On a family of relative quartic Thue inequalities

Abstract

We consider the relative Thue inequalities | X 4 − t 2 X 2 Y 2 + s 2 Y 4 | ⩽ | t | 2 − | s | 2 − 2 , where the parameters s and t and the solutions X and Y are integers in the same imaginary quadratic number field and t is sufficiently large with respect to s. Furthermore we study the specialization to s = 1 : | X 4 − t 2 X 2 Y 2 + Y 4 | ⩽ | t | 2 − 3 . We find all solutions to these Thue inequalities for | t | > 550 . Moreover we solve the relative Thue equations X 4 − t 2 X 2 Y 2 + Y 4 = μ for | t | > 245 , where the parameter t, the root of unity μ and the solutions X and Y are integers in the same imaginary quadratic number field. We solve these Thue inequalities respectively equations by using the method of Thue–Siegel.