Totally real Thue inequalities over imaginary quadratic fields

TL;DR

This paper presents an efficient method to reduce solving certain relative Thue inequalities over imaginary quadratic fields to solving absolute Thue inequalities over integers, demonstrated with a concrete example.

Contribution

The authors introduce a novel reduction technique for relative Thue inequalities over imaginary quadratic fields to absolute inequalities over integers.

Findings

Method effectively reduces problem complexity

Successfully applied to an explicit example

Facilitates solving Thue inequalities over imaginary quadratic fields

Abstract

Let $F(x,y)$ be an irreducible binary form of degree $\geq 3$ with integer coefficients and with real roots. Let $M$ be an imaginary quadratic field, with ring of integers $Z_M$. Let $K>0$. We describe an efficient method how to reduce the resolution of the relative Thue inequalities \[ |F(x,y)|\leq K \;\; (x,y\in Z_M) \] to the resolution of absolute Thue inequalities of type \[ |F(x,y)|\leq k \;\; (x,y\in Z). \] We illustrate our method with an explicit example.