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Let $M_i$ be fixed rectangular matrices with integer coefficients less than $n$. Consider the variety defined by the condition $$ \mathrm{rank}(\lambda_1M_1 + \lambda_2M_2 + ... + \lambda_kM_k) = 1. $$

Assume that there is an integral point $(\lambda_1, ..., \lambda_k)$ lying on this variety, can we then guarantee that there is an integral point with the height (maximal absolute value of the coordinates) bounded polynomially in $n$ (the exponent may depend on $k$ and the size of the matrices)?

P. S.

It might be related to the Birch-Davenport theorem, so with high probability this question is either known or out of reach for current methods (which would be ok to know), but I couldn't find a reference.

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    $\begingroup$ Are you allowing the degree of the polynomial (i.e. the exponent of $n$) to depend on $k$? $\endgroup$
    – Noam D. Elkies
    Commented May 13, 2015 at 16:29
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    $\begingroup$ Yes, and it may also depend on the size of the matrices. $\endgroup$
    – DmitryZ
    Commented May 14, 2015 at 9:07

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