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arXiv · 2303.15243

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

Benjamin Earp-Lynch·Bernadette Faye·Eva G. Goedhart·Ingrid Vukusic·Daniel P. Wisniewski
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Abstract

Let $t$ be any imaginary quadratic integer with $|t|\geq 100$. 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 $|F_t(X,Y)| \leq C|t|$ and $|F_t(X,Y)| \leq |t|^{2 -\varepsilon}$. 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.

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Benjamin Earp-Lynch, Bernadette Faye, Eva G. Goedhart, Ingrid Vukusic, Daniel P. Wisniewski. 2023-03-27. On a simple quartic family of Thue equations over imaginary quadratic number fields. https://arxiv.org/abs/2303.15243

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