Solving genus zero Diophantine equations with at most two infinite valuations (Q1600044)

Let \(f(X,Y)\) be an irreducible polynomial such that the curve defined by \(f(X,Y)=0\) has genus zero. Denote by \(C\) the projective curve \(F(X,Y,Z)=0\) obtained by homogenizing this curve, and denote by \(C_{\infty}\) the set of infinite valuations of the corresponding function field. It is well known that if \(|C_{\infty}|\geq 3\) then the equation \(f(X,Y)=0\) has only finitely many integer solutions. Bounds for the solutions of such curves were given by \textit{D. Poulakis} [Colloq. Math. 66, 1--7 (1993; Zbl 0817.11023), Acta Math. Hung. 23, 327--346 (2001)], and methods for solving these equations were described by \textit{D. Poulakis} and \textit{E. Voskos} [J. Symb. Comput. 30, 573--582 (2000; Zbl 0985.11051)]. In case \(|C_{\infty}|\leq 2\), studied in this paper, the curve may have infinitely many integer solutions. The authors give a practical solution to find these integer solutions. The method is based on the construction of a parametrization of the curve, the resolution of polynomial congruences and generalized Pellian equations. The paper is illustrated with interesting examples.