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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
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.
Received: 15.09.2020 Revised: 15.09.2020 Accepted: 21.09.2020
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
Linking options:
https://www.mathnet.ru/eng/danma9 https://www.mathnet.ru/eng/danma/v495/p48
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Abstract page: | 116 | Full-text PDF : | 26 | References: | 12 |
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