Periodicity of Jacobi-Perron algorithm
| Thesis title in Czech: | Periodičnost Jacobiho-Perronova algoritmu |
|---|---|
| Thesis title in English: | Periodicity of Jacobi-Perron algorithm |
| Key words: | Jacobi-Perronův algoritmus|řetězové zlomky|nerozložitelné prvky|kubická tělesa |
| English key words: | Jacobi-Perron algorithm|continued fractions|indecomposable elements|cubic fields |
| Academic year of topic announcement: | 2019/2020 |
| Thesis type: | diploma thesis |
| Thesis language: | angličtina |
| Department: | Department of Algebra (32-KA) |
| Supervisor: | doc. Mgr. Vítězslav Kala, Ph.D. |
| Author: | hidden - assigned and confirmed by the Study Dept. |
| Date of registration: | 28.12.2019 |
| Date of assignment: | 28.12.2019 |
| Confirmed by Study dept. on: | 11.02.2020 |
| Date and time of defence: | 23.06.2021 09:00 |
| Date of electronic submission: | 21.05.2021 |
| Date of submission of printed version: | 21.05.2021 |
| Date of proceeded defence: | 23.06.2021 |
| Opponents: | Ing. Tomáš Vávra, Ph.D. |
| Advisors: | Ing. Magdaléna Tinková, Ph.D. |
| Guidelines |
| The Jacobi-Perron algorithm (JPA) is one of the most important multidimensional generalizations of continued fractions. Of particular interest is the case when JPA is periodic, for then it can be used to construct a unit in the corresponding number field.
After writing up the basic theory of JPA, the student will focus on the properties of periodic JPAs with arithmetic applications, such as estimates of the coefficients and norms of the convergents, or the behavior of JPA expansions in families of number fields of small degree. |
| References |
| [1] L. Bernstein, The Jacobi-Perron algorithm – Its theory and application, Lecture Notes in Mathematics 207, Springer-Verlag, Berlin, New York, 1971.
[2] V. Kala, Norms of indecomposable integers in real quadratic fields, J. Number Theory 166, 193-207 (2016). [3] F. Schweiger, Multidimensional Continued Fractions, Oxford University Press, Oxford, 2000. [4] M. Tinková and P. Voutier, Indecomposable integers in real quadratic fields, 25 pp., J. Number Theory, to appear. [5] P. Voutier, Families of periodic Jacobi-Perron algorithms for all period lengths, J. Number Theory 168, 472–486 (2016). |