When do points in tropical projective space impose independent conditions?

A tropical hypersurface is the double-max locus of a polynomial. This seems a better description of a Cartier object than a Weil object. A refinement of this would be to split the terms of the polynomial into the positives and negatives and set them equal to each other. If you do this with the determinant of an $n \times n$ matrix, you get Gondran-Minoux singularity instead of ordinary tropical singularity. When you further apply this to points in projective space, you get an interesting criterion for independence, which is what I will talk about. This is joint work with a group of postdocs and grad students in Utah.