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Periodicity Criterion for Continued Fractions of Key Elements in Hyperelliptic Fieldsстатья
- Авторы: Platonov V.P., Fedorov G.V.
- Журнал: Doklady Mathematics
- Том: 106
- Номер: 2
- Год издания: 2022
- Издательство: Maik Nauka/Interperiodica Publishing
- Местоположение издательства: Russian Federation
- Первая страница: 262
- Последняя страница: 269
- DOI: 10.1134/S1064562422700223
- Аннотация: The periodicity and quasi-periodicity of functional continued fractions in the hyperelliptic field has a more complex nature than the periodicity of numerical continued fractions of elementsof quadratic fields. It is known that the periodicity of a continued fraction of constructed using thevaluation associated with a first-degree polynomial h is equivalent to the existence of nontrivial S-units in afield L of genus g and is equivalent to the existence of nontrivial torsion in the divisor class group. In this article, we find an exact interval of values of such that the elements have a periodic continued fraction expansion, where is a square-free polynomial of even degree. For polynomials f of odd degree,the periodicity problem for continued fractions of elements of the form was discussed in [5], where itwas proved that the length of the quasi-period does not exceed the degree of the fundamental S-unit of L. Forpolynomials f of even degree, the periodicity of continued fractions of elements of the form is a morecomplicated problem. This is underlined by an example we have found, namely, a polynomial f of degree 4for which the corresponding continued fraction has an abnormally long period. Earlier in [5], for elements ofa hyperelliptic field L, we found examples of continued fractions with a quasi-period length significantlyexceeding the degree of the fundamental S-unit of L.
- Добавил в систему: Федоров Глеб Владимирович