On a simple quartic family of Thue equations over imaginary quadratic number fields

Let $t$ be any imaginary quadratic integer with $\mathrm{\mid }t\mathrm{\mid }\ge 100$. We prove that the inequality $$\mathrm{\mid }{F}_t(X,Y)\mathrm{\mid }=\mathrm{\mid }{X}^4-t{X}^{3}Y-6X^2{Y}^2+tX{Y}^3+Y^4\mathrm{\mid }\le 1$$ has only trivial solutions $(x,y)$ in integers of the same imaginary quadratic number field as $t$. Moreover, we prove results on the inequalities $\mathrm{\mid }{F}_t(X,Y)\mathrm{\mid }\le C\mathrm{\mid }t\mathrm{\mid }$ and $\mathrm{\mid }{F}_t(X,Y)\mathrm{\mid }\le \mathrm{\mid }t{\mathrm{\mid }}^{2-\epsilon }$. These results follow from an approximation result that is based on the hypergeometric method. The proofs in this paper require a fair amount of computations, for which the code (in Sage) is provided.