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Matematicheskie Zametki, 2022, Volume 112, Issue 3, Pages 444–452
DOI: https://doi.org/10.4213/mzm13544 (Mi mzm13544) 		 
On the Parametrization of Hyperelliptic Fields with $S$-Units of Degrees 7 and 9

G. V. Fedorovab, V. S. Zhgoonab, M. M. Petruninab, Yu. N. Shteinikovab

a "Sirius" University, Sochi
b Scientific Research Institute for System Analysis of the Russian Academy of Sciences, Moscow
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DOI: https://doi.org/10.4213/mzm13544
Abstract: We show that if $k$ is an algebraically closed field with $\operatorname{char}k=0$, then the set of polynomials $f$ of degree $5$ such that the field $k(x)(\sqrt{f}\,)$ has a nontrivial $S$-unit of degree $7$ or $9$ and the continued fraction expansion of $\sqrt{f}/x$ is periodic is a one-parameter set corresponding to a rational curve with finitely many deleted points.
Keywords: hyperelliptic field, torsion point, rational curve, Gröbner basis.
Funding agency Grant number
Университет "Сириус" FMF-RND-2125
Ministry of Science and Higher Education of the Russian Federation FNEF-2022-0011
This work was supported in part by the Sirius University, project FMF-RND-2125 (Theorem 2) and in part by the Ministry of Science and Higher Education of the Russian Federation, grant no. FNEF-2022-0011 (Theorem 1).
Received: 13.04.2022
Revised: 26.04.2022
English version:
Mathematical Notes, 2022, Volume 112, Issue 3, Pages 451–457
DOI: https://doi.org/10.1134/S0001434622090139
Bibliographic databases:
Document Type: Article
UDC: 511.6
Language: Russian
Citation: G. V. Fedorov, V. S. Zhgoon, M. M. Petrunin, Yu. N. Shteinikov, “On the Parametrization of Hyperelliptic Fields with $S$-Units of Degrees 7 and 9”, Mat. Zametki, 112:3 (2022), 444–452; Math. Notes, 112:3 (2022), 451–457
Citation in format AMSBIB
\Bibitem{FedZhgPet22}
\by G.~V.~Fedorov, V.~S.~Zhgoon, M.~M.~Petrunin, Yu.~N.~Shteinikov
\paper On the Parametrization of Hyperelliptic Fields with $S$-Units of Degrees~7 and 9
\jour Mat. Zametki
\yr 2022
\vol 112
\issue 3
\pages 444--452
\mathnet{http://mi.mathnet.ru/mzm13544}
\crossref{https://doi.org/10.4213/mzm13544}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4538780}
\transl
\jour Math. Notes
\yr 2022
\vol 112
\issue 3
\pages 451--457
\crossref{https://doi.org/10.1134/S0001434622090139}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85141080920}
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