal {H}} that the height assumes on algebraic numbers of degree d with precisely k conjugates that lie inside the open unit disk. For both counts, we do not obtain an asymptotic, but only a rough order of growth, which arises from an asymptotic for the logarithm of the counting function; for the first count, even this rough order of growth exists only if k in {0,d} or gcd (k,d) = 1. We therefore study the behaviour in the case where 0< k < d and gcd (k,d) > 1 in more detail. We also count integer polynomials of fixed degree and fixed Mahler measure with a fixed number of complex zeroes inside the open unit disk (counted with multiplicities) and study the dynamical behaviour of the height function.
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