Abstract:
For an arbitrary odd-degree polynomial $ f $ over an arbitrary field of algebraic numbers $ \mathbb K $, the class of always quasiperiodic elements in $ \mathbb K((x)) $ of the form $ \frac{v + w \sqrt{f}}{u} $, where $ v, w, u \in \mathbb K[x] $, in the hyperelliptic field $ \mathbb K(x)(\sqrt{f}) $, has been determined. This class is characterized by certain relationships involving the polynomials $ u, v, w, $ and $ f $, as well as their degrees. The class is guaranteed to be nonempty if at least one quasiperiodic element exists in the hyperelliptic field. Furthermore, a specific subclass of always periodic elements has been identified within this broader class.
Keywords:hyperelliptic field, continued fractions, functional continued fractions, $S$-units, periodicity, quasiperiodicity, pseudoperiodicity.
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