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Mathematics of Computation

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On quartic Thue equations with trivial solutions

Author: R. J. Stroeker
Journal: Math. Comp. 52 (1989), 175-187
MSC: Primary 11D25
MathSciNet review: 946605
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Abstract: Let K be a quartic number field with negative absolute discriminant and let $ {\mathbf{L}} = {\mathbf{Q}}(\sqrt d )$ be its real quadratic subfield, with $ d \equiv 3\;\pmod 4$. Moreover, assume K to be embedded in the reals. Further, let $ \xi > 1$ generate the subgroup of units relative to L in the group of positive units of K. Under certain conditions, which can be explicitly checked, and for suitable linear forms $ X(u,v)$ and $ Y(u,v)$ with coefficients in $ {\mathbf{Z}}[\sqrt d ]$, the diophantine equation

$\displaystyle {\text{Norm}_{{\mathbf{K}}/{\mathbf{Q}}}}(X(u,v) + Y(u,v){\xi ^2}) = 1,$

which is a quartic Thue equation in the indeterminates u and v, has only trivial solutions, that is, solutions given by $ XY = 0$.

Information on a substantial number of equations of this type and their associated number fields is incorporated in a few tables.

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Article copyright: © Copyright 1989 American Mathematical Society

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