The best constant in Siegel's Lemma
Let \(k\) be an algebraic number field of degree \(d\) over \(\mathbb{Q}\). Denote by \(k_{\mathbb{A}}\) the adele ring of \(k\). For each place \(v\) of \(k\), denote by \(| \cdot | _v\) the absolute value on the completion \(k_v\) associated with \(v\), normalized such that it extends the standard absolute value on \(\mathbb{Q}\) if \(v\) is archimedean, and the standard \(p\)-adic absolute value if \(v\) lies above the prime \(p\). Further, denote by \(| | \cdot | | _v\) the norm on \(k_v^L\) \((L\geq 1)\) which extends the Euclidean norm on \(\mathbb{Q}^L\) if \(v\) is archimedean, and the maximum norm if \(v\) is non-archimedean. Lastly, let \(d_v:= [k_v: \mathbb{Q}_p]\) where \(p\) is the place (\(\infty\) or a prime) on \(\mathbb{Q}\) lying below \(v\). Then for an adelic matrix \(A\in \text{ GL}(L,k_{\mathbb{A}})\) one defines \(| A| :=\prod_v | \det A_v| _v^{d_v}\), where the product is taken over all places \(v\) of \(k\), and \(A_v\) is the component of \(A\) at \(v\), and the twisted height \(H_A({\mathbf x}):= \prod_v | | A_v{\mathbf x}| | _v^{d_v}\) for \({\mathbf x}\in k^L\). Thunder defines Hermite's constant \(\gamma_k(L)^{1/2}\) for \(k\) to be the infimum of all positive reals \(C\) with the property that for every \(A\in \text{ GL}(L,k_{{\mathbf A}})\) there is a non-zero point \({\mathbf x}\in k^L\) such that \(H_A({\mathbf x})\leq C| A| ^{1/L}\). This coincides with the classical Hermite's constant if \(k= \mathbb{Q}\). The author proves the following result: \vskip2pt\noindent Let \(A\in \text{ GL}(L,k_{{\mathbf A}})\). Then there exist \(L\) linearly independent points \({\mathbf x}_1,\ldots, {\mathbf x}_L\) in \(k^L\) such that \(\prod_{i=1}^L H_A({\mathbf x}_i)\leq \gamma_k(L)| A| \). This improves upon existing estimates with instead of \(\gamma_k(L)\) some explicit constant in terms of \(L\). The author shows that in his result \(\gamma_k(L)\) cannot be replaced by a smaller constant. Further he deduces some improvements of existing versions of Siegel's Lemma.
scientific article; zbMATH DE number 1998187