Groups of $S$-units in hyperelliptic fields and continued fractions

New methods for calculating fundamental $S$-units in hyperelliptic fields are found. Continued fractions in function fields are investigated. As an application, it is proved that if a valuation is defined by a linear polynomial, then a fundamental $S$-unit in a hyperelliptic field can be found by expanding certain elements into continued fractions.
Bibliography: 11 titles.