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Doklady Mathematics

, Volume 98, Issue 3, pp 641–645 | Cite as

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

  • V. P. PlatonovEmail author
  • V. S. Zhgoon
  • M. M. Petrunin
  • Yu. N. Shteinikov
Mathematics
  • 3 Downloads

Abstract

We prove the finiteness of the set of square-free polynomials fk[x] of odd degree distinct from 11 considered up to a natural equivalence relation for which the continued fraction expansion of the irrationality \(\sqrt {f\left( x \right)} \) 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.

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References

  1. 1.
    N. H. Abel, J. Reine Angew. Math. 1, 185–221 (1826).MathSciNetCrossRefGoogle Scholar
  2. 2.
    P. Tchebicheff, J. Math. Pures Appl. 2, 168–192 (1857).Google Scholar
  3. 3.
    V. P. Platonov, Russ. Math. Surv. 69 (1), 1–34 (2014).CrossRefGoogle Scholar
  4. 4.
    M. W. Schmidt, Acta Arith. 95 (2), 139–166 (2000).MathSciNetCrossRefGoogle Scholar
  5. 5.
    V. P. Platonov and M. M. Petrunin, Dokl. Math. 94 (2), 532–537 (2016).MathSciNetCrossRefGoogle Scholar
  6. 6.
    V. P. Platonov and M. M. Petrunin, Russ. Math. Surv. 71 (5), 973–975 (2016).CrossRefGoogle Scholar
  7. 7.
    M. M. Petrunin, Dokl. Math. 95 (3), 222–225 (2017).MathSciNetCrossRefGoogle Scholar
  8. 8.
    V. P. Platonov and G. V. Fedorov, Dokl. Math. 96 (1), 332–335 (2017).MathSciNetCrossRefGoogle Scholar
  9. 9.
    V. P. Platonov and G. V. Fedorov, Sb. Math. 209 (4), 519–559 (2018).MathSciNetCrossRefGoogle Scholar
  10. 10.
    V. P. Platonov and G. V. Fedorov, Dokl. Math. 95 (3), 254–258 (2017).MathSciNetCrossRefGoogle Scholar
  11. 11.
    V. P. Platonov and M. M. Petrunin, Dokl. Math. 92 (3), 667–669 (2015).MathSciNetCrossRefGoogle Scholar
  12. 12.
    V. V. Benyash-Krivets and V. P. Platonov, Sb. Math. 200 (11), 1587–1615 (2009).MathSciNetCrossRefGoogle Scholar
  13. 13.
    V. P. Platonov and M. M. Petrunin, Proc. Steklov Inst. Math. 302, (2018).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • V. P. Platonov
    • 1
    Email author
  • V. S. Zhgoon
    • 1
  • M. M. Petrunin
    • 1
  • Yu. N. Shteinikov
    • 1
  1. 1.Scientific Research Institute for System AnalysisRussian Academy of SciencesMoscowRussia

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