= Solution
For a <linear form> $L=a_1X_1+\cdots+a_NX_N$ over a <number field> $K$, define its height to be the <projective height> of its coefficient vector:
$$
H_p(L)=\prod_{v\in M_K}\max_j|a_j|_v^{d_v/[K:\mathbb Q]}.
$$
The <product formula> makes this independent of multiplying $L$ by a nonzero scalar.
The <Siegel lemma> says the following. Let $D=[K:\mathbb Q]$, let $L_1,\ldots,L_M$ be linear forms in $N$ variables with $DM<N$, and suppose $H_p(L_i)\leq\mathcal H$ for every $i$, where $\mathcal H\geq1$. Then there is a nonzero $\mathbf x\in\mathbb Z^N$ annihilated by all the forms and satisfying
$$
\lVert\mathbf x\rVert_\infty
\leq(N\mathcal H)^{DM/(N-DM)}.
$$
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