Bases of minimal vectors in tame lattices

TL;DR

This paper introduces tame lattices inspired by number theory and constructs sub-lattices with explicit bases of minimal vectors, advancing understanding of lattice structures.

Contribution

It defines tame lattices and provides a method to construct sub-lattices with explicit minimal vector bases, a novel approach in lattice theory.

Findings

Construction of a parametric family of sub-lattices with minimal vector bases

Explicit bases of minimal vectors are provided for each sub-lattice

The approach links lattice theory with number field trace pairings

Abstract

Motivated by the behavior of the trace pairing over tame cyclic number fields, we introduce the notion of tame lattices. Given an arbitrary non-trivial lattice $\mathcal{L}$ we construct a parametric family of full-rank sub-lattices $\{\mathcal{L}_{\alpha}\}$ of $\mathcal{L}$ such that whenever $\mathcal{L}$ is tame each $\mathcal{L}_{\alpha}$ has a basis of minimal vectors. Furthermore, for each $\mathcal{L}_{\alpha}$ in the family a basis of minimal vectors is explicitly constructed.