Small solutions of congruences (Q5906441)

Let \(K\) be an algebraic number field of degree \(n\) over \(\mathbb{Q}\) with ring of integers \(J\). Let \(Q_ j(\lambda) = \sum_{1 \leq q,r \leq s} \alpha_{qr} (j) \lambda_ q \lambda_ r\), where \(\alpha_{qr} (j) = \alpha_{rq} (j)\), \(1 \leq j \leq h\), be a system of \(h\) quadratic forms in \(\lambda = (\lambda_ 1, \dots, \lambda_ s)\) with coefficients in \(J\). Let \({\mathfrak a}\) be a nonzero integral ideal of \(K\). In this paper the author proves that for any \(\varepsilon>0\) there is a constant \(C>0\) depending on \(h,n,\varepsilon\) only such that if \(s\geq C\) the system of congruences \(Q_ j (\lambda) \equiv 0 \pmod {\mathfrak a}\), \(1 \leq j \leq h\) has a solution \(\lambda\) in \(J^ s\) with \(0<\max_ j | N (\lambda_ j) | \ll N ({\mathfrak a})^{\varepsilon + (1/2)}\). In the proof, the author first obtains a similar bound for small solutions of the system of congruences \(\sum^ s_{j=1} \alpha_{ij} \lambda^ k_ j \equiv 0 \pmod {\mathfrak a}\) where \(1 \leq i \leq h\) and \(k \geq 2\). Then he derives the above result on \(Q_ j (\lambda)\) by the method of algebraic diagonalization. The author's result improves a result by \textit{T. Cochrane} [Ill. J. Math. 31, 618-625 (1987; Zbl 0617.12001)] and generalizes a result of \textit{R. C. Baker} [Mathematika 27, 30-45 (1980; Zbl 0426.10020)].