The Indecomposable Solutions of Linear Congruences

TL;DR

This paper characterizes the minimal solutions of a specific linear congruence, identifies those reaching a known bound, and explores their asymptotic behavior, with implications for zero-sum sequences and invariant theory.

Contribution

It provides a complete characterization of indecomposable solutions to a linear congruence and analyzes their asymptotic properties, extending understanding in zero-sum sequence theory.

Findings

Characterization of solutions attaining Eggleton-Erdős bound

Asymptotic analysis of the number of indecomposable solutions

Connections to zero-sum sequences and invariant theory

Abstract

This article considers the minimal non-zero (= indecomposable) solutions of the linear congruence $1\cdot x_1 + \cdots + (m-1)\cdot x_{m-1} \equiv 0 \pmod m$ for unknown non-negative integers $x_1, \ldots, x_n$, and characterizes the solutions that attain the Eggleton-Erd\H{o}s bound. Furthermore it discusses the asymptotic behaviour of the number of indecomposable solutions. The results have direct interpretations in terms of zero-sum sequences and invariant theory.