Nerozložitelné binární kvadratické formy nad číselnými tělesy
| Thesis title in Czech: | Nerozložitelné binární kvadratické formy nad číselnými tělesy |
|---|---|
| Thesis title in English: | Non-decomposable binary quadratic forms over number fields |
| Key words: | totálne reálne teleso|nerozložiteľná kvadratická forma|nerozložiteľné prvky |
| English key words: | totally real field|non-decomposable quadratic form|indecomposable integer |
| Academic year of topic announcement: | 2024/2025 |
| Thesis type: | diploma thesis |
| Thesis language: | čeština |
| Department: | Department of Algebra (32-KA) |
| Supervisor: | Ing. Magdaléna Tinková, Ph.D. |
| Author: | hidden - assigned and confirmed by the Study Dept. |
| Date of registration: | 28.02.2025 |
| Date of assignment: | 28.02.2025 |
| Confirmed by Study dept. on: | 28.02.2025 |
| Date and time of defence: | 03.06.2025 08:00 |
| Date of electronic submission: | 30.04.2025 |
| Date of submission of printed version: | 30.04.2025 |
| Date of proceeded defence: | 03.06.2025 |
| Opponents: | Jongheun Yoon |
| Advisors: | doc. Mgr. Vítězslav Kala, Ph.D. |
| Guidelines |
| Non-decomposable quadratic forms are a useful tool in the study of n-universality of quadratic forms. In the thesis, the student will focus on such forms of rank 2 over families of totally real number fields of degrees 3 and 4, e.g., the simplest cubic, and real biquadratic fields. The student will prove that over these fields, there always exists a non-decomposable binary quadratic form, and estimate the number of such non-equivalent forms. |
| References |
| M. Tinková and P. Yatsyna, Non-decomposable quadratic forms over totally real number fields, preprint. arXiv:2502.05991
P. Erdös and C. Ko, On definite quadratic forms, which are not the sum of two definite or semi-definite forms, Acta Arith. 3, (1939) 102–122. J. Krásenský, M. Tinková and K. Zemková, There are no universal ternary quadratic forms over biquadratic fields, Proc. Edinb. Math. Soc. 63 (3), (2020) 861–912. V. Kala and M. Tinková, Universal quadratic forms, small norms and traces in families of number fields, Int. Math. Res. Not. IMRN 2023, (2023) 7541–7577. |