Balanced factorisations
Anton A. Klyachko, Anton N. Vassilyev

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.
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