Balanced factorisations

TL;DR

This paper explores the problem of factoring rational numbers and elements in various algebraic structures into products with sum zero, providing complete solutions in finite fields and some rings, and posing open questions.

Contribution

It offers a comprehensive solution to factorization problems with sum-zero conditions across multiple algebraic systems, extending prior work and identifying new open questions.

Findings

Complete solutions for finite fields

Factorization results in complex and real matrix algebras

Open questions in algebraic factorization

Abstract

Any rational number can be factored into a product of several rationals whose sum vanishes. This simple but nontrivial fact was suggested as a problem on a maths olympiad for high-school students. We completely solve similar questions in all finite fields and in some other rings, e.g., in the complex and real matrix algebras. Also, we state several open questions.