Non-negative integral matrices with given spectral radius and controlled dimension
A celebrated theorem of Douglas Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number p, we prove that there is an integral irreducible matrix with spectral radius p, and with...
| Main Author: | |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
Cambridge University Press
2021
|
| _version_ | 1826308538128924672 |
|---|---|
| author | Yazdi, M |
| author_facet | Yazdi, M |
| author_sort | Yazdi, M |
| collection | OXFORD |
| description | A celebrated theorem of Douglas Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number p, we prove that there is an integral irreducible matrix with spectral radius p, and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number p, there is an irreducible shift of finite type with entropy log(p) defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data. |
| first_indexed | 2024-03-07T07:20:58Z |
| format | Journal article |
| id | oxford-uuid:2864e645-e027-4cfd-93ca-fabcd9a696fd |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:20:58Z |
| publishDate | 2021 |
| publisher | Cambridge University Press |
| record_format | dspace |
| spelling | oxford-uuid:2864e645-e027-4cfd-93ca-fabcd9a696fd2022-10-06T11:35:04ZNon-negative integral matrices with given spectral radius and controlled dimensionJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:2864e645-e027-4cfd-93ca-fabcd9a696fdEnglishSymplectic ElementsCambridge University Press2021Yazdi, MA celebrated theorem of Douglas Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number p, we prove that there is an integral irreducible matrix with spectral radius p, and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number p, there is an irreducible shift of finite type with entropy log(p) defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data. |
| spellingShingle | Yazdi, M Non-negative integral matrices with given spectral radius and controlled dimension |
| title | Non-negative integral matrices with given spectral radius and controlled dimension |
| title_full | Non-negative integral matrices with given spectral radius and controlled dimension |
| title_fullStr | Non-negative integral matrices with given spectral radius and controlled dimension |
| title_full_unstemmed | Non-negative integral matrices with given spectral radius and controlled dimension |
| title_short | Non-negative integral matrices with given spectral radius and controlled dimension |
| title_sort | non negative integral matrices with given spectral radius and controlled dimension |
| work_keys_str_mv | AT yazdim nonnegativeintegralmatriceswithgivenspectralradiusandcontrolleddimension |