Continued fractions and \(S\)-units in function fields (Q542259)

Let \(k\) be an arbitrary field, \(k(x)\) the rational function field, \(\nu\in k[x]\) a polynomial of degree \(1\), \(k(x)_\nu\) the completion at \(\nu\), \(\beta\in k(x)_\nu\). The authors prove that if a fraction \(\frac ab\in k(x)\) is a best approximation to \(\beta \), then there exists a convergent \(\frac{p_n}{q_n}\) of \(\beta\) and a constant \(c\in k^*\) such that \(a=cp_n\) and \(b=cq_n\). They show how continued fractions can be applied to find fundamental \(S\)-units in hyperelliptic fields.