The \(4\times 4\) minors of a \(5\times n\) matrix are a tropical basis
One of the main difficulties in tropical geometry is caused by the fact that the intersection of the tropical hypersurfaces defined by a set of polynomials can be strictly larger than the tropical variety of the ideal that they generate. A particular case of this phenomenon is that the tropical rank of a matrix can be strictly smaller than its Kapranov rank. Indeed, it is known that there exists a 7-square matrix with tropical rank 3 and Kapranov rank 4 [\textit{M. Develin, F. Santos, B. Sturmfels}, in: Combinatorial and computational geometry. Cambridge: Cambridge University Press. Mathematical Sciences Research Institute Publications 52, 213--242 (2005; Zbl 1095.15001)]. However, in that paper it is proved that for \(4\times n\) matrices those two notions of rank coincide, and it is asked whether this holds for 5-square matrices as well. In the paper under review, the authors compute the space of 5-square matrices of tropical rank at most 3 and show that it coincides with the space of 5-square matrices of Kapranov rank at most 3, that is, the space of five labeled coplanar points in the tropical torus. They then prove that the Kapranov rank of every \(5\times n\) matrix equals its tropical rank; equivalently, that the \(4\times 4\) minors of a \(5\times n\) matrix of variables form a tropical basis. A less elegant proof of this result had previously been given in a preprint by the third author [\textit{E. Rubei}, On tropical and Kapranov ranks of tropical matrices, \url{arXiv:0712.3007}].
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