Periodic continued fractions and $S$-units with second degree valuations in hyperelliptic fields

Based on the method of continued fractions by now the problem of the existence and construction of nontrivial $S$-units is deeply studied in hyperelliptic fields in the case when the set $S$ consists of two linear valuations. This article is devoted to a more general problem, namely the problem of the existence and construction of fundamental $S$-units in hyperelliptic fields for sets $S$ containing valuations of the degree $2$. The key case when the set $S = S_h$ consists two conjugate valuations, connected with an irreducible polynomial $h$ of the degree $2$. The main results were obtained using the theory of generalized functional continued fractions in conjunction with the geometric approach to the problem of torsion in Jacobian varieties of hyperelliptic curves.
We have developed a theory of generalized functional continued fractions and the divisors of the hyperelliptic field associated with them, constructed with the help of valuations of the degree $2$. This theory allowed us to find new effective methods for searching and constructing fundamental $S_h$-units in hyperelliptic fields.
As a demonstration of the results, we consider in detail algorithm to search for fundamental $S_h$-units for hyperelliptic fields of genus $3$ over the field of rational numbers and give explicit computational examples of hyperelliptic fields $L = \mathbb{Q}(x)(\sqrt{f})$ for polynomials $f$ of degree $7$, possessing fundamental $S_h$-units of large powers.