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New Results on the Periodicity Problem for Continued Fractions of Elements of Hyperelliptic Fields

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Abstract

We study the problem of describing square-free polynomials \(f(x)\) of odd degree with periodic expansion of \(\sqrt{f(x)}\) into a functional continued fraction in \(k((x))\), where \(k\subseteq\overline{\mathbb Q}\). We obtain a complete description of such polynomials \(f(x)\) that does not depend on the field \(k\) and the degree of a polynomial, provided that the degree \(U\) of the fundamental \(S\)-unit of the corresponding hyperelliptic field \(k(x)(\sqrt{f(x)})\) either does not exceed \(12\) or is even and does not exceed \(20\).

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Funding

This work was performed within the state assignment for basic scientific research, project no. FNEF-2022-0011.

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Authors and Affiliations

  1. Scientific Research Institute for System Analysis of the Russian Academy of Sciences, Nakhimovskii prosp. 36, bld. 1, Moscow, 117218, Russia

    V. P. Platonov & M. M. Petrunin

  2. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    V. P. Platonov

Authors
  1. V. P. Platonov
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  2. M. M. Petrunin
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Correspondence to V. P. Platonov.

Additional information

Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2023, Vol. 320, pp. 278–286 https://doi.org/10.4213/tm4283.

To the blessed memory of Alexey Nikolaevich Parshin

Translated by I. Nikitin

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Platonov, V.P., Petrunin, M.M. New Results on the Periodicity Problem for Continued Fractions of Elements of Hyperelliptic Fields. Proc. Steklov Inst. Math. 320, 258–266 (2023). https://doi.org/10.1134/S008154382301011X

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  • DOI: https://doi.org/10.1134/S008154382301011X

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