Abstract
AbstractWe count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height $${\mathcal {H}}$$
H
with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to $${\mathcal {H}}$$
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 $$k \in \{0,d\}$$
k
∈
{
0
,
d
}
or $$\gcd (k,d) = 1$$
gcd
(
k
,
d
)
=
1
. We therefore study the behaviour in the case where $$0< k < d$$
0
<
k
<
d
and $$\gcd (k,d) > 1$$
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.
Funder
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
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