When are tropical multidegrees positive?

Abstract: We study the positivity of the tropical multidegrees of a tropical variety contained in a product of real vector spaces. These multidegrees are obtained by stably intersecting the tropical variety with pullbacks of positive tropical divisors. We introduce projection-purity and facet-selectability, two conditions under which positivity is determined by the dimensions of the natural projections, and the support of the tropical multidegrees is precisely the set of lattice points of a polymatroid base polytope. This extends He's theorem for translation-admissible tropical varieties. We also show that these conditions alone do not force the corresponding tropical volume polynomial to be Lorentzian. By contrast, for the augmented Bergman fan of any polymatroid, the positive multidegrees are supported precisely on the lattice points of the polymatroid base polytope, and the tropical volume polynomial is Lorentzian for every sequence of positive tropical divisors.