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On Some Properties of Continued Periodic Fractions with Small Length of Period Related with Hyperelliptic Fields and S-units
| Content Provider | Semantic Scholar |
|---|---|
| Author | Kuznetsov, Yu. V. Shteinikov, Yu. N. |
| Copyright Year | 2018 |
| Abstract | Let Q be a field of rational numbers, let Q(x) be the field of rational functions of one variable, and let f ∈ Q[x] be a squarefree polynomial of odd degree which is equal to 2g+1, g > 0. Suppose that for a polynomial h of degree 1 the discrete valuation νh which is uniquely defined on Q(x) has two nonequivalent extensions to the field L = Q(x)( √ f) and ν′ h is one of these extensions. We put S = {ν′ h, ν∞}, where ν∞ is an infinite valuation of the field L. In the paper [4] V. P. Platonov and M. M. Petrunin (see also [2]) obtained that a S-unit in L exists if and only if the element √ f hg+1 expands into the periodic infinite continuous functional fraction. In this paper we study continuous periodic fractions connected with this expansion. For some small values of the length of period and quasiperiod we obtained estimates for the degrees of corresponding fundamental S-units and some necessary conditions with which the elements of these fractions have to satisfy. Исследование выполнено за счет гранта Российского научного фонда (проект No 16-11-10111). О НЕКОТОРЫХ СВОЙСТВАХ НЕПРЕРЫВНЫХ ПЕРИОДИЧЕСКИХ ДРОБЕЙ . . . 261 In the proof we use the results obtained by V. P. Platonov and M. M. Petrunin in the paper [4]. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://www.chebsbornik.ru/jour/article/viewFile/397/359 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
