Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Integral Matrices of Fixed Rank over Number Fields

Nihar Gargava, Vlad Serban, Maryna Viazovska, Ilaria Viglino

TL;DR

The paper extends Katznelson's fixed-rank matrix counts from $\mathbb{Z}$-entries to matrices over the ring of integers $\mathcal{O}_K$ of a number field $K$, proving an asymptotic formula for admissible test functions with main term $\mathfrak{C}_{main} \cdot T^{kn d}$ and an explicit, controllable error. The leading constant is given by a sum over echelon matrices $D$ of $\mathfrak{D}(D)^{-n}$ times an integral $\int f(xD)\,dx$, a Rogers-type interpretation made rigorous via Schmidt's theorem. The work connects algebraic-lattice geometry with coding theory through $(\mathcal{P},r)$-Hecke neighbors (lifts of codes) and shows that moment estimates for Haar-random number-field lattices carry over to large discrete sets of module lattices, thereby bridging Rogers' integral formula with arithmetic lattices. Special attention is given to the case $k=1,d=1,n=m+1$, where a gap in Katznelson's proof is addressed and the log-term behavior explained. Overall, the results generalize classical fixed-rank counting to an arithmetic setting, illuminate Rogers-type constants, and enable applications to codes and lattice-based cryptography over number fields.

Abstract

We prove an asymptotic formula for the number of fixed rank matrices with integer coefficients over a number field K/Q and bounded norm. As an application, we derive an approximate Rogers integral formula for discrete sets of module lattices obtained from lifts of algebraic codes. This in turn implies that the moment estimates of random lattices with a number field structure also carry through for large enough discrete sets of module lattices.

Integral Matrices of Fixed Rank over Number Fields

TL;DR

The paper extends Katznelson's fixed-rank matrix counts from -entries to matrices over the ring of integers of a number field , proving an asymptotic formula for admissible test functions with main term and an explicit, controllable error. The leading constant is given by a sum over echelon matrices of times an integral , a Rogers-type interpretation made rigorous via Schmidt's theorem. The work connects algebraic-lattice geometry with coding theory through -Hecke neighbors (lifts of codes) and shows that moment estimates for Haar-random number-field lattices carry over to large discrete sets of module lattices, thereby bridging Rogers' integral formula with arithmetic lattices. Special attention is given to the case , where a gap in Katznelson's proof is addressed and the log-term behavior explained. Overall, the results generalize classical fixed-rank counting to an arithmetic setting, illuminate Rogers-type constants, and enable applications to codes and lattice-based cryptography over number fields.

Abstract

We prove an asymptotic formula for the number of fixed rank matrices with integer coefficients over a number field K/Q and bounded norm. As an application, we derive an approximate Rogers integral formula for discrete sets of module lattices obtained from lifts of algebraic codes. This in turn implies that the moment estimates of random lattices with a number field structure also carry through for large enough discrete sets of module lattices.

Paper Structure

This paper contains 23 sections, 29 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Connection to Rogers' integral formula
  3. Motivation from coding theory and cryptography
  4. The case of k=1,d=1,n=m+1
  5. A note on the implicit constants
  6. Preliminaries and Notations
  7. Lattices
  8. Covering radius and Voronoi domain
  9. Minkowski and Hadamard
  10. Hypothesis on test functions
  11. Summation by parts
  12. Number fields, Euclidean structure and algebraic integer lattice
  13. Matrices, subspaces and lattices
  14. A useful trijection
  15. Schmidt's theorem
  16. ...and 8 more sections

Key Result

Theorem 1

Let $f: \mathop{\mathrm{M}}\nolimits_{n \times m}( \mathbb{R}) \rightarrow \mathbb{R}$ be the indicator function of an origin-centered unit ball in the $l^{2}$-norm $\| \cdot \| : \mathop{\mathrm{M}}\nolimits_{n \times m}(\mathbb{R})\cong \mathbb{R}^{nm} \rightarrow \mathbb{R}$. Then, for some co where

Theorems & Definitions (64)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Theorem 4
  • Definition 5
  • Remark 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 54 more