On a family of relative quartic Thue inequalities Original Research Article

We consider the relative Thue inequalitiesX4−t2X2Y2+s2Y4less-than-or-equals, slantt2−s2−2, 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:X4−t2X2Y2+Y4less-than-or-equals, slantt2−3. We find all solutions to these Thue inequalities for image. Moreover we solve the relative Thue equationsX4−t2X2Y2+Y4=μ for image, 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.