
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. Platonov^{ab}, G. V. Fedorov^{bc} ^{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 quasiperiodicity 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 quasiperiod 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 quasiperiod 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.
DOI:
https://doi.org/10.22405/222683832018201248260
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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 248260
\mathnet{http://mi.mathnet.ru/cheb730}
\crossref{https://doi.org/10.22405/222683832018201248260}
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http://mi.mathnet.ru/eng/cheb730 http://mi.mathnet.ru/eng/cheb/v20/i1/p248
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This publication is cited in the following articles:

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

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