Integer solutions to decomposable form inequalities
Let \(k\) be an algebraic number field, \(S\) a finite set of places of \(k\) containing the infinite ones and \(\mathcal O_S\) the ring of \(S\)-integers of \(k\). For any place \(v\) let \(\| \cdot\| _v\) denote the normalized \(v\)-adic absolute value, and for \(\mathbf x = (x_1, \dots, x_m) \in k^m\) define the \(S\)-height by \(H_S (\mathbf x) = \prod_{\nu \in S} \max_{1 \leq i \leq m} \| x_i \| _v\). Let \(F \in k[X_1, \dots, X_m]\) be a decomposable form satisfying the general finiteness criteria of \textit{J. H. Evertse} and \textit{K. Győry} [Acta Arith. 50, No. 4, 357--379 (1988; Zbl 0595.10013)]. The main result of this paper shows that for any real constants \(\lambda < \frac 1 {m-1}\) and \(c>0\) there are -- up to multiplication by \(\mathcal O_S^\times\) -- only finitely many \(\mathbf x = (x_1, \dots, x_m) \in \mathcal O_S^m\) satisfying \[ 0 < \prod_{v \in S} \| F(\mathbf x) \| _v \leq c H_S^\lambda (\mathbf x)\;. \] The proof uses a refined form of Schmidt's subspace theorem, which is stated as Theorem 2.1 in this paper.
scientific article; zbMATH DE number 5009836
scientific article