Thue equations and CM-fields

TL;DR

This paper establishes an upper bound for solutions of certain Thue equations over totally real fields with CM-field roots and provides an algorithm for computing these solutions.

Contribution

It introduces a polynomial upper bound for solutions of Thue equations over totally real fields with CM-field roots and presents a new algorithm for finding all solutions.

Findings

Derived a polynomial upper bound for solutions

Developed an algorithm for computing solutions

Applied results to equations over CM-fields

Abstract

We obtain a polynomial type upper bound for the size of the integral solutions of Thue equations $F(X,Y) = b$ defined over a totally real number field $K$, assuming that $F(X,1)$ has a root $\alpha$ such that $K(\alpha)$ is a CM-field. Furthermore, we give an algorithm for the computation of the integral solutions of such an equation.