RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Chebyshevskii Sb.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Chebyshevskii Sb., 2019, Volume 20, Issue 1, Pages 248–260 (Mi cheb730)  

This article is cited in 1 scientific paper (total in 1 paper)

The criterion of periodicity of continued fractions of key elements in hyperelliptic fields

V. P. Platonovab, G. V. Fedorovbc

a Steklov Mathematical Institute (MIAN), Moscow
b Federal State Institution Scientific Research Institute for System Analysis of the Russian Academy of Sciences (SRISA)
c Moscow State University (MSU), Moscow

Abstract: The periodicity and quasi-periodicity of functional continued fractions in the hyperelliptic field $L = \mathbb{Q}(x)(\sqrt {f})$ has a more complex nature, than the periodicity of the numerical continued fractions of the elements of a quadratic fields. It is known that the periodicity of a continued fraction of the element $\sqrt{f}/h^{g + 1}$, constructed by valuation associated with a polynomial $h$ of first degree, is equivalent to the existence of nontrivial $S$-units in a field $L$ of the genus $g$ and is equivalent to the existence nontrivial torsion in a group of classes of divisors. This article has found an exact interval of values of $s \in \mathbb{Z}$ such that the elements $\sqrt {f}/h^s $ have a periodic decomposition into a continued fraction, where $f \in \mathbb{Q}[x] $ is a squarefree polynomial of even degree. For polynomials $f$ of odd degree, the problem of periodicity of continued fractions of elements of the form $\sqrt {f}/h^s $ are discussed in the article [5], and it is proved that the length of the quasi-period does not exceed degree of the fundamental $S$-unit of $L$. The problem of periodicity of continued fractions of elements of the form $\sqrt {f}/h^s$ for polynomials $f$ of even degree is more complicated. This is underlined by the example we found of a polynomial $f$ of degree $4$, for which the corresponding continued fractions have an abnormally large period length. Earlier in the article [5] we found examples of continued fractions of elements of the hyperelliptic field $L$ with a quasi-period length significantly exceeding the degree of the fundamental $S$-unit of $L$.

Keywords: continued fractions, fundamental units, $S$-units, torsion in the Jacobians, hyperelliptic fields, divisors, divisor class group.

Funding Agency Grant Number
Russian Academy of Sciences - Federal Agency for Scientific Organizations -19-119011590095-7
The publication was performed within the framework of the state assignment of SRISA (14 GP implementation of fundamental research) on the subject № 0065-2019-0011 (№-19-119011590095-7).


DOI: https://doi.org/10.22405/2226-8383-2018-20-1-248-260

Full text: PDF file (663 kB)
References: PDF file   HTML file

UDC: 511.6
Received: 02.02.2019
Accepted:10.04.2019

Citation: V. P. Platonov, G. V. Fedorov, “The criterion of periodicity of continued fractions of key elements in hyperelliptic fields”, Chebyshevskii Sb., 20:1 (2019), 248–260

Citation in format AMSBIB
\Bibitem{PlaFed19}
\by V.~P.~Platonov, G.~V.~Fedorov
\paper The criterion of periodicity of continued fractions of key elements in hyperelliptic fields
\jour Chebyshevskii Sb.
\yr 2019
\vol 20
\issue 1
\pages 248--260
\mathnet{http://mi.mathnet.ru/cheb730}
\crossref{https://doi.org/10.22405/2226-8383-2018-20-1-248-260}


Linking options:
  • http://mi.mathnet.ru/eng/cheb730
  • http://mi.mathnet.ru/eng/cheb/v20/i1/p248

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    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. P. Platonov, G. V. Fedorov, “On the problem of classification of periodic continued fractions in hyperelliptic fields”, Russian Math. Surveys, 75:4 (2020), 785–787  mathnet  crossref  crossref  isi
  • Number of views:
    This page:43
    Full text:5
    References:1

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020