When do the \(r\)-by-\(r\) minors of a matrix form a tropical basis? (Q388189)

Let \(A\) be a real matrix of size \(6\times n\) whose tropical rank is at most 3, i.e., among the \(4!\) four-term sums defining the tropical permanent of any 4-square submatrix, the minimum is attained at least twice. In the paper under review, it is proved that then \(A\) has Kapranov rank at most 3, i.e., \(A\) can be lifted to a matrix \(\tilde A\) of Hahn series such that \(\tilde A\) has rank at most 3.NEWLINENEWLINELet \(n\geq d\) be positive integers throughout what follows. It has been shown in [\textit{M. Develin} et al., in: Combinatorial and computational geometry. Cambridge: Cambridge University Press. Mathematical Sciences Research Institute Publications 52, 213--242 (2005; Zbl 1095.15001)] that the Kapranov rank of any \(d\times n\) matrix is at least its tropical rank, and the two ranks coincide if the tropical rank is at most 2 or the Kapranov rank is \(d\). Later, [\textit{M. Chan} et al., Linear Algebra Appl. 435, No. 7, 1598--1611 (2011; Zbl 1231.14053)] proved that the two ranks coincide if \(d\leq 5\). \textit{Ya. Shitov} in a previous work [``Example of a 6-by-6 matrix with different tropical and Kapranov ranks'', Vestnik Moskov. Univ. Ser. I 5, 58--61 (2011) (in Russian); English version available via \url{arXiv:1012.5507}] gave an example of a 6-square matrix with tropical rank 4 and Kapranov rank 5. As remarked in the paper under review, this trivially yields a \(d\times n\) matrix of tropical rank 4 and Kapranov rank 5 whenever \(d\geq 6\). Already in [Zbl 1095.15001], an example of of a \(7\)-square matrix with tropical rank 3 and Kapranov rank 4 was given. In the paper under review, it is explained how to modify these examples to get a \(d\times n\) matrix of tropical rank less than \(r\) but Kapranov rank at least \(r\) whenever \(4\leq r<d\geq 7\).NEWLINENEWLINEIt is also explained that the existence of such an example for given \(r\), \(d\), \(n\) is equivalent to saying that the \(r\)-square minors of a \(d\times n\) matrix of algebraically independent variables do not from a tropical basis of the ideal they generate. Thus, they do form a tropical basis if and only if \(r\leq 3\) or \(r=d\) or \(d\leq 5\) (known cases), or \(r=4\) and \(d=6\) (new case). This answers a question raised in [Zbl 1231.14053].