Some lattices from polynomial rings

TL;DR

This paper investigates specific lattice structures derived from polynomial rings over finite fields, focusing on their properties and algebraic structure related to kernel definitions of certain polynomial maps.

Contribution

It introduces a new class of lattices from polynomial ring kernels and analyzes their algebraic properties and potential applications.

Findings

Characterization of the lattice structure

Properties of the kernel of polynomial maps

Insights into algebraic and combinatorial aspects

Abstract

We study some properties of the lattices defined as the kernel of the map $(a_1,\ldots, a_{q-1}) \mapsto \prod (1-c_ix)^{a_i} \in (\mathbb{F}_q[x]/(x^e))^*$, where $\mathbb{F}_q^* = \{c_1,\ldots,c_{q-1}\}$.