Calculating “small” solutions of inhomogeneous relative Thue inequalities

Abstract

Thue equations and their relative and inhomogeneous extensions are well known in the literature. There exist methods, usually tedious methods, for the complete resolution of these equations. On the other hand our experiences show that such equations usually do not have extremely large solutions. Therefore in several applications it is useful to have a fast algorithm to calculate the "small" solutions of these equations. Under "small" solutions we mean the solutions, say, with absolute values or sizes ≤ 10 100 . Such algorithms were formerly constructed for Thue equations, relative Thue equations. The relative and inhomogeneous Thue equations have applications in solving index form equations and certain resultant form equations. It is also known that certain "totally real" relative Thue equations can be reduced to absolute Thue equations (equations over Z).As a common generalization of the above results, in our paper we develop a fast algorithm for calculating "small" solutions (say with sizes ≤ 10 100 ) of inhomogeneous relative Thue equations, more exactly of certain inequalities that generalize those equations. We shall show that in the "totally real" case these can similarly be reduced to absolute inhomogeneous Thue inequalities. We also give an application to solving certain resultant equations in the relative case.

Monogenity in Totally Complex Sextic Fields, Revisited

“…That method was further developed in [9]. However some ideas of [8] lead to a considerable improvement of that algorithm, what we are going to detail here. We also note that a parametric family of this type of number fields, consisting of composites of the simplest cubic fields and imaginary quadratic fields was studied in [10], but applying results on the connected simplest family of relative Thue equations, which counts as a more complicated approach.…”