Calculating "small" solutions of inhomogeneous relative Thue inequalities

TL;DR

This paper develops a fast algorithm for finding small solutions (up to size 10^100) of inhomogeneous relative Thue inequalities, which are useful in various number theory applications and can often be reduced to absolute cases.

Contribution

It introduces a new efficient algorithm for small solutions of inhomogeneous relative Thue inequalities, extending previous methods for Thue equations.

Findings

Algorithm effectively finds small solutions up to size 10^100.

Reduces totally real cases to absolute inhomogeneous Thue inequalities.

Applications include solving certain resultant equations in the relative case.

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 $\leq 10^{100}$. 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 $\Bbb Z$). As a common generalization of the above results, in our paper we develop a fast algorithm for calculating "small" solutions (say with sizes $\leq 10^{100}$) 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.