Solving effectively some families of Thue Diophantine equations

Abstract

Let $\alpha$ be an algebraic number of degree $d\ge 3$ and let $K$ be the algebraic number field $\Q(α)$. When $\epsilon$ is a unit of $K$ such that $\Q(α\varepsilon)=K$, we consider the irreducible polynomial $f_{\epsilon }(X)\in \mathbb{Z}[X]$ such that $f_{\epsilon }(\alpha \epsilon )=0$. Let $F_{\epsilon }(X,Y)$ be the irrreducible binary form of degree $d$ associated to $f_{\epsilon }(X)$ under the condition $F_{\epsilon }(X,1)=f_{\epsilon }(X)$. For each positive integer $m$, we want to exhibit an effective upper bound for the solutions $(x,y,\epsilon )$ of the diophantine inequation $|F_{\epsilon }(x,y)|\le m$. We achieve this goal by restricting ourselves to a subset of units $\epsilon$ which we prove to be sufficiently large as soon as the degree of $K$ is $\ge 4$.

Moscow Journal of Combinatorics and Number Theory, to appear

Cited by in corpus (1)

• Families of Thue equations associated with a rank one subgroup of the unit group of a number field