On a new absolute version of Siegel's lemma

TL;DR

This paper introduces a new absolute version of Siegel's lemma over number fields, providing bounds on basis vector heights with added sparsity and subspace bounds, using only linear algebra techniques.

Contribution

It presents a novel linear algebra-based approach to Siegel's lemma that yields absolute bounds and sparsity conditions, avoiding traditional geometry of numbers methods.

Findings

Bounds on basis vector heights are independent of the field of definition.

Basis vectors can be chosen to satisfy sparsity conditions.

Provides bounds on heights of subspaces generated by basis vectors.

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 $k^N$, $N \geq 2$. 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.