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

Abstract

Twisting a binary form $F_0(X,Y)\in \mathbb{Z}[X,Y]$ of degree $d\ge 3$ by powers ${\upsilon }^a$ ($a\in \mathbb{Z}$) of an algebraic unit $\upsilon$ gives rise to a binary form $F_a(X,Y)\in \mathbb{Z}[X,Y]$. More precisely, when $K$ is a number field of degree $d$, ${\sigma }_1,{\sigma }_2,\dots ,{\sigma }_d$ the embeddings of $K$ into $\mathbb{C}$, $\alpha$ a nonzero element in $K$, $a_0\in \mathbb{Z}$, $a_0>0$ and $$F_0(X,Y)=a_0\prod _{i=1}^d(X-{\sigma }_i(\alpha )Y),$$ then for $a\in \mathbb{Z}$ we set $$F_a(X,Y)=a_0\prod _{i=1}^d(X-{\sigma }_i(\alpha {\upsilon }^a)Y).$$ Given $m\ge 0$, our main result is an effective upper bound for the solutions $(x,y,a)\in {\mathbb{Z}}^3$ of the Diophantine inequalities $$0<|F_a(x,y)|\le m$$ for which $xy\ne 0$ and $\mathbb{Q}(\alpha {\upsilon }^a)=K$. Our estimate involves an effectively computable constant depending only on $d$; it is explicit in terms of $m$, in terms of the heights of $F_0$ and of $\upsilon$, and in terms of the regulator of the number field $K$.

Submitted. 22 pages

References in corpus (1)

• Solving effectively some families of Thue Diophantine equations