Solving simultaneously Thue equations in the almost totally imaginary case

Abstract

Let $\alpha$ be an algebraic number of degree $d\ge 3$ having at most one real conjugate and let $K$ be the algebraic number field $\mathbf{Q}(\alpha )$. For any unit $\epsilon$ of $K$ such that $\mathbf{Q}(\alpha \epsilon )=K$, we consider the irreducible polynomial $f_{\epsilon }(X)\in \mathbf{Z}[X]$ such that $f_{\epsilon }(\alpha \epsilon )=0$. Let $F_{\epsilon }(X,Y)=Y^d{f}_{\epsilon }(X/Y)\in \mathbf{Z}[X,Y]$ be the associated binary form. For each positive integer $m$, we exhibit an effectively computable bound for the solutions $(x,y,\epsilon )$ of the diophantine equation $|F_{\epsilon }(x,y)|\le m$.

Proceedings of the International Meeting on Number Theory HRI 2011, in honor of R. Balasubramanian. To appear