An algorithm for solving a quartic Diophantine equation satisfying Runge's condition (Q2175593)

The classical method of \textit{ C. Runge} [J. Reine Angew. Math. 100, 425--435 (1887; JFM 19.0076.03)] is well known in diophantine number theory, see also \textit{ P. G. Walsh} [Acta Arith. 62, No. 2, 157--172 (1992; Zbl 0769.11017)]. The scope of equations that can be solved using this method is although restricted, but it extends also to surprising cases. Moreover, the resolution of these equations might be very efficient. For cubic equations \textit{ N. N. Osipov} and \textit{B. V. Gulnova} [J. Sib. Fed. Univ. Math. Phys. 11, No. 2, 137--147 (2018; Zbl 1460.11150)] gave a practical algorithm which is extended in the present paper to certain quartic equations by the authors. Moreover, the algorithm is implemented in the computer algebra system PARI/GP. For the entire collection see [Zbl 1428.68016].