Calculating "small" solutions of relative Thue equations

TL;DR

This paper develops a fast algorithm based on LLL reduction to efficiently find small solutions of relative Thue equations, which are important in solving Diophantine equations, improving computational practicality.

Contribution

It introduces a novel LLL-based algorithm for calculating small solutions of relative Thue equations, extending previous methods for classical Thue equations.

Findings

Algorithm effectively finds small solutions under 10^{100}

Method is demonstrated with explicit examples

Potential applications in solving Diophantine equations

Abstract

Diophantine equations can often be reduced to various types of classical Thue equations. These equations usually have only very small solutions, on the other hand to compute all solutions (i.e. to prove the non-existence of large solutions) is a time consuming procedure. Therefore it is very practical to have a fast algorithm to calculate the "small" solutions, especially if "small" means less than e.g. $10^{100}$. Such an algorithm was constructed by A.Peth\H o in 1987 based on continued fractions. In the present paper we construct a similar type of fast algorithm to calculate "small" solutions of {\it relative} Thue equations. Our method is based on the LLL reduction algorithm. We illustrate the method with explicit examples. The algorithm has several applications.