On a family of relative quartic Thue inequalities

We consider the relative Thue inequalities|X4−t2X2Y2+s2Y4|⩽|t|2−|s|2−2,|X4−t2X2Y2+s2Y4|⩽|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=1s=1:|X4−t2X2Y2+Y4|⩽|t|2−3.|X4−t2X2Y2+Y4|⩽|t|2−3. We find all solutions to these Thue inequalities for |t|>550. Moreover we solve the relative Thue equationsX4−t2X2Y2+Y4=μX4−t2X2Y2+Y4=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.