Thesis (Selection of subject)Thesis (Selection of subject)(version: 285)
Assignment details
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Aritmetika číselných těles a zobecněné řetězové zlomky
Thesis title in Czech: Aritmetika číselných těles a zobecněné řetězové zlomky
Thesis title in English: Arithmetics of number fields and generalized continued fractions
Academic year of topic announcement: 2016/2017
Type of assignment: dissertation
Thesis language:
Department: Department of Algebra (32-KA)
Supervisor: Mgr. Vítězslav Kala, Ph.D.
Author: hidden - assigned and confirmed by the Study Dept.
Date of registration: 20.09.2017
Date of assignment: 20.09.2017
Confirmed by Study dept. on: 03.10.2017
Guidelines
The aim of the thesis is to study the arithmetics of number fields, in particular the structure of totally positive additively indecomposable integers. In the case of real quadratic fields, these can be nicely characterized in terms of periodic continued fractions, which then has applications to the study of universal quadratic forms. However, the classical theory no longer works for higher degree number fields, so the student will investigate and develop other methods based on generalized continued fractions or numeration systems, e.g., the Jacobi-Perron algorithm or continued fractions with algebraic coefficients.
References
Valentin Blomer and Vitezslav Kala, Number fields without n-ary universal quadratic forms, Math. Proc. Cambridge Philos. Soc. 159 (2015), no. 2, 239–252.

Vítezslav Kala, Norms of indecomposable integers in real quadratic fields, J. Number Theory 166 (2016), 193-207.

Vítezslav Kala, Universal quadratic forms and elements of small norm in real quadratic fields, Bull. Aust. Math. Soc. 94 (2016), 7-14.

Fritz Schweiger, Multidimensional Continued Fractions, Oxford University Press, 2000.

Julien Bernat, Continued fractions and numeration in the Fibonacci base, Discrete Math. 306 (2006), no. 22, 2828–2850.

Leon Bernstein, The Jacobi-Perron Algorithm Its Theory and Application, Lecture Notes in Mathematics, Volume 207 (1971).
 
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