Positivity of tropical multidegrees

Let \(\Gamma \) $\mathrm{\Gamma }$ be a tropical variety in \({\mathbb {R}}^m={\mathbb {R}}^{m_1}\times \cdots \times {\mathbb {R}}^{m_k}\) ${\mathbb{R}}^m={\mathbb{R}}^{m_1}\times \cdots \times {\mathbb{R}}^{m_k}$ . We show that, under a certain condition, the positivity of the stable intersection of \(\Gamma \) $\mathrm{\Gamma }$ with certain tropical varieties pulled back from each \({\mathbb {R}}^{m_i}\) ${\mathbb{R}}^{m_i}$ is governed by the dimensions of the images of \(\Gamma \) $\mathrm{\Gamma }$ under all possible projections from \({\mathbb {R}}^m\) ${\mathbb{R}}^m$ . As an application, we give a tropical proof of the criterion of the positivity of the multidegrees of a closed subscheme of a multi-projective space, carried out in the paper Castillo et al. (Adv Math 374:107382, 2020).