A degree problem for two algebraic numbers and their sum
| Autor: | Paulius Drungilas, Chris Smyth, Artūras Dubickas |
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| Rok vydání: | 2021 |
| Předmět: |
Discrete mathematics
Algebraic number Abc degree problem Degree (graph theory) Mathematics::Number Theory sum-feasible General Mathematics Multiplicative function Prime number Algebraic number field 11R32 Combinatorics Integer 11R04 $abc$ degree problem Sum-feasible Algebraic number sum-feasible abc degree problem Mathematics |
| Zdroj: | Recercat: Dipósit de la Recerca de Catalunya Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) Recercat. Dipósit de la Recerca de Catalunya instname Smyth, C, Dubickas, A & Drungilas, P 2012, ' A degree problem for two algebraic numbers and their sum ', Publicacions Matemàtiques, vol. 56, pp. 413-448 . Dipòsit Digital de Documents de la UAB Universitat Autònoma de Barcelona Publicacions Matemàtiques; Vol. 56, Núm. 2 (2012); p. 413-448 Publ. Mat. 56, no. 2 (2012), 413-448 |
| Popis: | For all but one positive integer triplet $(a,b,c)$ with $a\leqslant b\leqslant c$ and $b\leqslant 6$, we decide whether there are algebraic numbers $\alpha$, $\beta$ and $\gamma$ of degrees $a$, $b$ and $c$, respectively, such that $\alpha+\beta+\gamma=0$. The undecided case $(6,6,8)$ will be included in another paper. These results imply, for example, that the sum of two algebraic numbers of degree $6$ can be of degree $15$ but cannot be of degree $10$. We also show that if a positive integer triplet $(a,b,c)$ satisfies a certain triangle-like inequality with respect to every prime number then there exist algebraic numbers $\alpha$, $\beta$, $\gamma$ of degrees $a$, $b$, $c$ such that $\alpha+\beta+\gamma=0$. We also solve a similar problem for all $(a,b,c)$ with $a\leqslant b\leqslant c$ and $b\leqslant 6$ by finding for which $a$, $b$, $c$ there exist number fields of degrees $a$ and $b$ such that their compositum has degree $c$. Further, we have some results on the multiplicative version of the first problem, asking for which triplets $(a,b,c)$ there are algebraic numbers $\alpha$, $\beta$ and $\gamma$ of degrees $a$, $b$ and $c$, respectively, such that $\alpha\beta\gamma=1$. |
| Databáze: | OpenAIRE |
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