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arXiv · 2012.09085

On the frequency of height values

Gabriel Andreas Dill
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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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Gabriel Andreas Dill. 2020-12-16. On the frequency of height values. https://doi.org/10.1007/s40993-021-00261-1

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