I want to show a proposition related to Siegel's lemma:
Let $K$ be a number field of degree $d$, $L_j = \sum_{i} a_{ij}X_i \in \mathcal{O}_K[X_1, \cdots, X_N] (j= 1, \cdots, M)$ linear forms in $K$ which is all independent, $A = H_K(\cdots, a_{ij}, \cdots)$, and assume $M \lt N$. Then there exists nonzero $x \in \mathcal{O}_K^N$ such that for all $j$, $L_j(x) = 0$ and $$ \prod_{v | \infty} \max_i |x_i|_v \le (cNA)^{\frac{M}{N-M}}, $$ where $c$ is a constant which depends only on $K$ and an integral basis of $K$.
In my text, Siegel's lemma says similar statement except that $M \lt N$ resp. $x \in \mathcal{O}_K^N$ resp. $ \prod_{v | \infty} \max_i |x_i|_v$ resp. $M/(N-M)$ in the proposition above, is replaced by $dM \lt N$ resp. $x \in \mathbb{Z}^N$ resp. $\max_i|x_i|$ resp $dM/(N-dM)$. I think that Siegel's lemma in my text can't be extended to the proposition above.
And theorem 7 in this pdf also says the similar statement, but I think that it can't be extended, too. So please show the proposition above, or give me some references.
Thank you very much!
I used lectures notes from Jan-Hendrik Evertse. You can find here page 50 for the Siegel's lemma for number fields and its proof.