Geodesic
On the Parametrization of Hyperelliptic Fields with $S$-Units of Degrees~7 and 9
Matematičeskie zametki, Tome 112 (2022) no. 3, pp. 444-452

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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, rational curve, Gröbner basis.
Mots-clés : torsion point 
@article{MZM_2022_112_3_a11,
     author = {G. V. Fedorov and V. S. Zhgoon and M. M. Petrunin and Yu. N. Shteinikov},
     title = {On the {Parametrization} of {Hyperelliptic} {Fields} with $S${-Units} of {Degrees~7} and 9},
     journal = {Matemati\v{c}eskie zametki},
     pages = {444--452},
     publisher = {mathdoc},
     volume = {112},
     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_112_3_a11/}
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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. Matematičeskie zametki, Tome 112 (2022) no. 3, pp. 444-452. http://geodesic.mathdoc.fr/item/MZM_2022_112_3_a11/