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引用次数: 0
摘要
摘要 我们在数域 k 上建立了一个新版本的西格尔(Siegel)定理,为 \(k^N\) , \(N \ge 2\) 子空间的基向量的最大高度提供了一个约束。除了小高属性之外,我们得到的基向量还满足一定的稀疏性条件。此外,我们还对这些基向量的子集合所产生的所有可能子空间的高度给出了一个非难约束。我们的边界是绝对的,因为它们不依赖于定义域。我们的方法的主要新颖之处在于,它只使用线性代数,而不依赖于数的几何或以前有关这一主题的著作中使用的迪里希特盒原理。
We establish a new version of Siegel’s lemma over a number field k, providing a bound on the maximum of heights of basis vectors of a subspace of \(k^N\), \(N \ge 2\). In addition to the small-height property, the basis vectors we obtain satisfy certain sparsity condition. Further, we produce a nontrivial bound on the heights of all the possible subspaces generated by subcollections of these basis vectors. Our bounds are absolute in the sense that they do not depend on the field of definition. The main novelty of our method is that it uses only linear algebra and does not rely on the geometry of numbers or the Dirichlet box principle employed in the previous works on this subject.
期刊介绍:
Research in the Mathematical Sciences is an international, peer-reviewed hybrid journal covering the full scope of Theoretical Mathematics, Applied Mathematics, and Theoretical Computer Science. The Mission of the Journal is to publish high-quality original articles that make a significant contribution to the research areas of both theoretical and applied mathematics and theoretical computer science.
This journal is an efficient enterprise where the editors play a central role in soliciting the best research papers, and where editorial decisions are reached in a timely fashion. Research in the Mathematical Sciences does not have a length restriction and encourages the submission of longer articles in which more complex and detailed analysis and proofing of theorems is required. It also publishes shorter research communications (Letters) covering nascent research in some of the hottest areas of mathematical research. This journal will publish the highest quality papers in all of the traditional areas of applied and theoretical areas of mathematics and computer science, and it will actively seek to publish seminal papers in the most emerging and interdisciplinary areas in all of the mathematical sciences. Research in the Mathematical Sciences wishes to lead the way by promoting the highest quality research of this type.