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Simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. (English) Zbl 1479.11054

For \(t\in\mathbb{Z}\), set \[F_t^{(4)}(x, y)=x^4-tx^3y-6x^2y^2+txy^3+y^4\] and \[F_t^{(6)}(x, y)=x^6-2tx^5y-(5t+15)x^4y^2-20x^3y^3+5tx^2y^4+(2t+6)xy^5+y^6. \] Let \(m\) be a square-free positive integer, set \(M=\mathbb{Q}(i\sqrt{m})\) and denote the ring of integers of \(M\) by \(\mathbb{Z}_M\). These two forms correspond to the families of quartic and sextic fields.
For all \(t\in\mathbb{Z}\setminus\{-3, 0, 3\}\), the authors give all solutions \((x, y)\in\mathbb{Z}_M^2\) of the relative quartic Thue equations \(|F_t^{(4)}(x, y)|\le 1\).
For all \(t\in\mathbb{Z}\setminus\{-8, -3, 0, 5\}\), the authors give all solutions \((x, y)\in\mathbb{Z}_M^2\) of the relative sextic Thue equations \(|F_t^{(6)}(x, y)|\le 1\).
In both results, for the excluded values of \(t\), the forms are reducible over \(\mathbb{Z}\).
The method consists of a reduction to the absolute case proved by the authors [I. Gaál et al., Glas. Mat., III. Ser. 53, No. 2, 229–238 (2018; Zbl 1455.11052)] and applying the results for the absolute case proved by [G. Lettl et al., Trans. Am. Math. Soc. 351, No. 5, 1871–1894 (1999; Zbl 0920.11041)].

MSC:

11D59 Thue-Mahler equations
11D57 Multiplicative and norm form equations

Software:

Magma; Maple
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Full Text: DOI arXiv

References:

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