On a simple quartic family of Thue equations over imaginary quadratic number fields
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 in...
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Zusammenfassung: | 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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DOI: | 10.48550/arxiv.2303.15243 |