Details
| Original language | English |
|---|---|
| Article number | 33 |
| Journal | Research in Number Theory |
| Volume | 7 |
| Issue number | 2 |
| Publication status | Published - 26 Apr 2021 |
| Externally published | Yes |
Abstract
We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height H with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to 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∈ { 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.
Keywords
- Counting, Height, Mahler measure
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Research in Number Theory, Vol. 7, No. 2, 33, 26.04.2021.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - On the frequency of height values
AU - Dill, Gabriel A.
N1 - Publisher Copyright: © 2021, The Author(s).
PY - 2021/4/26
Y1 - 2021/4/26
N2 - We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height H with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to 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∈ { 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.
AB - We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height H with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to 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∈ { 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.
KW - Counting
KW - Height
KW - Mahler measure
UR - http://www.scopus.com/inward/record.url?scp=85116989796&partnerID=8YFLogxK
U2 - 10.1007/s40993-021-00261-1
DO - 10.1007/s40993-021-00261-1
M3 - Article
VL - 7
JO - Research in Number Theory
JF - Research in Number Theory
IS - 2
M1 - 33
ER -