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Doklady Rossijskoj Akademii Nauk. Mathematika, Informatika, Processy Upravlenia, 2020, Volume 495, Pages 48–54
DOI: https://doi.org/10.31857/S2686954320060119
(Mi danma9)
 

This article is cited in 7 scientific papers (total in 7 papers)

MATHEMATICS

On the finiteness of the number of expansions into a continued fraction of $\sqrt f$ for cubic polynomials over algebraic number fields

V. P. Platonovab, M. M. Petrunina

a Scientific Research Institute for System Analysis of the Russian Academy of Sciences, Moscow
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Full-text PDF (171 kB) Citations (7)
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Abstract: We obtain a complete description of cubic polynomials f over algebraic number fields $\mathbb K$ of degree 3 over $\mathbb Q$ for which the continued fraction expansion of $\sqrt f$ in the field of formal power series $\mathbb K((x))$ is periodic. We also prove a finiteness theorem for cubic polynomials $f\in K[x]$ with a periodic expansion of $\sqrt f$ for extensions of $\mathbb Q$ of degree at most 6. Additionally, we give a complete description of such polynomials $f$ over an arbitrary field corresponding to elliptic fields with a torsion point of order $N\ge30$.
Keywords: elliptic field, $S$-units, continued fractions, periodicity, modular curves, torsion point.
Funding agency Grant number
Ministry of Education and Science of the Russian Federation 0065-2019-0011
This work was performed within the state assignment to basic scientific research, project no. 0065-2019-0011.
Received: 15.09.2020
Revised: 15.09.2020
Accepted: 21.09.2020
English version:
Doklady Mathematics, 2020, Volume 102, Issue 3, Pages 487–492
DOI: https://doi.org/10.1134/S1064562420060137
Bibliographic databases:
Document Type: Article
UDC: 511.6
Language: Russian
Citation: V. P. Platonov, M. M. Petrunin, “On the finiteness of the number of expansions into a continued fraction of $\sqrt f$ for cubic polynomials over algebraic number fields”, Dokl. RAN. Math. Inf. Proc. Upr., 495 (2020), 48–54; Dokl. Math., 102:3 (2020), 487–492
Citation in format AMSBIB
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  • This publication is cited in the following 7 articles:
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    Doklady Rossijskoj Akademii Nauk. Mathematika, Informatika, Processy Upravlenia Doklady Rossijskoj Akademii Nauk. Mathematika, Informatika, Processy Upravlenia
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