Small zeros of quadratic forms over number fields. II

Vaaler, Jeffrey D.

doi:10.1090/S0002-9947-1989-0940914-7

Small zeros of quadratic forms over number fields. II
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Trans. Amer. Math. Soc. 313 (1989), 671-686 Request permission

Abstract:

Let $F$ be a nontrivial quadratic form in $N$ variables with coefficients in a number field $k$ and let $\mathcal {Z}$ be a subspace of ${k^N}$ of dimension $M,1 \leq M \leq N$. If $F$ restricted to $\mathcal {Z}$ vanishes on a subspace of dimension $L,1 \leq L < M$, and if the rank of $F$ restricted to $\mathcal {Z}$ is greater than $M - L$, then we show that $F$ must vanish on $M - L + 1$ distinct subspaces ${\mathcal {X}_0},{\mathcal {X}_1}, \ldots ,{\mathcal {X}_{M - L}}$ in $\mathcal {Z}$ each of which has dimension $L$. Moreover, we show that for each pair ${\mathcal {X}_0},{\mathcal {X}_1},1 \leq l \leq M - L$, the product of their heights $H({\mathcal {X}_0})H({\mathcal {X}_1})$ is relatively small. Our results generalize recent work of Schlickewei and Schmidt.

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