On a new absolute version of Siegel’s lemma

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY Accounts of Chemical Research Pub Date : 2024-02-15 DOI:10.1007/s40687-024-00422-5

引用次数: 0

Abstract

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.

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关于西格尔定理的新绝对版本

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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.

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