Small zeros of quadratic forms over number fields

Let $F$ be a nontrivial quadratic form in $N$ variables with coefficients in a number field $k$ and let $A$ be a $K \times N$ matrix over $k$. We show that if the simultaneous equations $F({\mathbf{x}}) = 0$ and $A{\mathbf{x}} = 0$ hold on a subspace $\mathfrak{X}$ of dimension $L$ and $L$ is maximal, then such a subspace $\mathfrak{X}$ can be found with the height of $\mathfrak{X}$ relatively small. In particular, the height of $\mathfrak{X}$ can be explicitly bounded by an expression depending on the height of $F$ and the height of $A$. We use methods from geometry of numbers over adèle spaces and local to global techniques which generalize recent work of H. P. Schlickewei.