On the Finiteness of Hyperelliptic Fields with the Special Properties and Periodic Expansion of √f

,; Petrunin, M.; Zhgun, Vladimir; Shteynikov, Yuriy

doi:10.31857/S086956520003431-7

Home

Doklady Akademii nauk

Issue 6

On the Finiteness of Hyperelliptic Fields with the Special Properties and Periodic Expansion of √f

Annotation
Reviews
References

On the Finiteness of Hyperelliptic Fields with the Special Properties and Periodic Expansion of √f

PII

S086956520003431-7-1

DOI

10.31857/S086956520003431-7

Publication type

Article

Status

Published

Authors

Affiliation: Scientific Research Institute for System Analysis, RAS
Address: Russian Federation

M. Petrunin

Affiliation: Scientific Research Institute for System Analysis, RAS
Address: Russian Federation

Vladimir Zhgun

Affiliation: Scientific Research Institute for System Analysis, RAS
Address: Russian Federation

Yuriy Shteynikov

Affiliation: Scientific Research Institute for System Analysis, RAS
Address: Russian Federation

Journal name

Doklady Akademii nauk

Edition

Volume 483 Issue 6

Pages

609-613

Abstract

We prove the finiteness of the set of square-free polynomials f ∈ k[x] of odd degree, distinct from 11, considered up to a natural equivalence relation for which the continued fraction expansion of the irrationality √f(x) in k((x)) is periodic and the corresponding hyperelliptic field k(x)(√f) contains an S-unit of degree 11. Moreover it was proved for k = Q that there are no polynomials of odd degree distinct from 9 and 11, satisfying the conditions mentioned above.

Keywords

Received

26.12.2018

Publication date

26.12.2018

Cite Download pdf To download PDF you should sign in

GOST	, Petrunin M., Zhgun V., Shteynikov Y. On the Finiteness of Hyperelliptic Fields with the Special Properties and Periodic Expansion of √f // Doklady Akademii nauk – 2018. – V. 483. – Issue 6 C. 609-613 [Electronic resource]. URL: http://ras.jes.su/dan/s086956520003431-7-1-en (circulation date: 06.05.2024). DOI: 10.31857/S086956520003431-7
MLA	, , Petrunin, M., Zhgun, Vladimir, Shteynikov, Yuriy "On the Finiteness of Hyperelliptic Fields with the Special Properties and Periodic Expansion of √f." Doklady Akademii nauk 483.6 (2018):609-613. DOI: 10.31857/S086956520003431-7
APA	, Petrunin M., Zhgun V., Shteynikov Y. (2018). On the Finiteness of Hyperelliptic Fields with the Special Properties and Periodic Expansion of √f. Doklady Akademii nauk 483(6), pp.609-613 DOI: 10.31857/S086956520003431-7

Размещенный ниже текст является ознакомительной версией и может не соответствовать печатной

Readers community rating: votes 0

1. Abel N.H. Ueber die Integration der Dierential-Formel pdx=vR wenn R und p ganze Functionen sind // Journal fur die reine und angewandte Mathematik. 1826. Vol. 1. P. 185221.

2. Tchebiche P. Sur l'integration des dierentielles qui contiennent une racine carree d'un polynome du troisieme ou du quatrieme degre' // Journal des math. pures et appl. 1857. Vol. 2. P. 168192.

3. Platonov V. P. Teoretiko-chislovye svojstva giperehllipticheskikh polej i problema krucheniya v yakobianakh giperehllipticheskikh krivykh nad polem ratsional'nykh chisel // UMN. 2014. T. 69:1, № 415. S. 338.

4. Schmidt Wolfgang M. On continued fractions and Diophantine approximation in power series elds // Acta arithmetica. 2000. Vol. 95, no. 2. P. 139166.

5. Platonov V. P., Petrunin M. M. S-edinitsy v giperehllipticheskikh polyakh i periodichnost' nepreryvnykh drobej // DAN. 2016. T. 470, № 3. S. 260265.

6. Platonov V. P., Petrunin M. M. S-edinitsy i periodichnost' v kvadratichnykh funktsional'nykh polyakh // UMN. 2016. T. 71, № 5. S. 181182.

7. Petrunin M. M. S-edinitsy i periodichnost' kvadratnogo kornya v giperehllipticheskikh polyakh. 2017. T. 474, № 2. S. 155158.

8. Platonov V. P., Fedorov G. V. O periodichnosti nepreryvnykh drobej v ehllipticheskikh polyakh // DAN. 2017. T. 475, № 2. S. 133136.

9. Platonov V. P., Fedorov G. V. O probleme periodichnosti nepreryvnykh drobej v giperehllipticheskikh polyakh // Matematicheskij sbornik. 2018. T. 4, № 209. S. 5494.

10. Platonov V. P., Fedorov G. V. O periodichnosti nepreryvnykh drobej v giperehllipticheskikh polyakh // DAN. 2018. T. 474, № 5. S. 540544.

11. Platonov V. P., Petrunin M. M. Fundamental'nye S-edinitsy v giperehllipticheskikh polyakh i problema krucheniya v yakobianakh giperehllipticheskikh krivykh // DAN. 2015. T. 465, № 1. S. 2325.

12. Benyash-Krivets V. V., Platonov V. P. Gruppy S-edinits v giperehllipticheskikh polyakh i nepreryvnye drobi // Matematicheskij sbornik. 2009. T. 200, № 11. S. 1544.

13. Platonov V. P., Petrunin M.M. Gruppy S-edinits i problema periodichnosti nepreryvnykh drobej v giperehllipticheskikh polyakh // Sbornik statej, Tr. MIAN, 2018. T. 302, S.