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А.A.Karatsuba's 80th Birthday Conference in Number Theory and Applications
May 27, 2017 11:35–12:05, Moscow, Department of Mechanics and Mathematics, Lomonosov Moscow State University
 


On the periodicity of continued fractions in elliptic fields

G. V. Fedorovab

a Institute for System Research, Russian Academy of Sciences
b Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
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G. V. Fedorov


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Abstract: The paper [1] raised the question of the finiteness of the number squarefree polynomials $f \in \mathbb{Q}[h]$ of fixed degree with periodic continued fraction expansion of $\sqrt{f\mathstrut}$ in the field $\mathbb{Q}((h))$, for which the fields $\mathbb{Q}(h)(\sqrt{f\mathstrut})$ are not isomorphic to one another and to fields of the form $\mathbb{Q}(h)(\sqrt{ch^{n} + 1})$, where $c \in \mathbb{Q}^{\ast}$, $n \in \mathbb{N}$. In the joint paper, V.P. Platonov and G.V. Fedorov [2] obtained a positive answer to this question for elliptic fields $\mathbb{Q}(h)(\sqrt{f\mathstrut})$, $\deg f = 3$. The report will present the results of the article [2].
[1] V.P. Platonov, G.V. Fedorov, On the periodicity of continued fractions in elliptic fields. Doklady Mathematics. 2017 (to appear).
[2] V.P. Platonov, G.V. Fedorov, On the periodicity of continued fractions in hyperelliptic fields. Doklady Mathematics. 2017 (to appear).

Language: English

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