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On a new absolute version of Siegel's lemma

Maxwell Forst, Lenny Fukshansky

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.

On a new absolute version of Siegel's lemma

Abstract

We establish a new version of Siegel's lemma over a number field , providing a bound on the maximum of heights of basis vectors of a subspace of , . 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.
Paper Structure (4 sections, 7 theorems, 98 equations)

This paper contains 4 sections, 7 theorems, 98 equations.

Table of Contents

  1. Introduction and statement of results
  2. Notation and heights
  3. Proof of Theorem \ref{['new_siegel']}
  4. Proof of Theorem \ref{['many_bases']}

Key Result

Theorem 1.1

Let ${\mathcal{Z}} = Ak^L \subseteq k^N$ be a subspace of dimension $L$ where $1 \leq L < N$ and $A$ is an $N \times L$ basis matrix for ${\mathcal{Z}}$. Let $B$ be a full rank $L \times L$ submatrix of $A$, and write for the column vectors of the matrix $AB^{-1}$. Then $\boldsymbol \omega_1, \dots, \boldsymbol \omega_L$ is another basis for ${\mathcal{Z}}$ over $k$, which consists of $(N-L+1)$-s

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • proof : Proof of Theorem \ref{['new_siegel']}
  • Example 1
  • Theorem 3.1
  • proof
  • ...and 3 more