On the resolution of relative Thue equations (Q2759111)

Let \(M \subset K\) be algebraic number fields with rings of integers \({\mathbb Z}_M\) and \({\mathbb Z}_K\), and assume that \([K:M]\geq 3\). Let \(K=M(\alpha)\) for some \(\alpha\in{\mathbb Z}_K\), \(\mu\in {\mathbb Z}_M\) be an algebraic integer and \(\eta\) be a unit in \({\mathbb Z}_M\). The authors consider the relative Thue equation \(N_{K/M}(X-\alpha Y)=\eta\mu\) in \(X\), \(Y\in{\mathbb Z}_M\).NEWLINENEWLINENEWLINEA solution \(X-\alpha Y\) is usually written as \(\nu \varepsilon_1^{a_1}\cdots\varepsilon_r^{a_r}\) for fundamental units \(\varepsilon_1\), \dots, \(\varepsilon_r\), integer exponents \(a_1\), \dots, \(a_r\), and a \(\nu\in{\mathbb Z}_K\) coming from a finite set of non-associated numbers of norm \(\mu\).NEWLINENEWLINENEWLINEIt is well known how (huge) upper bounds for the \(a_i\) can be obtained by using the theory of linear forms in logarithms and how to reduce these bounds using the LLL-reduction algorithm.NEWLINENEWLINENEWLINEHowever, even the final step of enumerating all possible solutions when the upper bound for the \(a_i\) has been reduced to \(100\), for example, is nontrivial for larger unit rank \(r\). Sieve methods are frequently employed. NEWLINENEWLINENEWLINEThe authors of the present paper show how an enumeration procedure due to \textit{K. Wildanger} [Über das Lösen von Einheiten- und Indexformgleichungen in algebraischen Zahlkörpern mit einer Anwendung auf die Bestimmung aller ganzen Punkte einer Mordellschen Kurve, Dissertation, Technical University Berlin (1997; Zbl 0912.11061)] -- which is based on the ellipsoid method of Fincke and Pohst -- can be adapted to solve this final enumeration step.NEWLINENEWLINENEWLINENumerical examples are given. Applications include the calculation of power integral bases of algebraic number fields.