Complete solution of a family of quartic Thue equations
This is the second example of a family of quartic Thue equations, depending on an integer parameter \(a\), which is completely solved for all values of \(a\). The first analogous example is due to \textit{M. Mignotte}, the second named author and \textit{R. Roth} [Math. Comput. 65, 341-354 (1996; see the following review)]. The authors solve completely the quartic Thue equations \[ F_a (x, y)= x^4- ax^3 y- 6x^2 y^2+ axy^3+ y^4= c, \qquad c\in \{\pm 1, \pm 4\} \] for all values of \(a\). Their method is much in the spirit of the aforementioned paper but, certainly, less technical -- due to the nice properties of the number fields defined by a root of \(F_a (X, 1)\) -- and by far less demanding in computer assistance. These features make the present paper more easily readable than that of Mignotte et al, which means that, if one wants to understand the underlying method in both these papers, one should probably start form the one under review. An interesting proposition characterizing the units of the maximal order of the corresponding quartic fields is also proved.
scientific article; zbMATH DE number 878391
scientific article