Nonnegative rank depends on the field II
  We provide an example of a $21\times 21$ matrix with nonnegative integer entries which can be written as a sum of $19$ nonnegative rank-one matrices but not as a sum of $19$ rational nonnegative rank-one matrices. This gives a solution for a problem posed by Cohen and Rothblum in 1993. 
Yaroslav Shitov
math.CO