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Periodicity of Jacobi-Perron algorithm
Název práce v češtině: Periodičnost Jacobiho-Perronova algoritmu Periodicity of Jacobi-Perron algorithm Jacobi-Perronův algoritmus|řetězové zlomky|nerozložitelné prvky|kubická tělesa Jacobi-Perron algorithm|continued fractions|indecomposable elements|cubic fields 2019/2020 diplomová práce angličtina Katedra algebry (32-KA) doc. Mgr. Vítězslav Kala, Ph.D. skrytý - zadáno a potvrzeno stud. odd. 28.12.2019 28.12.2019 11.02.2020 23.06.2021 09:00 21.05.2021 21.05.2021 23.06.2021 Ing. Tomáš Vávra, Ph.D. Ing. Magdaléna Tinková, Ph.D.
 Zásady pro vypracování 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.
 Seznam odborné literatury [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).