Thomas' family of Thue equations over imaginary quadratic fields
Let \(t\) be an integer, \(\mu=\pm 1\) and consider the solutions of \[ x^3-(t-1)x^2y-(t+2)xy^2-y^3=\mu \] in \(x,y\in\mathbb Z\). In his celebrated result \textit{E. Thomas} [J. Number Theory 34, 235--250 (1990; Zbl 0697.10011)] proved that if the parameter \(t\) is large enough, then the equation has only trivial solutions. In the present paper the authors extend Thomas's result to the case of imaginary quadratic ground fields: this is the first result of this type, hence the paper has a particular importance in this respect. Let \(t\) be an integer of an imaginary quadratic number field, \(\mu\) a root of unity in the same field and consider the solutions \((x,y)\) of the above equation in integers of this field. The authors prove that if \(|t|\) is large enough or the discriminant of the quadratic number field is large enough (in absolute value) or \(\text{Re}\;t=-1/2\), then the equation has only trivial solutions with \(|x|,|y|\leq 1\) (with some exceptional cases stated explicitly).
scientific article; zbMATH DE number 1886759
scientific article