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

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 ÀÀÀÀ-À19-119011590095-7 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 (№ÀÀÀÀ-À19-119011590095-7).

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

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UDC: 511.6
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} 

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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
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