Small solutions to systems of linear congruences over number fields (Q1359138)

Let \(a,b,m>0\) be integers. By the Aubry-Thue theorem the congruence \[ ax+by\equiv 0 \pmod m \] has a nontrivial integer solution with \(\max(|x|,|y|)\leq m^{1/2}\). In 1951 \textit{A. Brauer} and \textit{R. L. Reynolds} [Can. J. Math. 3, 367-374 (1951; Zbl 0042.26801) gave an extension of this result to systems of linear congruences in several variables. If \(A\) is an \(M\) by \(N\) matrix with rational integer entries and \(\text{rank} (A)=M<N\), the set of vectors \(x\in \mathbb{Z}^N\) satisfying \(Ax\in(m\mathbb{Z})^M\) is a lattice. The purpose of the paper is to give upper bounds for \(N\) linearly independent vectors in the lattice that are relatively short. The question is investigated over arbitrary number fields, and the bound is shown to be sharp.