Balanced Factorizations

Anton A. Klyachko, Anton N. Vassilyev
2016 The American mathematical monthly  
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.
doi:10.4169/amer.math.monthly.123.10.989 fatcat:wbxryzrg2nca3cwyb2yui7dmfi