Bounds for the size of integral points on curves of genus zero
The author gives improved bounds for the size of integral points on curves of genus zero defined over an algebraic number field in the case when there are at least three points at infinity on the curve. The bounds are of the form \[ \max \{ H_K(x),H_K(y) \} < \exp \{ \Omega \cdot D^A \cdot H_K(F)^B \} \] where \(K\) is the number field and \(D\) is the absolute value of its discriminant, \(F(X,Y)=0\) is the equation of the curve, and \(H_K\) denotes the height function. There are five cases, the simplest to state is when the curve has an ordinary singular point at infinity of multiplicity \(\geq 3\). Then \(\Omega = d^{14dN^4} N^{52dN^7}\), where \(d\) is the degree of \(K\) over the rationals and \(N\) is the degree of \(F\), \(A = 4N^6\) and \(B =13N^6\). In the proof the author uses his earlier results on the size of integral solutions of Thue equations [J. Aust. Math. Soc., Ser. A 63, 145-164 (1997; Zbl 0893.11010)] and on effective parametrization of curves of genus zero [Acta Arith. 86, 51-90 (1998; Zbl 0934.11035)].
scientific article; zbMATH DE number 1822360