Inhomogeneous norm form equations in two dominating variables over function fields
Let k be an algebraically closed field of characteristic 0 and denote by k(z) the field of rational functions over k. Let \({\mathfrak L}\), \({\mathfrak M}\) denote finite extension fields of k(z) such that \({\mathfrak L}({\mathfrak M}\) and [\({\mathfrak M} : {\mathfrak L}]\geq 3\). Further \({\mathfrak K}\) denote the smallest normal extension of \({\mathfrak L}\) containing \({\mathfrak M}\). We shall denote by g the genus of \({\mathfrak K}\) and put \(G=\max (0,2g-2)\), \(\Omega_{{\mathfrak K}}\) the set of all (additive) valuations on \({\mathfrak K}\). For any non-zero \(\alpha\in {\mathfrak K}\) let \(H_{{\mathfrak K}}(\alpha)=-\sum_{v\in \Omega_{{\mathfrak K}}}\min \{0,v(\alpha)\}\) be the additive height of \(\alpha\). Let \({\mathbb{Z}}_{{\mathfrak K},S}\) be the ring of S-integers of \({\mathfrak K}\). \({\mathbb{Z}}_{{\mathfrak M},S}={\mathbb{Z}}_{{\mathfrak K},S}\cap {\mathfrak M}\), \({\mathbb{Z}}_{{\mathfrak L},S}={\mathbb{Z}}_{{\mathfrak K},S}\cap {\mathfrak L}\), and \(| S|\) the cardinality of S. Finally, let \(\alpha \in {\mathbb{Z}}_{{\mathfrak K},S}\) be a primitive element of \({\mathfrak M}\) over \({\mathfrak L}\) with \(H_{{\mathfrak K}}(\alpha)\leq A\) and let \(0\neq \mu \in {\mathfrak L}.\) The following theorem is proved. If \(x,y\in {\mathbb{Z}}_{{\mathfrak L},S}\) and \(\lambda \in {\mathbb{Z}}_{{\mathfrak M},S}\) are solutions of the equation \[ Nm_{{\mathfrak M}/{\mathfrak L}}(x+\alpha y+\lambda)=\mu \] and \(H_{{\mathfrak K}}(\lambda)\leq c \max \{H_{{\mathfrak L}}(x),H_{{\mathfrak L}}(y)\}\), where \(c=[{\mathfrak K} : {\mathfrak L}]/400\) and \(H_{{\mathfrak L}}(\cdot)=H_{{\mathfrak K}}(\cdot)/[{\mathfrak K} : {\mathfrak L}]\), then we have \[ \max \{H_{{\mathfrak L}}(x), H_{{\mathfrak L}}(y)\}\quad <\quad \frac{64}{[{\mathfrak K\setminus : {\mathfrak L}}]}(| S| +G+2H_{{\mathfrak K}}(\mu)+25A). \] The proof of this result is based on the recent paper of \textit{R. C. Mason} [J. Number Theory, 22, 190-207 (1986; Zbl 0578.10021)].
solutions of inhomogeneous norm form equations over function fields
scientific article; zbMATH DE number 4021233
scientific article