Cargando…
On the frequency of height values
We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height [Formula: see text] with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to [Formula: see text] that the height assumes on algebraic numbers of degree d with...
Autor principal: | |
---|---|
Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Springer International Publishing
2021
|
Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8550142/ https://www.ncbi.nlm.nih.gov/pubmed/34723094 http://dx.doi.org/10.1007/s40993-021-00261-1 |
_version_ | 1784590899228966912 |
---|---|
author | Dill, Gabriel A. |
author_facet | Dill, Gabriel A. |
author_sort | Dill, Gabriel A. |
collection | PubMed |
description | We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height [Formula: see text] with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to [Formula: see text] 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 [Formula: see text] or [Formula: see text] . We therefore study the behaviour in the case where [Formula: see text] and [Formula: see text] 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. |
format | Online Article Text |
id | pubmed-8550142 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2021 |
publisher | Springer International Publishing |
record_format | MEDLINE/PubMed |
spelling | pubmed-85501422021-10-29 On the frequency of height values Dill, Gabriel A. Res Number Theory Research We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height [Formula: see text] with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to [Formula: see text] 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 [Formula: see text] or [Formula: see text] . We therefore study the behaviour in the case where [Formula: see text] and [Formula: see text] 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. Springer International Publishing 2021-04-26 2021 /pmc/articles/PMC8550142/ /pubmed/34723094 http://dx.doi.org/10.1007/s40993-021-00261-1 Text en © The Author(s) 2021 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) . |
spellingShingle | Research Dill, Gabriel A. On the frequency of height values |
title | On the frequency of height values |
title_full | On the frequency of height values |
title_fullStr | On the frequency of height values |
title_full_unstemmed | On the frequency of height values |
title_short | On the frequency of height values |
title_sort | on the frequency of height values |
topic | Research |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8550142/ https://www.ncbi.nlm.nih.gov/pubmed/34723094 http://dx.doi.org/10.1007/s40993-021-00261-1 |
work_keys_str_mv | AT dillgabriela onthefrequencyofheightvalues |