AI 中文总结
该论文研究热带多重度的正性，引入投影纯性与 facet 可选择性条件，推广 He 的定理，同时指出增广 Bergman 扇对应的热带体积多项式具有洛伦兹型性质。
AI 中文摘要
我们研究包含在实向量空间乘积中的热带簇的热带多重度的正性。这些多重度是通过将热带簇与正热带除子的拉回进行稳定相交得到的。我们引入投影纯性和 facet 可选择性这两个条件，在这些条件下，正性由自然投影的维数决定，且热带多重度的支撑恰好是一个多面体基多面体的格点集合。这推广了 He 关于平移可允许热带簇的定理。我们还表明，仅这些条件并不强制对应的热带体积多项式为洛伦兹型。相比之下，对于任意多面体的增广 Bergman 扇，正多重度恰好支撑在多面体基多面体的格点上，且对于每一组正热带除子序列，热带体积多项式均为洛伦兹型。
英文摘要
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.
