Balanced factorisations
arXiv:1506.01571 · doi:10.4169/amer.math.monthly.123.10.989
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.
7 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm