  We show that determining Kapranov rank of tropical matrices is not only
NP-hard over any infinite field but also if solving Diophantine equations over
the rational numbers is undecidable, then determining Kapranov rank over the
rational numbers is undecidable. We prove that Kapranov rank of tropical
matrices is not bounded in terms of tropical rank, answering a question of
Develin, Santos, and Sturmfels.
