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

Abstract

Let $t$ be any imaginary quadratic integer with $|t|\ge 100$. We prove that the inequality \[ |F_t(X,Y)| = | X^4 - t X^3 Y - 6 X^2 Y^2 + t X Y^3 + Y^4 | \leq 1 \] has only trivial solutions $(x,y)$ in integers of the same imaginary quadratic number field as $t$. Moreover, we prove results on the inequalities $|F_t(X,Y)|\le C|t|$ and $|F_t(X,Y)|\le |t{|}^{2-\epsilon }$. 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.

27 pages