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A classical theorem of Cassels states that if a homogenous quadratic form $Q$ has an integer zero, then there is a zero of small height (bounded solely by the coefficients and number of variables). Birch and Davenport extended this result to an arbitrary lattice $\Lambda$.

Now suppose there is an integer lattice $\Lambda$ in $\mathbb{R}^{kl}$ and it is known that there is a vector $v \in \Lambda$ of the form $$ v =(\lambda_1 w, \lambda_2 w, \ldots, \lambda_kw), $$ where $\lambda_i$ are integers and $w$ is an integer vector in $\mathbb{R}^l$. Of course, such a condition is equivalent to multiple quadratic relations $X_iX_j - X_kX_l = 0$. I am wondering if one can then guarantee the existence of such $v$ with bounded height (polynomial in $\det(\Lambda))$)?

It seems that in general there are no meaningful generalisations of Cassels' theorem to simultaneous quadratic forms.

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