On the frequency of height values

TL;DR

This paper investigates the distribution and frequency of algebraic numbers with fixed degree and height, focusing on the number of conjugates inside the unit disk, and explores related counts of polynomials with specified properties.

Contribution

It provides new insights into the counting of algebraic numbers and polynomials with fixed degree, height, and conjugate properties, especially in cases where gcd(k,d) > 1.

Findings

Counts are obtained for algebraic numbers with conjugates inside the unit disk.

The growth order of these counts is characterized for certain cases.

Analysis of polynomial zeroes and height dynamics related to algebraic numbers.

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.