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Singular systems of linear forms over global function fields

Gukyeong Bang, Taehyeong Kim, Seonhee Lim

Abstract

In this paper, we consider singular systems of linear forms over global function fields of class number one and give an upper bound for the Hausdorff dimension of the set of singular systems of linear forms by constructing an appropriate Margulis height function on the space of lattices over global function fields.

Singular systems of linear forms over global function fields

Abstract

In this paper, we consider singular systems of linear forms over global function fields of class number one and give an upper bound for the Hausdorff dimension of the set of singular systems of linear forms by constructing an appropriate Margulis height function on the space of lattices over global function fields.
Paper Structure (9 sections, 15 theorems, 194 equations)

This paper contains 9 sections, 15 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Global function fields
  4. Homogeneous dynamics
  5. Exterior algebra
  6. Key linear algebra estimates
  7. the Hodge duality and the proof of Proposition \ref{['LinAlgEst']}
  8. Construction of Margulis functions
  9. Covering estimates using Margulis function

Key Result

Theorem 1.1

Assume that $R_\nu$ is a principal ideal domain. For $m,n \in \mathbb{Z}_{\geq 1}$,

Theorems & Definitions (39)

  • Theorem 1.1
  • Lemma 2.1: Dani's correspondence
  • proof
  • Proposition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Remark 3.4
  • proof : Proofs of Theorem \ref{['ThmWedgeUpperBound']} and Lemma \ref{['LemWedgeUpperBound']}
  • proof : Proof of Claim 1
  • proof : Proof of Claim 2
  • ...and 29 more