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Mat. Sb., 2009, Volume 200, Number 11, Pages 15–44 (Mi msb7587)  

This article is cited in 20 scientific papers (total in 22 papers)

Groups of $S$-units in hyperelliptic fields and continued fractions

V. V. Benyash-Krivetsa, V. P. Platonovb

a Belarusian State University
b Scientific Research Institute for System Studies of RAS

Abstract: New methods for calculating fundamental $S$-units in hyperelliptic fields are found. Continued fractions in function fields are investigated. As an application, it is proved that if a valuation is defined by a linear polynomial, then a fundamental $S$-unit in a hyperelliptic field can be found by expanding certain elements into continued fractions.
Bibliography: 11 titles.

Keywords: $S$-units, valuations, hyperelliptic fields, continued fractions, best approximations.

DOI: https://doi.org/10.4213/sm7587

Full text: PDF file (632 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2009, 200:11, 1587–1615

Bibliographic databases:

UDC: 511.6
MSC: Primary 11R58; Secondary 11A55, 11R27
Received: 10.06.2009

Citation: V. V. Benyash-Krivets, V. P. Platonov, “Groups of $S$-units in hyperelliptic fields and continued fractions”, Mat. Sb., 200:11 (2009), 15–44; Sb. Math., 200:11 (2009), 1587–1615

Citation in format AMSBIB
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  • http://mi.mathnet.ru/eng/msb/v200/i11/p15

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. V. Benyash-Krivets, V. P. Platonov, “A new local-global principle for quadratic functional fields”, Dokl. Math., 82:1 (2010), 531–534  crossref  mathscinet  zmath  isi  elib  elib  scopus
    2. V. P. Platonov, “Arifmeticheskie svoistva funktsionalnykh kvadratichnykh polei i kruchenie v yakobianakh”, Chebyshevskii sb., 11:1 (2010), 234–238  mathnet  mathscinet
    3. Yu. V. Kuznetsov, M. M. Petrunin, “Bystryi algoritm proverki vyrozhdennosti gankelevykh matrits”, Chebyshevskii sb., 12:2 (2011), 60–67  mathnet  mathscinet
    4. V. P. Platonov, “Number-theoretic properties of hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves over the rational number field”, Russian Math. Surveys, 69:1 (2014), 1–34  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. S. I. Adian, V. V. Benyash-Krivets, V. M. Buchstaber, E. I. Zelmanov, V. V. Kozlov, G. A. Margulis, S. P. Novikov, A. N. Parshin, G. Prasad, A. S. Rapinchuk, L. D. Faddeev, V. I. Chernousov, “Vladimir Petrovich Platonov (on his 75th birthday)”, Russian Math. Surveys, 70:1 (2015), 197–201  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. S. I. Adyan, V. V. Benyash-Krivets, V. M. Bukhshtaber, E. I. Zelmanov, V. V. Kozlov, G. A. Margulis, S. P. Novikov, A. N. Parshin, G. Prasad, A. S. Rapinchuk, L. D. Faddeev, V. I. Chernousov, “Vladimir Petrovich Platonov (k 75-letiyu so dnya rozhdeniya)”, Chebyshevskii sb., 16:4 (2015), 6–10  mathnet
    7. M. M. Petrunin, “Vychislenie fundamentalnykh $S$-edinits v giperellipticheskikh polyakh roda $2$ i problema krucheniya v yakobianakh giperellipticheskikh krivykh”, Chebyshevskii sb., 16:4 (2015), 250–283  mathnet  elib
    8. Platonov V.P., Petrunin M.M., “Fundamental S-units in hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves”, Dokl. Math., 92:3 (2015), 667–669  mathnet  crossref  mathscinet  zmath  isi  scopus
    9. Platonov V.P., Fedorov G.V., “S-units and periodicity of continued fractions in hyperelliptic fields”, Dokl. Math., 92:3 (2015), 752–756  mathnet  crossref  mathscinet  zmath  isi  scopus
    10. V. P. Platonov, M. M. Petrunin, “$S$-Units and periodicity in quadratic function fields”, Russian Math. Surveys, 71:5 (2016), 973–975  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. Platonov V.P., Petrunin M.M., “S-units in hyperelliptic fields and periodicity of continued fractions”, Dokl. Math., 94:2 (2016), 532–537  mathnet  crossref  mathscinet  zmath  isi  scopus
    12. Platonov V.P., Zhgoon V.S., Fedorov G.V., “Continued rational fractions in hyperelliptic fields and the Mumford representation”, Dokl. Math., 94:3 (2016), 692–696  mathnet  crossref  mathscinet  zmath  isi  elib  scopus
    13. Fedorov G.V., “on the Periodicity of Continued Fractions in Hyperelliptic Fields”, Advances in Dynamical Systems and Control, Studies in Systems Decision and Control, 69, eds. Sadovnichiy V., Zgurovsky M., Springer Int Publishing Ag, 2016, 141–157  crossref  mathscinet  zmath  isi  scopus
    14. Petrunin M.M., “S-Units and Periodicity of Square Root in Hyperelliptic Fields”, Dokl. Math., 95:3 (2017), 222–225  crossref  mathscinet  zmath  isi  scopus
    15. Platonov V.P., Fedorov G.V., “on the Periodicity of Continued Fractions in Hyperelliptic Fields”, Dokl. Math., 95:3 (2017), 254–258  mathnet  crossref  zmath  isi  scopus
    16. Platonov V.P., Fedorov G.V., “on the Periodicity of Continued Fractions in Elliptic Fields”, Dokl. Math., 96:1 (2017), 332–335  mathnet  crossref  mathscinet  zmath  isi  scopus
    17. V. S. Zhgun, “Obobschennye yakobiany i nepreryvnye drobi v giperellipticheskikh polyakh”, Chebyshevskii sb., 18:4 (2017), 209–221  mathnet  crossref
    18. Yu. V. Kuznetsov, Yu. N. Shteinikov, “O nekotorykh svoistvakh nepreryvnykh periodicheskikh drobei s nebolshoi dlinoi perioda, svyazannykh s giperellipticheskimi polyami i $S$-edinitsami”, Chebyshevskii sb., 18:4 (2017), 261–268  mathnet  crossref
    19. V. P. Platonov, G. V. Fedorov, “On the problem of periodicity of continued fractions in hyperelliptic fields”, Sb. Math., 209:4 (2018), 519–559  mathnet  crossref  crossref  adsnasa  isi  elib
    20. V. P. Platonov, M. M. Petrunin, “Groups of $S$-units and the problem of periodicity of continued fractions in hyperelliptic fields”, Proc. Steklov Inst. Math., 302 (2018), 336–357  mathnet  crossref  crossref  isi  elib
    21. Platonov V.P., Zhgoon V.S., Fedorov G.V., “On the Periodicity of Continued Fractions in Hyperelliptic Fields Over Quadratic Constant Field”, Dokl. Math., 98:2 (2018), 430–434  crossref  mathscinet  zmath  isi  scopus
    22. V. P. Platonov, V. S. Zhgoon, M. M. Petrunin, Yu. N. Shteinikov, “On the finiteness of hyperelliptic fields with special properties and periodic expansion of $\sqrt f$”, Dokl. Math., 98:3 (2018), 641–645  mathnet  crossref  crossref  zmath  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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