= Siegel lemma
{c}
{wiki=Siegel's_lemma}
Let $K$ be a <number field> of degree $D$, and let $L_1,\ldots,L_M$ be <linear form>[linear forms] in $N>DM$ variables with coefficients in $K$ and <projective height> at most $\mathcal H\geq1$. There is a nonzero $\mathbf x\in\mathbb Z^N$ satisfying every $L_j(\mathbf x)=0$ and
$$
\lVert\mathbf x\rVert_\infty
\leq(N\mathcal H)^{DM/(N-DM)}.
$$
For $K=\mathbb Q$, this follows by applying the <pigeonhole principle> to the images of the integer box $\{0,\ldots,Y\}^N$ under $(L_1,\ldots,L_M)$. Expanding coefficients in a rational basis of $K$ gives the factor $D$ in the general count.
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