On the nonnegative ranks of matrices in Puiseux series fields

Abstract

The positive part $C^{+}$ of the field $C\left\{\right\{t\left\}\right\}$ consists of Puiseux series with positive real leading terms. Answering a question of Yu, we show that, if M is an $m\times n$ matrix with entries in $C^{+}$ and rank two, then there are an $m\times 2$ matrix A and $2\times n$ matrix B with entries in $C^{+}$ such that $M=AB$. We discuss the problem in larger ranks and answer a further question arisen in a work of Brandenburg, Loho, and Sinn.