V. P. Platonov, M. M. Petrunin, V. S. Zhgoon, Yu. N. Shteinikov, “On the Finiteness of Hyperelliptic Fields with Special Properties and Periodic Expansion of $\sqrt f$”, Doklady Akademii Nauk, 2018, Volume 483, Number 6, Pages 609

Abstract: We prove the finiteness of the set of square-free polynomials

f \in k [x]

$f \in k[x]$ of odd degree distinct from 11 considered up to a natural equivalence relation for which the continued fraction expansion of the irrationality

\sqrt{f (x)}

$\sqrt{f(x)}$ in

k ((x))

$k((x))$ is periodic and the corresponding hyperelliptic field

k (x) (\sqrt{f})

$k(x)(\sqrt f)$ contains an

S

$S$ -unit of degree 11. Moreover, it was proved for

$k = \mathbb{Q}$ that there are no polynomials of odd degree distinct from 9 and 11 satisfying the conditions mentioned above.

This publication is cited in the following 12 articles:

V. P. Platonov, V. S. Zhgun, G. V. Fedorov, “Teoremy konechnosti dlya obobschennykh yakobianov s netrivialnym krucheniem”, Matem. sb., 216:4 (2025), 113–131
V. P. Platonov, M. M. Petrunin, “New Results on the Periodicity Problem for Continued Fractions of Elements of Hyperelliptic Fields”, Proc. Steklov Inst. Math., 320 (2023), 258–266
G. V. Fedorov, “Continued Fractions and the Classification Problem for Elliptic Fields Over Quadratic Fields of Constants”, Math. Notes, 114:6 (2023), 1195–1211
V. P. Platonov, V. S. Zhgoon, M. M. Petrunin, “On the problem of periodicity of continued fraction expansions of $\sqrt{f}$ for cubic polynomials $f$ over algebraic number fields”, Sb. Math., 213:3 (2022), 412–442
V. P. Platonov, G. V. Fedorov, “On the classification problem for polynomials $f$ with a periodic continued fraction expansion of $\sqrt{f}$ in hyperelliptic fields”, Izv. Math., 85:5 (2021), 972–1007
V. P. Platonov, M. M. Petrunin, Yu. N. Shteinikov, “On the periodicity problem for the continued fraction expansion of elements of hyperelliptic fields with fundamental $S$-units of degree at most 11”, Dokl. Math., 104:5 (2021), 258–263
G. V. Fedorov, “On $S$-units for valuations of the second degree in hyperelliptic fields”, Izv. Math., 84:2 (2020), 392–435
S. I. Adian, V. M. Buchstaber, E. I. Zelmanov, S. V. Kislyakov, V. V. Kozlov, Yu. V. Matiyasevich, S. P. Novikov, D. O. Orlov, A. N. Parshin, V. L. Popov, D. V. Treschev, “Vladimir Petrovich Platonov (on his 80th birthday)”, Russian Math. Surveys, 75:2 (2020), 387–391
V. P. Platonov, M. M. Petrunin, Yu. N. Shteinikov, “Periodicheskie elementy $\sqrt{f}$ v ellipticheskikh polyakh s polem konstant nulevoi kharakteristiki”, Chebyshevskii sb., 21:1 (2020), 273–296
G. V. Fedorov, “O semeistvakh giperellipticheskikh krivykh nad polem ratsionalnykh chisel, yakobiany kotorykh soderzhat tochki krucheniya dannykh poryadkov”, Chebyshevskii sb., 21:1 (2020), 322–340
V. P. Platonov, V. S. Zhgoon, M. M. Petrunin, “On the problem of periodicity of continued fraction expansions of $\sqrt{f}$ for cubic polynomials over number fields”, Dokl. Math., 102:1 (2020), 288–292
V. P. Platonov, M. M. Petrunin, Yu. N. Shteinikov, “On the Finiteness of the Number of Elliptic Fields with Given Degrees of $S$-Units and Periodic Expansion of $\sqrt f$”, Dokl. Math., 100:2 (2019), 1–5