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\(S\)-units and periodicity of continued fractions in hyperelliptic fields. (English. Russian original) Zbl 1337.11040

Dokl. Math. 92, No. 3, 752-756 (2015); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 465, No. 5, 537-541 (2015).
Let \(K\) be a field of characteristic different from \(2\), \(f\in K[x]\) be a square free polynomial such that \(\deg f=2s+1\) where \(s\geq 1\), \(h \in K[x]\) such that \(\deg h=1\) and the lowest coefficient of the polynomial \(f=f(h)\) differs from zero and is \(s\) a complete square in the field \(K\). We denote by \(v_h\) the valuation corresponding to \(h\) in \(K(x)\). Suppose that \(v_h\) has two extensions, \(v^{-}_{h}\) and \(v^{+}_h\), to the field \(L=K(x)(\sqrt f)\). We set \(S=\{v^-_h,v_\infty\}\) where \(v_\infty\) is the infinite valuation of the field \(L\). By \(O_S\) we denote the ring of \(S\)-integer elements of \(L\) and \(O_S^\ast\) is called the group of \(S\)-units. In this paper, the authors, study a relationship between the problem of the existence of non trivial \(S\)-units in the field \(L\) and the periodicity of the continued fraction expansion of certain key element of \(L\) (as \(\sqrt f\)).

MSC:

11G16 Elliptic and modular units
11A55 Continued fractions
11R27 Units and factorization
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References:

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