On the Periodicity Problem for the Continued Fraction Expansion of Elements of Hyperelliptic Fields with Fundamental S-Units of Degree at Most 11

Academician of the RAS V. P. Platonov^a^,^b^,*, M. M. Petrunin ^a^,**, and Yu. N. Shteinikov^a^,***

^aScientific Research Institute for System Analysis, Russian Academy of Sciences, Moscow, 117218 Russia

^bSteklov Mathematical Institute, Russian Academy of Sciences, Moscow, 119991 Russia

Correspondence to: *e-mail: platonov@mi-ras.ru
Correspondence to: **e-mail: petrushkin@yandex.ru
Correspondence to: ***e-mail: yuriisht@yandex.ru

Received 26 August, 2021

Abstract—We solve the problem of describing square-free polynomials $f(x) \in k[x]$ with a periodic expansion of $\sqrt {f(x)} $ into a functional continued fraction in $k((x))$, where k is a number field and the degree of the corresponding fundamental $S$-unit of the hyperelliptic field $k(x)(\sqrt {f(x)} )$ is less than or equal to 11.

Keywords: hyperelliptic field, S-units, continued fractions, periodicity, torsion points

DOI: 10.1134/S1064562421050082